diff --git a/docs/metrics.md b/docs/metrics.md index 361a52a..42b795c 100644 --- a/docs/metrics.md +++ b/docs/metrics.md @@ -61,7 +61,7 @@ This document provides comprehensive information about all quality metrics used | Type | Metric | Description | Paper Source | Evaluation Results | Examples | |------|--------|-------------|--------------|-------------------|----------| | `QUALITY_BAD_COMPLETENESS` | RuleLineEndWithEllipsis, RuleLineEndWithTerminal, RuleSentenceNumber, RuleWordNumber | Checks whether the ratio of lines ending with ellipsis is below threshold; Checks whether the ratio of lines ending w... | [RedPajama: an Open Dataset for Training Large Language Models](https://github.com/togethercomputer/RedPajama-Data) (Together Computer, 2023) | [📊 See Results](eval/rule/slimpajama_data_evaluated_by_rule.md) | N/A | -| `QUALITY_BAD_EFFECTIVENESS` | RuleDoi, RuleIsbn, RuleAbnormalChar, RuleAbnormalHtml, RuleAlphaWords, RuleAudioDataFormat, RuleCharNumber, RuleColonEnd, RuleContentNull, RuleContentShort, RuleContentShortMultiLan, RuleEnterAndSpace, RuleEnterMore, RuleEnterRatioMore, RuleHtmlEntity, RuleHtmlTag, RuleInvisibleChar, RuleImageDataFormat, RuleLatexSpecialChar, RuleLineJavascriptCount, RuleLoremIpsum, RuleMeanWordLength, RuleNlpDataFormat, RuleSftDataFormat, RuleSpaceMore, RuleSpecialCharacter, RuleStopWord, RuleSymbolWordRatio, RuleVedioDataFormat, RuleOnlyUrl | Check whether the string is in the correct format of the doi; Check whether the string is in the correct format of th... | Internal Implementation | N/A | N/A | +| `QUALITY_BAD_EFFECTIVENESS` | RuleDoi, RuleIsbn, RuleAbnormalChar, RuleAbnormalHtml, RuleAlphaWords, RuleAudioDataFormat, RuleCharNumber, RuleColonEnd, RuleContentNull, RuleContentShort, RuleContentShortMultiLan, RuleEnterAndSpace, RuleEnterMore, RuleEnterRatioMore, RuleHtmlEntity, RuleHtmlTag, RuleInvisibleChar, RuleImageDataFormat, RuleLatexSpecialChar, RuleLineJavascriptCount, RuleLoremIpsum, RuleMeanWordLength, RuleNlpDataFormat, RuleSftDataFormat, RuleSpaceMore, RuleSpecialCharacter, RuleStopWord, RuleSymbolWordRatio, RuleVedioDataFormat, RuleOnlyUrl, RuleDictConsistency | Check whether the string is in the correct format of the doi; Check whether the string is in the correct format of th... | Internal Implementation | N/A | N/A | | `QUALITY_BAD_FLUENCY` | RuleAbnormalNumber, RuleCharSplit, RuleNoPunc, RuleWordSplit, RuleWordStuck | Checks PDF content for abnormal book page or index numbers that disrupt text flow; Checks PDF content for abnormal ch... | [RedPajama: an Open Dataset for Training Large Language Models](https://github.com/togethercomputer/RedPajama-Data) (Together Computer, 2023) | [📊 See Results](eval/rule/slimpajama_data_evaluated_by_rule.md) | N/A | | `QUALITY_BAD_RELEVANCE` | RuleHeadWordAr, RuleHeadWordCs, RuleHeadWordHu, RuleHeadWordKo, RuleHeadWordRu, RuleHeadWordSr, RuleHeadWordTh, RuleHeadWordVi, RulePatternSearch, RuleWatermark | Checks whether Arabic content contains irrelevant tail source information; Checks whether Czech content contains irre... | [RedPajama: an Open Dataset for Training Large Language Models](https://github.com/togethercomputer/RedPajama-Data) (Together Computer, 2023) | [📊 See Results](eval/rule/slimpajama_data_evaluated_by_rule.md) | N/A | | `QUALITY_BAD_SECURITY` | RuleIDCard, RuleUnsafeWords, RulePIIDetection | Checks whether content contains ID card information; Checks whether content contains unsafe words; Detects Personal I... | [RedPajama: an Open Dataset for Training Large Language Models](https://github.com/togethercomputer/RedPajama-Data) (Together Computer, 2023) | [📊 See Results](eval/rule/slimpajama_data_evaluated_by_rule.md) | N/A | diff --git a/docs/mineru_xunlianying/code.json b/docs/mineru_xunlianying/code.json new file mode 100644 index 0000000..821c645 --- /dev/null +++ b/docs/mineru_xunlianying/code.json @@ -0,0 +1,7 @@ +{ + "id": "code_01", + "url": "https://www.w3school.com.cn/python/numpy_creating_arrays.asp", + "html": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNumPy 简介\n\n\n
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\nw3school 在线教程\n
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Python 教程

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文件处理

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Python NumPy

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机器学习

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Python MySQL

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Python MongoDB

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Python 参考手册

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模块参考手册

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Python How To

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Python 实例

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NumPy 简介

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创建 NumPy ndarray 对象

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NumPy 用于处理数组。 NumPy 中的数组对象称为 ndarray。

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我们可以使用 array() 函数创建一个 NumPy ndarray 对象。

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实例

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import numpy as np \n\narr = np.array([1, 2, 3, 4, 5])\n\nprint(arr)\n\nprint(type(arr))\n
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运行实例

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type(): 这个内置的 Python 函数告诉我们传递给它的对象的类型。像上面的代码一样，它表明 arr 是 numpy.ndarray 类型。

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要创建 ndarray，我们可以将列表、元组或任何类似数组的对象传递给 array() 方法，然后它将被转换为 ndarray：

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实例

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使用元组创建 NumPy 数组：

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import numpy as np \n\narr = np.array((1, 2, 3, 4, 5))\n\nprint(arr)\n
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运行实例

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数组中的维

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数组中的维是数组深度（嵌套数组）的一个级别。

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嵌套数组：指的是将数组作为元素的数组。

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0-D 数组

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0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。

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实例

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用值 61 创建 0-D 数组：

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import numpy as np\n\narr = np.array(61)\n\nprint(arr)\n
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运行实例

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1-D 数组

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其元素为 0-D 数组的数组，称为一维或 1-D 数组。

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这是最常见和基础的数组。

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实例

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创建包含值 1、2、3、4、5、6 的 1-D 数组：

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import numpy as np\n\narr = np.array([1, 2, 3, 4, 5, 6])\n\nprint(arr)\n
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运行实例

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2-D 数组

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其元素为 1-D 数组的数组，称为 2-D 数组。

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它们通常用于表示矩阵或二阶张量。

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NumPy 有一个专门用于矩阵运算的完整子模块 numpy.mat。

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实例

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创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：

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import numpy as np\n\narr = np.array([[1, 2, 3], [4, 5, 6]])\n\nprint(arr)\n
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运行实例

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3-D 数组

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其元素为 2-D 数组的数组，称为 3-D 数组。

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实例

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用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：

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import numpy as np\n\narr = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(arr)\n
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运行实例

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检查维数？

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NumPy 数组提供了 ndim 属性，该属性返回一个整数，该整数会告诉我们数组有多少维。

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实例

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检查数组有多少维：

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import numpy as np\n\na = np.array(42)\nb = np.array([1, 2, 3, 4, 5])\nc = np.array([[1, 2, 3], [4, 5, 6]])\nd = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(a.ndim) \nprint(b.ndim) \nprint(c.ndim) \nprint(d.ndim)\n
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运行实例

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更高维的数组

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数组可以拥有任意数量的维。

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在创建数组时，可以使用 ndmin 参数定义维数。

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实例

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创建一个有 5 个维度的数组，并验证它拥有 5 个维度：

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import numpy as np\n\narr = np.array([1, 2, 3, 4], ndmin=5)\n\nprint(arr)\nprint('number of dimensions :', arr.ndim)\n
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运行实例

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在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。

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\nW3School 简体中文版提供的内容仅用于培训和测试，不保证内容的正确性。通过使用本站内容随之而来的风险与本站无关。版权所有，保留一切权利。\n

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\n关于 W3School\n帮助 W3School\n使用条款\n隐私条款\n技术支持：赢科\n蒙ICP备06004630号\n

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", + "m_content": "# NumPy 简介\n\n## 创建 NumPy ndarray 对象\n\nNumPy 用于处理数组。 NumPy 中的数组对象称为 `ndarray` 。\n\n我们可以使用 `array()` 函数创建一个 NumPy `ndarray` 对象。\n\n### 实例\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4, 5])\n\nprint(arr)\n\nprint(type(arr))\n```\n\ntype(): 这个内置的 Python 函数告诉我们传递给它的对象的类型。像上面的代码一样，它表明 `arr` 是 `numpy.ndarray` 类型。\n\n要创建 `ndarray` ，我们可以将列表、元组或任何类似数组的对象传递给 `array()` 方法，然后它将被转换为 `ndarray` ：\n\n### 实例\n\n使用元组创建 NumPy 数组：\n\n```python\nimport numpy as np\n\narr = np.array((1, 2, 3, 4, 5))\n\nprint(arr)\n```\n\n## 数组中的维\n\n数组中的维是数组深度（嵌套数组）的一个级别。\n\n嵌套数组： 指的是将数组作为元素的数组。\n\n## 0-D 数组\n\n0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。\n\n### 实例\n\n用值 61 创建 0-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array(61)\n\nprint(arr)\n```\n\n## 1-D 数组\n\n其元素为 0-D 数组的数组，称为一维或 1-D 数组。\n\n这是最常见和基础的数组。\n\n### 实例\n\n创建包含值 1、2、3、4、5、6 的 1-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4, 5, 6])\n\nprint(arr)\n```\n\n## 2-D 数组\n\n其元素为 1-D 数组的数组，称为 2-D 数组。\n\n它们通常用于表示矩阵或二阶张量。\n\nNumPy 有一个专门用于矩阵运算的完整子模块 `numpy.mat` 。\n\n### 实例\n\n创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array([[1, 2, 3], [4, 5, 6]])\n\nprint(arr)\n```\n\n## 3-D 数组\n\n其元素为 2-D 数组的数组，称为 3-D 数组。\n\n### 实例\n\n用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：\n\n```python\nimport numpy as np\n\narr = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(arr)\n```\n\n## 检查维数？\n\nNumPy 数组提供了 `ndim` 属性，该属性返回一个整数，该整数会告诉我们数组有多少维。\n\n### 实例\n\n检查数组有多少维：\n\n```python\nimport numpy as np\n\na = np.array(42)\nb = np.array([1, 2, 3, 4, 5])\nc = np.array([[1, 2, 3], [4, 5, 6]])\nd = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(a.ndim)\nprint(b.ndim)\nprint(c.ndim)\nprint(d.ndim)\n```\n\n## 更高维的数组\n\n数组可以拥有任意数量的维。\n\n在创建数组时，可以使用 `ndmin` 参数定义维数。\n\n### 实例\n\n创建一个有 5 个维度的数组，并验证它拥有 5 个维度：\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4], ndmin=5)\n\nprint(arr)\nprint('number of dimensions :', arr.ndim)\n```\n\n在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。", + "t_content": "# NumPy 简介\n\n## 创建 NumPy ndarray 对象\n\n`ndarray`\n\n\n`array()`\n\n`ndarray`\n\n\n### 实例\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4, 5]) print(arr ) print(type(arr ))\n\ntype():`arr`\n\n`numpy.ndarray`\n\n\n`ndarray`\n\n`array()`\n\n`ndarray`\n\n\n### 实例\n\n使用元组创建 NumPy 数组：\n\nimportnumpy asnp arr =np .array ((1, 2, 3, 4, 5)) print(arr )\n\n## 数组中的维\n\n数组中的维是数组深度（嵌套数组）的一个级别。\n\n嵌套数组：\n\n## 0-D 数组\n\n0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。\n\n### 实例\n\n用值 61 创建 0-D 数组：\n\nimportnumpy asnp arr =np .array (61) print(arr )\n\n## 1-D 数组\n\n其元素为 0-D 数组的数组，称为一维或 1-D 数组。\n\n这是最常见和基础的数组。\n\n### 实例\n\n创建包含值 1、2、3、4、5、6 的 1-D 数组：\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4, 5, 6]) print(arr )\n\n## 2-D 数组\n\n其元素为 1-D 数组的数组，称为 2-D 数组。\n\n它们通常用于表示矩阵或二阶张量。\n\n`numpy.mat`\n\n\n### 实例\n\n创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：\n\nimportnumpy asnp arr =np .array ([[1, 2, 3], [4, 5, 6]]) print(arr )\n\n## 3-D 数组\n\n其元素为 2-D 数组的数组，称为 3-D 数组。\n\n### 实例\n\n用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：\n\nimportnumpy asnp arr =np .array ([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]]) print(arr )\n\n## 检查维数？\n\n`ndim`\n\n\n### 实例\n\n检查数组有多少维：\n\nimportnumpy asnp a =np .array (42)b =np .array ([1, 2, 3, 4, 5])c =np .array ([[1, 2, 3], [4, 5, 6]])d =np .array ([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]]) print(a .ndim ) print(b .ndim ) print(c .ndim ) print(d .ndim )\n\n## 更高维的数组\n\n数组可以拥有任意数量的维。\n\n`ndmin`\n\n\n### 实例\n\n创建一个有 5 个维度的数组，并验证它拥有 5 个维度：\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4],ndmin =5) print(arr ) print('number of dimensions :',arr .ndim )\n\n在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。" +} diff --git a/docs/mineru_xunlianying/demo.jsonl b/docs/mineru_xunlianying/demo.jsonl new file mode 100644 index 0000000..5c83ad1 --- /dev/null +++ b/docs/mineru_xunlianying/demo.jsonl @@ -0,0 +1,3 @@ +{"id":"code_01","url":"https://www.w3school.com.cn/python/numpy_creating_arrays.asp","html":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNumPy 简介\n\n\n
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\nw3school 在线教程\n
\n\n\n
\n\n
\n
\n
\n\n
\n
\n

Python 教程

\n\n

文件处理

\n\n

Python NumPy

\n\n

机器学习

\n\n

Python MySQL

\n\n

Python MongoDB

\n\n

Python 参考手册

\n\n

模块参考手册

\n\n

Python How To

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Python 实例

\n\n
\n
\n
\n

NumPy 简介

\n
\n\n
\n
\n

创建 NumPy ndarray 对象

\n

NumPy 用于处理数组。 NumPy 中的数组对象称为 ndarray。

\n

我们可以使用 array() 函数创建一个 NumPy ndarray 对象。

\n

实例

\n
import numpy as np \n\narr = np.array([1, 2, 3, 4, 5])\n\nprint(arr)\n\nprint(type(arr))\n
\n

运行实例

\n

type(): 这个内置的 Python 函数告诉我们传递给它的对象的类型。像上面的代码一样，它表明 arr 是 numpy.ndarray 类型。

\n

要创建 ndarray，我们可以将列表、元组或任何类似数组的对象传递给 array() 方法，然后它将被转换为 ndarray：

\n

实例

\n

使用元组创建 NumPy 数组：

\n
import numpy as np \n\narr = np.array((1, 2, 3, 4, 5))\n\nprint(arr)\n
\n

运行实例

\n
\n
\n

数组中的维

\n

数组中的维是数组深度（嵌套数组）的一个级别。

\n

嵌套数组：指的是将数组作为元素的数组。

\n
\n
\n

0-D 数组

\n

0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。

\n

实例

\n

用值 61 创建 0-D 数组：

\n
import numpy as np\n\narr = np.array(61)\n\nprint(arr)\n
\n

运行实例

\n
\n
\n

1-D 数组

\n

其元素为 0-D 数组的数组，称为一维或 1-D 数组。

\n

这是最常见和基础的数组。

\n

实例

\n

创建包含值 1、2、3、4、5、6 的 1-D 数组：

\n
import numpy as np\n\narr = np.array([1, 2, 3, 4, 5, 6])\n\nprint(arr)\n
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运行实例

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2-D 数组

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其元素为 1-D 数组的数组，称为 2-D 数组。

\n

它们通常用于表示矩阵或二阶张量。

\n

NumPy 有一个专门用于矩阵运算的完整子模块 numpy.mat。

\n

实例

\n

创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：

\n
import numpy as np\n\narr = np.array([[1, 2, 3], [4, 5, 6]])\n\nprint(arr)\n
\n

运行实例

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3-D 数组

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其元素为 2-D 数组的数组，称为 3-D 数组。

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实例

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用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：

\n
import numpy as np\n\narr = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(arr)\n
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运行实例

\n
\n
\n

检查维数？

\n

NumPy 数组提供了 ndim 属性，该属性返回一个整数，该整数会告诉我们数组有多少维。

\n

实例

\n

检查数组有多少维：

\n
import numpy as np\n\na = np.array(42)\nb = np.array([1, 2, 3, 4, 5])\nc = np.array([[1, 2, 3], [4, 5, 6]])\nd = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(a.ndim) \nprint(b.ndim) \nprint(c.ndim) \nprint(d.ndim)\n
\n

运行实例

\n
\n
\n

更高维的数组

\n

数组可以拥有任意数量的维。

\n

在创建数组时，可以使用 ndmin 参数定义维数。

\n

实例

\n

创建一个有 5 个维度的数组，并验证它拥有 5 个维度：

\n
import numpy as np\n\narr = np.array([1, 2, 3, 4], ndmin=5)\n\nprint(arr)\nprint('number of dimensions :', arr.ndim)\n
\n

运行实例

\n

在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。

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","m_content":"# NumPy 简介\n\n## 创建 NumPy ndarray 对象\n\nNumPy 用于处理数组。 NumPy 中的数组对象称为 `ndarray` 。\n\n我们可以使用 `array()` 函数创建一个 NumPy `ndarray` 对象。\n\n### 实例\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4, 5])\n\nprint(arr)\n\nprint(type(arr))\n```\n\ntype(): 这个内置的 Python 函数告诉我们传递给它的对象的类型。像上面的代码一样，它表明 `arr` 是 `numpy.ndarray` 类型。\n\n要创建 `ndarray` ，我们可以将列表、元组或任何类似数组的对象传递给 `array()` 方法，然后它将被转换为 `ndarray` ：\n\n### 实例\n\n使用元组创建 NumPy 数组：\n\n```python\nimport numpy as np\n\narr = np.array((1, 2, 3, 4, 5))\n\nprint(arr)\n```\n\n## 数组中的维\n\n数组中的维是数组深度（嵌套数组）的一个级别。\n\n嵌套数组： 指的是将数组作为元素的数组。\n\n## 0-D 数组\n\n0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。\n\n### 实例\n\n用值 61 创建 0-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array(61)\n\nprint(arr)\n```\n\n## 1-D 数组\n\n其元素为 0-D 数组的数组，称为一维或 1-D 数组。\n\n这是最常见和基础的数组。\n\n### 实例\n\n创建包含值 1、2、3、4、5、6 的 1-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4, 5, 6])\n\nprint(arr)\n```\n\n## 2-D 数组\n\n其元素为 1-D 数组的数组，称为 2-D 数组。\n\n它们通常用于表示矩阵或二阶张量。\n\nNumPy 有一个专门用于矩阵运算的完整子模块 `numpy.mat` 。\n\n### 实例\n\n创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：\n\n```python\nimport numpy as np\n\narr = np.array([[1, 2, 3], [4, 5, 6]])\n\nprint(arr)\n```\n\n## 3-D 数组\n\n其元素为 2-D 数组的数组，称为 3-D 数组。\n\n### 实例\n\n用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：\n\n```python\nimport numpy as np\n\narr = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(arr)\n```\n\n## 检查维数？\n\nNumPy 数组提供了 `ndim` 属性，该属性返回一个整数，该整数会告诉我们数组有多少维。\n\n### 实例\n\n检查数组有多少维：\n\n```python\nimport numpy as np\n\na = np.array(42)\nb = np.array([1, 2, 3, 4, 5])\nc = np.array([[1, 2, 3], [4, 5, 6]])\nd = np.array([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]])\n\nprint(a.ndim)\nprint(b.ndim)\nprint(c.ndim)\nprint(d.ndim)\n```\n\n## 更高维的数组\n\n数组可以拥有任意数量的维。\n\n在创建数组时，可以使用 `ndmin` 参数定义维数。\n\n### 实例\n\n创建一个有 5 个维度的数组，并验证它拥有 5 个维度：\n\n```python\nimport numpy as np\n\narr = np.array([1, 2, 3, 4], ndmin=5)\n\nprint(arr)\nprint('number of dimensions :', arr.ndim)\n```\n\n在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。","t_content":"# NumPy 简介\n\n## 创建 NumPy ndarray 对象\n\n`ndarray`\n\n\n`array()`\n\n`ndarray`\n\n\n### 实例\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4, 5]) print(arr ) print(type(arr ))\n\ntype():`arr`\n\n`numpy.ndarray`\n\n\n`ndarray`\n\n`array()`\n\n`ndarray`\n\n\n### 实例\n\n使用元组创建 NumPy 数组：\n\nimportnumpy asnp arr =np .array ((1, 2, 3, 4, 5)) print(arr )\n\n## 数组中的维\n\n数组中的维是数组深度（嵌套数组）的一个级别。\n\n嵌套数组：\n\n## 0-D 数组\n\n0-D 数组，或标量（Scalars），是数组中的元素。数组中的每个值都是一个 0-D 数组。\n\n### 实例\n\n用值 61 创建 0-D 数组：\n\nimportnumpy asnp arr =np .array (61) print(arr )\n\n## 1-D 数组\n\n其元素为 0-D 数组的数组，称为一维或 1-D 数组。\n\n这是最常见和基础的数组。\n\n### 实例\n\n创建包含值 1、2、3、4、5、6 的 1-D 数组：\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4, 5, 6]) print(arr )\n\n## 2-D 数组\n\n其元素为 1-D 数组的数组，称为 2-D 数组。\n\n它们通常用于表示矩阵或二阶张量。\n\n`numpy.mat`\n\n\n### 实例\n\n创建包含值 1、2、3 和 4、5、6 两个数组的 2-D 数组：\n\nimportnumpy asnp arr =np .array ([[1, 2, 3], [4, 5, 6]]) print(arr )\n\n## 3-D 数组\n\n其元素为 2-D 数组的数组，称为 3-D 数组。\n\n### 实例\n\n用两个 2-D 数组创建一个 3-D 数组，这两个数组均包含值 1、2、3 和 4、5、6 的两个数组：\n\nimportnumpy asnp arr =np .array ([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]]) print(arr )\n\n## 检查维数？\n\n`ndim`\n\n\n### 实例\n\n检查数组有多少维：\n\nimportnumpy asnp a =np .array (42)b =np .array ([1, 2, 3, 4, 5])c =np .array ([[1, 2, 3], [4, 5, 6]])d =np .array ([[[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]]]) print(a .ndim ) print(b .ndim ) print(c .ndim ) print(d .ndim )\n\n## 更高维的数组\n\n数组可以拥有任意数量的维。\n\n`ndmin`\n\n\n### 实例\n\n创建一个有 5 个维度的数组，并验证它拥有 5 个维度：\n\nimportnumpy asnp arr =np .array ([1, 2, 3, 4],ndmin =5) print(arr ) print('number of dimensions :',arr .ndim )\n\n在此数组中，最里面的维度（第 5 个 dim）有 4 个元素，第 4 个 dim 有 1 个元素作为向量，第 3 个 dim 具有 1 个元素是与向量的矩阵，第 2 个 dim 有 1 个元素是 3D 数组，而第 1 个 dim 有 1 个元素，该元素是 4D 数组。"} +{"id":"math_01","url":"https://terrytao.wordpress.com/2008/05/16/285g-lecture-12-high-curvature-regions-of-ricci-flow-and-%ce%ba-solutions/","html":"\n\n\n\t\n\t285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions | What's new\n\t\n\t\n\n\n\n\n\n\n\n\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\t\n\t\t\t\t\t\n\t\t\t\n\n
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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

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285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions

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16 May, 2008 in 285G - poincare conjecture, math.DG | Tags: canonical neighbourhoods, compactness and contradiction, gradient shrinking solitons, kappa-solutions | by Terence Tao\t\t

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In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a \"\\kappa\"-noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the \"\\kappa\"-solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the \"\\kappa\"-noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of \"\\kappa\"-solutions. A secondary compactness argument lets us obtain that control of \"\\kappa\"-solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.

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[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, \"\\kappa\"-noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]

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The next few lectures will be devoted to the analysis of \"\\kappa\"-solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a \"\\kappa\"-solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about \"\\kappa\"-solutions.

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Our treatment here is based primarily on the book of Morgan and Tian.

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— Definition of a \"\\kappa\"-solution —

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We fix a small number \"\\kappa (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a \"\\kappa\"-solution:

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Definition 1. (\"\\kappa\"-solutions) A \"\\kappa\"-solution is a Ricci flow \"t which is

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  1. Ancient, in the sense that t ranges on the interval \"(-\\infty,0]\";
    2. \n
  2. Complete and connected (i.e. (M,g(t)) is complete and connected for every t);
    3. \n
  3. Non-negative Riemann curvature, i.e. \"\\hbox{Riem}: is positive semidefinite at all points in spacetime;
    4. \n
  4. Bounded curvature, thus \"\\sup_{(t,x);
    5. \n
  5. \"\\kappa\"-noncollapsed (see Definition 1 of Lecture 7) at every point \"(t_0,x_0)\" in spacetime and at every scale \"r_0;
    6. \n
  6. Non-flat, i.e. the curvature is non-zero at at least one point in spacetime.
    7. \n
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This laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.

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Remark 1. If a d-dimensional Riemann manifold is both flat (thus \"\\hbox{Riem}=0\") and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space \"{\\Bbb. Thus the non-flat condition is only excluding the trivial Ricci flow \"M with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere. \"\\diamond\"

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Remark 2. In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and \"\\hbox{Ric}\" are non-negative, and that \"|\\hbox{Riem}|_g, and \"R. Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of \"\\kappa\"-solutions are controlled by the scalar curvature. \"\\diamond\"

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Now we discuss examples (and non-examples) of \"\\kappa\"-solutions.

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Example 1. Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a \"\\kappa\"-solution for sufficiently small \"\\kappa\" if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere \"S^d\" with the standard metric is a gradient shrinking solution and will generate a \"\\kappa\"-solution for any \"d and sufficiently small \"\\kappa, which we will refer to as the shrinking round sphere \"\\kappa\"-solution. \"\\diamond\"

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Exercse 1. Show that the Cartesian product of two \"\\kappa\"-solutions is again a \"\\kappa\"-solution (with a smaller value of \"\\kappa\"), as is the Cartesian product of a \"\\kappa\"-solution. Thus for instance the product \"S^2 of the shrinking round 2-sphere and the Euclidean line is a \"\\kappa\"-solution, which we refer to as the shrinking round 3-cylinder \"S^2. \"\\diamond\"

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Example 2. In one dimension, there are no \"\\kappa\"-solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is not a \"\\kappa\"-solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere \"S^2\" is \"\\kappa\"-solution, as discussed above. We can quotient this by the obvious \"{\\Bbb action to also get a shrinking round projective plane \"\\Bbb{RP}^2\" as a \"\\kappa\"-solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional \"\\kappa\"-solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder \"S^1 is not a \"\\kappa\"-solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a \"\\kappa\"-solution due to it being collapsed at large scales. \"\\diamond\"
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Example 3. In three dimensions, we begin to get significantly more variety amongst \"\\kappa\"-solutions. We have the round shrinking 3-sphere \"S^3\", but also all the quotients \"S^3/\\Gamma\" of such round spheres by free finite group actions (including the projective space \"{\\Bbb, but with many other examples. We refer to these examples as round shrinking 3-spherical space forms. We have also seen the shrinking round cylinder \"S^2; there are also finite quotients of this example such as shrinking round projective cylinder \"\\Bbb{RP}^2, or the quotient of the cylinder by the orientation-preserving free involution \"(\\omega,z). We refer to these examples as the unoriented and oriented quotients of the shrinking round 3-cylinder respectively. The oriented quotient can be viewed as a half-cylinder \"S^2 capped off with a punctured \"\\Bbb{RP}^3\" (and the whole manifold is in fact homeomorphic to a punctured \"\\Bbb{RP}^3\"). \"\\diamond\"

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Example 4. One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking \"S^2\" or shrinking \"{\\Bbb example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as C-components (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders \"S^2 capped off with what is topologically either a punctured \"\\Bbb{RP}^3\" or punctured \"S^3\" (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as \"C\"-capped strong \"\\varepsilon\"-tubes (we will define this term precisely later). One can also consider doubly \"C\"-capped strong \"\\varepsilon\"-tubes, in which an approximately round finite cylinder \"S^2 is capped off at both ends by either a punctured \"\\Bbb{RP}^3\" or punctured \"S^3\"; such manifolds then become homeomorphic to either \"S^3\" or \"{\\Bbb. (Note we need to cap off any ends that show up in order to keep the manifold M complete.) \"\\diamond\"

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An important theorem of Perelman shows that these examples of \"\\kappa\"-solutions are in fact the only ones:

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Theorem 1. (Perelman classification theorem, imprecise version) Every 3-dimensional \"\\kappa\"-solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):

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  1. A shrinking round 3-sphere \"S^3\" (or shrinking round spherical space form \"S^3/\\Gamma\");
    2. \n
  2. A shrinking round 3-cylinder \"S^2, the quotient \"\\Bbb{RP}^2, or one of its quotients (either oriented or unoriented);
    3. \n
  3. A C-component;
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  4. A C-capped strong \"\\varepsilon\"-tube;
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  5. A doubly C-capped strong \"\\varepsilon\"-tube.
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We will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).

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Remark 3. At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong \"\\varepsilon\"-tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong \"\\varepsilon\"-tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional \"\\kappa\"-solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as \"S^1, or as the product of the cigar soliton and a line) or 3-dimensional (as in \"{\\Bbb) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not \"\\kappa\"-solutions. The unoriented quotiented cylinder \"\\Bbb{RP}^2 also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of \"\\Bbb{RP}^2 (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur. \"\\diamond\"

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Remark 4. In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round \"S^2\" with the trivial plane \"{\\Bbb. This is a \"\\kappa\"-solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).

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The classification theorem lets one understand the geometry of neighbourhoods of any given point in a \"\\kappa\"-solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):

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Definition 2. (Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.

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  1. We say that U is an \"\\varepsilon\"-neck if it is close (in a smooth topology) to a round cylinder \"S^2, with x well in the middle of of this cylinder;
    2. \n
  2. We say that U is a C-component if U is diffeomorphic to \"S^3\" or \"\\Bbb{RP}^3\" (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.
    3. \n
  3. We say that U is \"\\varepsilon\"-round if it is close (in a smooth topology) to a round sphere \"S^3\" or spherical space form \"S^3/\\Gamma\" (i.e. it is close to a constant curvature manifold).
    4. \n
  4. We say that U is a \"(C,\\varepsilon)\"-cap if it consists of an \"\\varepsilon\"-neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured \"{\\Bbb and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)
    5. \n
  5. We say that U is a canonical neighbourhood of x if it is one of the above four types.
    6. \n
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When the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.

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Using Theorem 1 (and defining all terms precisely), one can easily show the following important statement:

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Corollary 1 (Canonical neighbourhood theorem for \"\\kappa\"-solitons, informal version) Every point in a 3-dimensional \"\\kappa\"-solution that does not contain an embedded copy of \"\\Bbb{RP}^2\" with trivial normal bundle is contained in a canonical neighbourhood.

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The next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.

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— High curvature regions of Ricci flows —

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Corollary 1 is an assertion about \"\\kappa\"-solutions only, but it implies an important property about more general Ricci flows:

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Theorem 2. (Canonical neighbourhood for Ricci flows, informal version) Let \"t be a Ricci flow of compact 3-manifolds on a time interval \"{}[0,T)\", without any embedded copy of \"\\Bbb{RP}^2\" with trivial normal bundle. Then every point \"(t,x) with sufficiently large scalar curvature is contained in a canonical neighbourhood.

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(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)

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The importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)

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Theorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:

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  1. Suppose for contradiction that Theorem 2 failed. Then one could find a sequence \"(t_n,x_n) of points with \"R(t_n,x_n) which were not contained in canonical neighbourhoods.
    2. \n
  2. M, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.
    3. \n
  3. On any compact time interval \"{}[0,t], the scalar curvature is necessarily bounded, and thus \"t_n. As a consequence, if we define the rescaled Ricci flows \"g^{(n)}(t), where \"L_n is the natural length scale associated to the scalar curvature at \"(t_n,x_n)\", then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds \"(M,g^{(n)}(t))\" may cease to be compact, but will remain complete.
    4. \n
  4. Because of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3), we expect the rescaled flows \"t to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).
    5. \n
  5. If we can pick the points \"(t_n,x_n)\" suitably (so that the scalar curvature \"R(t_n,x_n)\" is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows \"t have bounded curvature in the limit.
    6. \n
  6. Since \"\\kappa\"-noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7) should ensure that the rescaled flows remain \"\\kappa\"-noncollapsed in the limit.
    7. \n
  7. Since the rescaled scalar curvature at the base point \"x_n\" of \"(M,g^{(n)})\" is equal to 1 by construction, any limiting flow will be non-flat.
    8. \n
  8. Various compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow \"(M^{(\\infty)},. This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.
    9. \n
  9. Applying 2-8, we see that the limiting flow \"(M^{(\\infty)}, is a \"\\kappa\"-solution.
    10. \n
  10. Applying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points \"(t_n,x_n)\" also lie in canonical neighbourhoods for sufficiently large n, a contradiction.
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There are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for \"\\kappa\"-solutions which we will deduce in later lectures. For Step 5, the problem is that the points \"(t_n,x_n)\" that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow \"(M^{(\\infty)}, ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point \"(t_n,x_n)\" (much further away than the natural length scale \"L_n\", in particular), so that it does not show up in the limiting flow \"(M^{(\\infty)},. We will discuss these issues in more detail in later lectures.

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— Benchmarks in controlling \"\\kappa\"-solutions —

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As mentioned earlier, the next few lectures will be focused on controlling \"\\kappa\"-solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.

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Proposition 1. (Consequences of Hamilton’s Harnack inequality) Let \"t be a \"\\kappa\"-solution. Then \"R(t,x)\" is a non-decreasing function of time. Furthermore, for any \"(t_0,x_0), we have the pointwise inequalities

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\"\\displaystyle (1)

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and

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\"\\displaystyle (2)

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on \"(-\\infty,t_0), where of course \"\\tau is the backwards time variable.

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These inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of \"\\kappa\"-solutions, as well as the Ricci flow equation \"\\dot of course.

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Now one can handle the two-dimensional case:

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Proposition 2. (Classification of 2-dimensional \"\\kappa\"-solutions) The only two-dimensional \"\\kappa\"-solutions are the round shrinking 2-spheres.

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This proposition relies on first studying a certain asymptotic limit of the \"\\kappa\"-solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original \"\\kappa\"-solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.

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Turning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.

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Proposition 3. (Asymptotic curvature) Let \"t be a 3-dimensional \"\\kappa\" solution which is not compact. Then for any time \"t and any base point \"p, we have \"\\limsup_{x.

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The proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.

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The increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:

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Proposition 4. (Asymptotic volume collapse) Let \"t be a 3-dimensional \"\\kappa\" solution which is not compact. Then for any time \"t and any base point \"p, we have \"\\limsup_{r.

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Note that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.

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An important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball \"B_{g(0)}(p,r)\" at time zero on which the volume is large (e.g. larger than \"\\nu for some \"\\nu), one has bounded normalised curvature, thus \"R on this ball. This fact helps us deduce

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Theorem 3. (Perelman compactness theorem, informal version) The space of all pointed \"\\kappa\"-solutions (allowing \"\\kappa to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.

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One corollary of this compactness is that there is in fact a universal \"\\kappa_0 such that every \"\\kappa\"-solution is a \"\\kappa_0\"-solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on \"\\kappa\"-solutions which come from the previous volume analysis.

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The proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.

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9 comments

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16 May, 2008 at 12:40 pm

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Anton Fonarev

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Dear prof. Tao,

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I mentioned a small misprint. There’s no closing bracket in Theorem 3 after \"\\kappa.

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16 May, 2008 at 12:44 pm

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Terence Tao

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Thanks for the correction!

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18 May, 2008 at 7:32 pm

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This past two weeks… « It’s Equal, but It’s Different…

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[…] 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume, 285G, Lecture 11: κ-noncollapsing via Perelman reduced volume, 285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions […]

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19 May, 2008 at 10:13 am

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285G, Lecture 13: Li-Yau-Hamilton Harnack inequalities and κ-solutions « What’s new

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[…] The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the -geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send ). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the -solutions introduced in the previous lecture. […]

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19 May, 2008 at 12:57 pm

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Sylvain Maillot

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Another minor correction: in Example 2, you should delete the words
\n“shrinking round” before \"S^1\\times : this manifold
\nis flat, so it does not shrink by Ricci flow (and I wouldn’t call it “round”.)

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A more serious point: I am surprised to see five cases in the classification of
\n\"\\kappa\"-solutions (Theorem 1), since in Perelman there are only four.
\nWhat is exactly the difference between case 3 “C-component” and case 5
\n“doubly \"C\"-capped strong \"\\varepsilon\"-tube” ?

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19 May, 2008 at 2:39 pm

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Terence Tao

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Dear Sylvain: Thanks again for the corrections!

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Regarding Theorem 1, I am basing this on Theorem 9.93 of Morgan-Tian (though I am grouping the cases a little differently). As I understand it, a C-component and a doubly C-capped strong epsilon-tube are topologically the same, but geometrically rather different; in the former the sectional curvature is bounded above and below everywhere, whereas in the latter the sectional curvature can be very small in the epsilon-tube component of the manifold (in planes parallel to the direction of the tube, of course). Also, if we normalise the maximal curvature to be 1, then C-components need to have diameter comparable to 1, but doubly C-capped epsilon-tubes can have arbitrarily large diameter.

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Perhaps the key difference here between Morgan-Tian and Perelman is that in Morgan-Tian one tries to classify the global structure of kappa-solutions, even though for applications it is only the local structure (i.e. the canonical neighbourhoods) which is of importance. Thus, for instance, if one takes a doubly C-capped strong epsilon-tube, the canonical neighbourhood of any given point in such an object would be either an epsilon-neck, epsilon-cap, or a closed manifold in Perelman’s notation, depending on whether the tube is long or short and whether the point lies in the middle of the manifold or near one of the ends.

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[In Morgan-Tian, the relevant canonical neighbourhood theorem is Corollary 9.94; the definition of canonical neighbourhoods on page 232 (and reproduced here as Definition 2) has exactly four cases and matches up with the corresponding notion in Perelman’s second paper.]

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20 May, 2008 at 10:12 am

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Sylvain Maillot

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OK. In fact, if you take a \"\\kappa\"-solution which is a \"C\"-component at time zero and look into its past, then you will see a doubly capped neck, where the neck part becomes longer and longer as time goes to negative infinity.

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This means that you can merge those two cases if you add time-shifting in your equivalence relation (in addition to isometry and rescaling). I would say that your statement is more a classification of time-zero slices of \"\\kappa\"-solutions than a classification of \"\\kappa\"-solutions strictly speaking.

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Well, all this is irrelevant to the Poincare conjecture anyway…

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2 June, 2008 at 4:12 pm

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285G, Lecture 17: The structure of κ-solutions « What’s new

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[…] 2 June, 2008 in 285G – poincare conjecture, math.DG by Terence Tao Tags: compactness and contradiction, geometric limits, gradient shrinking solitons, kappa-solutions, necks Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the previous lecture, we now use this classification, combined with extensive use of compactness and contradiction arguments, as well as the comparison geometry of complete Riemannian manifolds of non-negative curvature, to understand the structure of -solutions in these dimensions, with the aim being to state and prove precise versions of Theorem 1 and Corollary 1 from Lecture 12. […]

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4 June, 2008 at 8:37 am

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285G, Lecture 18: The structure of high-curvature regions of Ricci flow « What’s new

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[…] now use them to describe the structure of high curvature regions of Ricci flow, as promised back in Lecture 12, in particular controlling their geometry and topology to the extent that surgery will be applied, […]

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\n\t \n\n\n\n","m_content":"In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a $\\kappa$ -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the $\\kappa$ -solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the $\\kappa$ -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of $\\kappa$ -solutions. A secondary compactness argument lets us obtain that control of $\\kappa$ -solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.\n\n[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, $\\kappa$ -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]\n\nThe next few lectures will be devoted to the analysis of $\\kappa$ -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a $\\kappa$ -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about $\\kappa$ -solutions.\n\nOur treatment here is based primarily on the book of Morgan and Tian.\n\n— Definition of a $\\kappa$ -solution —\n\nWe fix a small number $\\kappa > 0$ (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a $\\kappa$ -solution:\n\nDefinition 1. ( $\\kappa$ -solutions) A $\\kappa$ -solution is a Ricci flow $t \\mapsto (M,g(t))$ which is\n\n1. Ancient , in the sense that t ranges on the interval $(-\\infty,0]$ ;\n2. Complete and connected (i.e. (M,g(t)) is complete and connected for every t);\n3. Non-negative Riemann curvature , i.e. $\\hbox{Riem}: \\bigwedge^2 TM \\to \\bigwedge^2 TM$ is positive semidefinite at all points in spacetime;\n4. Bounded curvature , thus $\\sup_{(t,x) \\in (-\\infty,0] \\times M} |\\hbox{Riem}|_g < +\\infty$ ;\n5. $\\kappa$ -noncollapsed (see Definition 1 of Lecture 7 ) at every point $(t_0,x_0)$ in spacetime and at every scale $r_0 > 0$ ;\n6. Non-flat , i.e. the curvature is non-zero at at least one point in spacetime.\n\nThis laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.\n\nRemark 1. If a d-dimensional Riemann manifold is both flat (thus $\\hbox{Riem}=0$ ) and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space ${\\Bbb R}^d$ . Thus the non-flat condition is only excluding the trivial Ricci flow $M = {\\Bbb R}^d$ with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere. $\\diamond$\n\nRemark 2. In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and $\\hbox{Ric}$ are non-negative, and that $|\\hbox{Riem}|_g, |\\hbox{Ric}|_g = O(R)$ and $R = O_d(1)$ . Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of $\\kappa$ -solutions are controlled by the scalar curvature. $\\diamond$\n\nNow we discuss examples (and non-examples) of $\\kappa$ -solutions.\n\nExample 1. Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a $\\kappa$ -solution for sufficiently small $\\kappa$ if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere $S^d$ with the standard metric is a gradient shrinking solution and will generate a $\\kappa$ -solution for any $d \\geq 2$ and sufficiently small $\\kappa > 0$ , which we will refer to as the shrinking round sphere $\\kappa$ -solution. $\\diamond$\n\nExercse 1. Show that the Cartesian product of two $\\kappa$ -solutions is again a $\\kappa$ -solution (with a smaller value of $\\kappa$ ), as is the Cartesian product of a $\\kappa$ -solution. Thus for instance the product $S^2 \\times {\\Bbb R}$ of the shrinking round 2-sphere and the Euclidean line is a $\\kappa$ -solution, which we refer to as the shrinking round 3-cylinder $S^2 \\times {\\Bbb R}$ . $\\diamond$\n\nExample 2. In one dimension, there are no $\\kappa$ -solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is not a $\\kappa$ -solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere $S^2$ is $\\kappa$ -solution, as discussed above. We can quotient this by the obvious ${\\Bbb Z}/2$ action to also get a shrinking round projective plane $\\Bbb{RP}^2$ as a $\\kappa$ -solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional $\\kappa$ -solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder $S^1 \\times {\\Bbb R}$ is not a $\\kappa$ -solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a $\\kappa$ -solution due to it being collapsed at large scales. $\\diamond$\n\nExample 3. In three dimensions, we begin to get significantly more variety amongst $\\kappa$ -solutions. We have the round shrinking 3-sphere $S^3$ , but also all the quotients $S^3/\\Gamma$ of such round spheres by free finite group actions (including the projective space ${\\Bbb RP}^3$ , but with many other examples. We refer to these examples as round shrinking 3-spherical space forms. We have also seen the shrinking round cylinder $S^2 \\times {\\Bbb R}$ ; there are also finite quotients of this example such as shrinking round projective cylinder $\\Bbb{RP}^2 \\times {\\Bbb R}$ , or the quotient of the cylinder by the orientation-preserving free involution $(\\omega,z) \\mapsto (-\\omega,-z)$ . We refer to these examples as the unoriented and oriented quotients of the shrinking round 3-cylinder respectively. The oriented quotient can be viewed as a half-cylinder $S^2 \\times [1,+\\infty)$ capped off with a punctured $\\Bbb{RP}^3$ (and the whole manifold is in fact homeomorphic to a punctured $\\Bbb{RP}^3$ ). $\\diamond$\n\nExample 4. One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking $S^2$ or shrinking ${\\Bbb RP}^3$ example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as C-components (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders $S^2 \\times [1,+\\infty)$ capped off with what is topologically either a punctured $\\Bbb{RP}^3$ or punctured $S^3$ (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as $C$ -capped strong $\\varepsilon$ -tubes (we will define this term precisely later). One can also consider doubly $C$ -capped strong $\\varepsilon$ -tubes, in which an approximately round finite cylinder $S^2 \\times [-T,T]$ is capped off at both ends by either a punctured $\\Bbb{RP}^3$ or punctured $S^3$ ; such manifolds then become homeomorphic to either $S^3$ or ${\\Bbb RP}^3$ . (Note we need to cap off any ends that show up in order to keep the manifold M complete.) $\\diamond$\n\nAn important theorem of Perelman shows that these examples of $\\kappa$ -solutions are in fact the only ones:\n\nTheorem 1. (Perelman classification theorem, imprecise version) Every 3-dimensional $\\kappa$ -solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):\n\n1. A shrinking round 3-sphere $S^3$ (or shrinking round spherical space form $S^3/\\Gamma$ );\n2. A shrinking round 3-cylinder $S^2 \\times {\\Bbb R}$ , the quotient $\\Bbb{RP}^2 \\times {\\Bbb R}$ , or one of its quotients (either oriented or unoriented);\n3. A C-component;\n4. A C-capped strong $\\varepsilon$ -tube;\n5. A doubly C-capped strong $\\varepsilon$ -tube.\n\nWe will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).\n\nRemark 3. At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong $\\varepsilon$ -tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong $\\varepsilon$ -tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional $\\kappa$ -solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as $S^1 \\times {\\Bbb R}^2$ , or as the product of the cigar soliton and a line) or 3-dimensional (as in ${\\Bbb R}^3$ ) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not $\\kappa$ -solutions. The unoriented quotiented cylinder $\\Bbb{RP}^2 \\times {\\Bbb R}$ also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of $\\Bbb{RP}^2 \\times {\\Bbb R}$ (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur. $\\diamond$\n\nRemark 4. In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round $S^2$ with the trivial plane ${\\Bbb R}^2$ . This is a $\\kappa$ -solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).\n\nThe classification theorem lets one understand the geometry of neighbourhoods of any given point in a $\\kappa$ -solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):\n\nDefinition 2. (Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.\n\n1. We say that U is an $\\varepsilon$ -neck if it is close (in a smooth topology) to a round cylinder $S^2 \\times (-R,R)$ , with x well in the middle of of this cylinder;\n2. We say that U is a C-component if U is diffeomorphic to $S^3$ or $\\Bbb{RP}^3$ (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.\n3. We say that U is $\\varepsilon$ -round if it is close (in a smooth topology) to a round sphere $S^3$ or spherical space form $S^3/\\Gamma$ (i.e. it is close to a constant curvature manifold).\n4. We say that U is a $(C,\\varepsilon)$ -cap if it consists of an $\\varepsilon$ -neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured ${\\Bbb RP}^3$ and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)\n5. We say that U is a canonical neighbourhood of x if it is one of the above four types.\n\nWhen the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.\n\nUsing Theorem 1 (and defining all terms precisely), one can easily show the following important statement:\n\nCorollary 1 (Canonical neighbourhood theorem for $\\kappa$ -solitons, informal version) Every point in a 3-dimensional $\\kappa$ -solution that does not contain an embedded copy of $\\Bbb{RP}^2$ with trivial normal bundle is contained in a canonical neighbourhood.\n\nThe next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.\n\n— High curvature regions of Ricci flows —\n\nCorollary 1 is an assertion about $\\kappa$ -solutions only, but it implies an important property about more general Ricci flows:\n\nTheorem 2. (Canonical neighbourhood for Ricci flows, informal version) Let $t \\mapsto (M,g)$ be a Ricci flow of compact 3-manifolds on a time interval ${}[0,T)$ , without any embedded copy of $\\Bbb{RP}^2$ with trivial normal bundle. Then every point $(t,x) \\in [0,T) \\times M$ with sufficiently large scalar curvature is contained in a canonical neighbourhood.\n\n(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)\n\nThe importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)\n\nTheorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:\n\n1. Suppose for contradiction that Theorem 2 failed. Then one could find a sequence $(t_n,x_n) \\in [0,T) \\times M$ of points with $R(t_n,x_n) \\to +\\infty$ which were not contained in canonical neighbourhoods.\n2. M, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.\n3. On any compact time interval ${}[0,t] \\times M$ , the scalar curvature is necessarily bounded, and thus $t_n \\to T$ . As a consequence, if we define the rescaled Ricci flows $g^{(n)}(t) = \\frac{1}{L_n^2} g( t_n + L_n^2 t )$ , where $L_n := R(t_n,x_n)^{-1/2}$ is the natural length scale associated to the scalar curvature at $(t_n,x_n)$ , then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds $(M,g^{(n)}(t))$ may cease to be compact, but will remain complete.\n4. Because of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3 ), we expect the rescaled flows $t \\mapsto (M, g^{(n)}(t))$ to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).\n5. If we can pick the points $(t_n,x_n)$ suitably (so that the scalar curvature $R(t_n,x_n)$ is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows $t \\mapsto (M, g^{(n)}(t))$ have bounded curvature in the limit.\n6. Since $\\kappa$ -noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7 ) should ensure that the rescaled flows remain $\\kappa$ -noncollapsed in the limit.\n7. Since the rescaled scalar curvature at the base point $x_n$ of $(M,g^{(n)})$ is equal to 1 by construction, any limiting flow will be non-flat.\n8. Various compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ . This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.\n9. Applying 2-8, we see that the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ is a $\\kappa$ -solution.\n10. Applying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points $(t_n,x_n)$ also lie in canonical neighbourhoods for sufficiently large n, a contradiction.\n\nThere are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for $\\kappa$ -solutions which we will deduce in later lectures. For Step 5, the problem is that the points $(t_n,x_n)$ that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point $(t_n,x_n)$ (much further away than the natural length scale $L_n$ , in particular), so that it does not show up in the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ . We will discuss these issues in more detail in later lectures.\n\n— Benchmarks in controlling $\\kappa$ -solutions —\n\nAs mentioned earlier, the next few lectures will be focused on controlling $\\kappa$ -solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.\n\nProposition 1. (Consequences of Hamilton’s Harnack inequality) Let $t \\mapsto (M,g(t))$ be a $\\kappa$ -solution. Then $R(t,x)$ is a non-decreasing function of time. Furthermore, for any $(t_0,x_0) \\in (-\\infty,0] \\times M$ , we have the pointwise inequalities\n\n$\\displaystyle |\\nabla l_{(t_0,x_0)}|^2 + R \\leq \\frac{3 l_{(t_0,x_0)}}{\\tau}$ (1)\n\nand\n\n$\\displaystyle -2 \\frac{l_{(t_0,x_0)}}{\\tau} \\leq \\frac{\\partial l_{(t_0,x_0)}}{\\partial \\tau} \\leq \\frac{l_{(t_0,x_0)}}{\\tau}$ (2)\n\non $(-\\infty,t_0) \\times M$ , where of course $\\tau := t_0 - t$ is the backwards time variable.\n\nThese inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of $\\kappa$ -solutions, as well as the Ricci flow equation $\\dot g = -2 \\hbox{Ric}$ of course.\n\nNow one can handle the two-dimensional case:\n\nProposition 2.(Classification of 2-dimensional $\\kappa$ -solutions) The only two-dimensional $\\kappa$ -solutions are the round shrinking 2-spheres.\n\nThis proposition relies on first studying a certain asymptotic limit of the $\\kappa$ -solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original $\\kappa$ -solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.\n\nTurning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.\n\nProposition 3. (Asymptotic curvature) Let $t \\mapsto (M,g(t))$ be a 3-dimensional $\\kappa$ solution which is not compact. Then for any time $t \\in (-\\infty,0)$ and any base point $p \\in M$ , we have $\\limsup_{x \\to \\infty} R(t,x) d_{g(t)}(x,p)^2 = +\\infty$ .\n\nThe proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.\n\nThe increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:\n\nProposition 4. (Asymptotic volume collapse) Let $t \\mapsto (M,g(t))$ be a 3-dimensional $\\kappa$ solution which is not compact. Then for any time $t \\in (-\\infty,0)$ and any base point $p \\in M$ , we have $\\limsup_{r \\to +\\infty} \\hbox{Vol}_{g(t)}( B_{g(t)})(p,r) ) / r^3 = 0$ .\n\nNote that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.\n\nAn important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball $B_{g(0)}(p,r)$ at time zero on which the volume is large (e.g. larger than $\\nu r^3$ for some $\\nu > 0$ ), one has bounded normalised curvature, thus $R = O_\\nu( 1 / r^2 )$ on this ball. This fact helps us deduce\n\nTheorem 3. (Perelman compactness theorem, informal version) The space of all pointed $\\kappa$ -solutions (allowing $\\kappa > 0$ to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.\n\nOne corollary of this compactness is that there is in fact a universal $\\kappa_0 > 0$ such that every $\\kappa$ -solution is a $\\kappa_0$ -solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on $\\kappa$ -solutions which come from the previous volume analysis.\n\nThe proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.","t_content":"In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the *-solutions* to the Ricci flow equation; we will be able to use compactness arguments (as well as the -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of -solutions. A secondary compactness argument lets us obtain that control of -solutions from control of an even more special type of solution, the *gradient shrinking solitons* that we already encountered in Lecture 8.\n\n[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]\n\nThe next few lectures will be devoted to the analysis of -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the *canonical neighbourhood theorem* for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about -solutions.\n\nOur treatment here is based primarily on the book of Morgan and Tian.\n\n\n— Definition of a -solution —\n\nWe fix a small number (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a -solution:\n\n\nDefinition 1.(-solutions) A-solutionis a Ricci flow which is\n-\nAncient, in the sense that t ranges on the interval ;\n-\nComplete and connected(i.e. (M,g(t)) is complete and connected for every t);\n-\nNon-negative Riemann curvature, i.e. is positive semidefinite at all points in spacetime;\n-\nBounded curvature, thus ;\n-\n-noncollapsed(see Definition 1 of Lecture 7) at every point in spacetime and at every scale ;\n-\nNon-flat, i.e. the curvature is non-zero at at least one point in spacetime.\nThis laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.\n\n**Remark 1. ** If a d-dimensional Riemann manifold is both flat (thus ) and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space . Thus the non-flat condition is only excluding the *trivial Ricci flow* with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere.\n**Remark 2.** In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and are non-negative, and that and . Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of -solutions are controlled by the scalar curvature.\nNow we discuss examples (and non-examples) of -solutions.\n\n**Example 1.** Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a -solution for sufficiently small if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere with the standard metric is a gradient shrinking solution and will generate a -solution for any and sufficiently small , which we will refer to as the *shrinking round sphere* -solution.\n**Exercse 1.** Show that the Cartesian product of two -solutions is again a -solution (with a smaller value of ), as is the Cartesian product of a -solution. Thus for instance the product of the shrinking round 2-sphere and the Euclidean line is a -solution, which we refer to as the *shrinking round 3-cylinder* .\n**Example 2.** In one dimension, there are no -solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is *not* a -solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere is -solution, as discussed above. We can quotient this by the obvious action to also get a shrinking round projective plane as a -solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional -solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder is not a -solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a -solution due to it being collapsed at large scales.\n**Example 3.** In three dimensions, we begin to get significantly more variety amongst -solutions. We have the round shrinking 3-sphere , but also all the quotients of such round spheres by free finite group actions (including the projective space , but with many other examples. We refer to these examples as *round shrinking 3-spherical space forms*. We have also seen the shrinking round cylinder ; there are also finite quotients of this example such as shrinking round projective cylinder , or the quotient of the cylinder by the orientation-preserving free involution . We refer to these examples as the *unoriented and oriented quotients of the shrinking round 3-cylinder* respectively. The oriented quotient can be viewed as a half-cylinder capped off with a punctured (and the whole manifold is in fact homeomorphic to a punctured ).\n**Example 4. **One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking or shrinking example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as *C-components* (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders capped off with what is topologically either a punctured or punctured (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as -capped strong -tubes (we will define this term precisely later). One can also consider *doubly -capped strong -tubes*, in which an approximately round finite cylinder is capped off at both ends by either a punctured or punctured ; such manifolds then become homeomorphic to either or . (Note we need to cap off any ends that show up in order to keep the manifold M complete.)\nAn important theorem of Perelman shows that these examples of -solutions are in fact the only ones:\n\n\nTheorem 1.(Perelman classification theorem, imprecise version) Every 3-dimensional -solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):\nA shrinking round 3-sphere (or shrinking round spherical space form );\n-\nA shrinking round 3-cylinder , the quotient , or one of its quotients (either oriented or unoriented);\n-\nA C-component;\n-\nA C-capped strong -tube;\n-\nA doubly C-capped strong -tube.\n-\nWe will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).\n\n**Remark 3.** At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong -tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong -tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional -solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as , or as the product of the cigar soliton and a line) or 3-dimensional (as in ) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not -solutions. The unoriented quotiented cylinder also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur.\n**Remark 4.** In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round with the trivial plane . This is a -solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).\nThe classification theorem lets one understand the geometry of neighbourhoods of any given point in a -solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):\n\n\nDefinition 2.(Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.\nWe say that U is an\n-\n-neckif it is close (in a smooth topology) to a round cylinder , with x well in the middle of of this cylinder;We say that U is a\n-\nC-componentif U is diffeomorphic to or (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.We say that U is\n-\n-roundif it is close (in a smooth topology) to a round sphere or spherical space form (i.e. it is close to a constant curvature manifold).We say that U is a\n-\n-capif it consists of an -neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)We say that U is a\n-\ncanonical neighbourhoodof x if it is one of the above four types.When the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.\n\nUsing Theorem 1 (and defining all terms precisely), one can easily show the following important statement:\n\nCorollary 1(Canonical neighbourhood theorem for -solitons, informal version) Every point in a 3-dimensional -solution that does not contain an embedded copy of with trivial normal bundle is contained in a canonical neighbourhood.\nThe next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.\n\n— High curvature regions of Ricci flows —\n\nCorollary 1 is an assertion about -solutions only, but it implies an important property about more general Ricci flows:\n\nTheorem 2.(Canonical neighbourhood for Ricci flows, informal version) Let be a Ricci flow of compact 3-manifolds on a time interval , without any embedded copy of with trivial normal bundle. Then every point with sufficiently large scalar curvature is contained in a canonical neighbourhood.\n(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)\n\nThe importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)\n\nTheorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:\n\nSuppose for contradiction that Theorem 2 failed. Then one could find a sequence of points with which were not contained in canonical neighbourhoods.\n-\nM, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.\n-\nOn any compact time interval , the scalar curvature is necessarily bounded, and thus . As a consequence, if we define the rescaled Ricci flows , where is the natural length scale associated to the scalar curvature at , then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds may cease to be compact, but will remain complete.\n-\nBecause of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3), we expect the rescaled flows to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).\n-\nIf we can pick the points suitably (so that the scalar curvature is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows have bounded curvature in the limit.\n-\nSince -noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7) should ensure that the rescaled flows remain -noncollapsed in the limit.\n-\nSince the rescaled scalar curvature at the base point of is equal to 1 by construction, any limiting flow will be non-flat.\n-\nVarious compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow . This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.\n-\nApplying 2-8, we see that the limiting flow is a -solution.\n-\nApplying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points also lie in canonical neighbourhoods for sufficiently large n, a contradiction.\n-\nThere are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for -solutions which we will deduce in later lectures. For Step 5, the problem is that the points that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point (much further away than the natural length scale , in particular), so that it does not show up in the limiting flow . We will discuss these issues in more detail in later lectures.\n\n— Benchmarks in controlling -solutions —\n\nAs mentioned earlier, the next few lectures will be focused on controlling -solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.\n\n\nProposition 1.(Consequences of Hamilton’s Harnack inequality) Let be a -solution. Then is a non-decreasing function of time. Furthermore, for any , we have the pointwise inequalities(1)\n\nand\n\n(2)\n\non , where of course is the backwards time variable.\n\nThese inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of -solutions, as well as the Ricci flow equation of course.\n\nNow one can handle the two-dimensional case:\n\nProposition 2.(Classification of 2-dimensional -solutions) The only two-dimensional -solutions are the round shrinking 2-spheres.\nThis proposition relies on first studying a certain asymptotic limit of the -solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original -solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.\n\nTurning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.\n\nProposition 3.(Asymptotic curvature) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .\nThe proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.\n\nThe increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:\n\n\nProposition 4.(Asymptotic volume collapse) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .\nNote that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.\n\nAn important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball at time zero on which the volume is large (e.g. larger than for some ), one has bounded normalised curvature, thus on this ball. This fact helps us deduce\n\nTheorem 3.(Perelman compactness theorem, informal version) The space of all pointed -solutions (allowing to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.\nOne corollary of this compactness is that there is in fact a universal such that every -solution is a -solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on -solutions which come from the previous volume analysis.\n\nThe proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.\n"} +{"id":"table_01","url":"https://nauticalstudies.org/category/psychico/","html":"\n\n\n\n\n\npsychico – Institute of International Nautical Studies\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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Κολλεγιο Αθηνων-Ψυχικου – Ναυτιλιακά Προγραμματα Ενηλικων

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Το Κολλέγιο Ψυχικού-Αθηνών πρωτοπορεί και δημιουργεί 15 ναυτιλιακά προγράμματα Ενηλίκων για επαγγελματίες που ενδιαφέρονται να πιστοποιηθούν στον χώρο της Ναυτιλίας. Τα προγράμματα προσφέρονται τόσο εξ αποστάσεως όσο και στο campus στο Π.Ψυχικό και παρέχουν πιστοποίηση στους εξής τομείς:

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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

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285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions

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16 May, 2008 in 285G - poincare conjecture, math.DG | Tags: canonical neighbourhoods, compactness and contradiction, gradient shrinking solitons, kappa-solutions | by Terence Tao\t\t

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In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a \"\\kappa\"-noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the \"\\kappa\"-solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the \"\\kappa\"-noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of \"\\kappa\"-solutions. A secondary compactness argument lets us obtain that control of \"\\kappa\"-solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.

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[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, \"\\kappa\"-noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]

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The next few lectures will be devoted to the analysis of \"\\kappa\"-solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a \"\\kappa\"-solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about \"\\kappa\"-solutions.

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Our treatment here is based primarily on the book of Morgan and Tian.

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— Definition of a \"\\kappa\"-solution —

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We fix a small number \"\\kappa (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a \"\\kappa\"-solution:

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Definition 1. (\"\\kappa\"-solutions) A \"\\kappa\"-solution is a Ricci flow \"t which is

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  1. Ancient, in the sense that t ranges on the interval \"(-\\infty,0]\";
    2. \n
  2. Complete and connected (i.e. (M,g(t)) is complete and connected for every t);
    3. \n
  3. Non-negative Riemann curvature, i.e. \"\\hbox{Riem}: is positive semidefinite at all points in spacetime;
    4. \n
  4. Bounded curvature, thus \"\\sup_{(t,x);
    5. \n
  5. \"\\kappa\"-noncollapsed (see Definition 1 of Lecture 7) at every point \"(t_0,x_0)\" in spacetime and at every scale \"r_0;
    6. \n
  6. Non-flat, i.e. the curvature is non-zero at at least one point in spacetime.
    7. \n
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This laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.

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Remark 1. If a d-dimensional Riemann manifold is both flat (thus \"\\hbox{Riem}=0\") and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space \"{\\Bbb. Thus the non-flat condition is only excluding the trivial Ricci flow \"M with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere. \"\\diamond\"

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Remark 2. In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and \"\\hbox{Ric}\" are non-negative, and that \"|\\hbox{Riem}|_g, and \"R. Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of \"\\kappa\"-solutions are controlled by the scalar curvature. \"\\diamond\"

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Now we discuss examples (and non-examples) of \"\\kappa\"-solutions.

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Example 1. Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a \"\\kappa\"-solution for sufficiently small \"\\kappa\" if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere \"S^d\" with the standard metric is a gradient shrinking solution and will generate a \"\\kappa\"-solution for any \"d and sufficiently small \"\\kappa, which we will refer to as the shrinking round sphere \"\\kappa\"-solution. \"\\diamond\"

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Exercse 1. Show that the Cartesian product of two \"\\kappa\"-solutions is again a \"\\kappa\"-solution (with a smaller value of \"\\kappa\"), as is the Cartesian product of a \"\\kappa\"-solution. Thus for instance the product \"S^2 of the shrinking round 2-sphere and the Euclidean line is a \"\\kappa\"-solution, which we refer to as the shrinking round 3-cylinder \"S^2. \"\\diamond\"

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Example 2. In one dimension, there are no \"\\kappa\"-solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is not a \"\\kappa\"-solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere \"S^2\" is \"\\kappa\"-solution, as discussed above. We can quotient this by the obvious \"{\\Bbb action to also get a shrinking round projective plane \"\\Bbb{RP}^2\" as a \"\\kappa\"-solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional \"\\kappa\"-solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder \"S^1 is not a \"\\kappa\"-solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a \"\\kappa\"-solution due to it being collapsed at large scales. \"\\diamond\"
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Example 3. In three dimensions, we begin to get significantly more variety amongst \"\\kappa\"-solutions. We have the round shrinking 3-sphere \"S^3\", but also all the quotients \"S^3/\\Gamma\" of such round spheres by free finite group actions (including the projective space \"{\\Bbb, but with many other examples. We refer to these examples as round shrinking 3-spherical space forms. We have also seen the shrinking round cylinder \"S^2; there are also finite quotients of this example such as shrinking round projective cylinder \"\\Bbb{RP}^2, or the quotient of the cylinder by the orientation-preserving free involution \"(\\omega,z). We refer to these examples as the unoriented and oriented quotients of the shrinking round 3-cylinder respectively. The oriented quotient can be viewed as a half-cylinder \"S^2 capped off with a punctured \"\\Bbb{RP}^3\" (and the whole manifold is in fact homeomorphic to a punctured \"\\Bbb{RP}^3\"). \"\\diamond\"

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Example 4. One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking \"S^2\" or shrinking \"{\\Bbb example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as C-components (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders \"S^2 capped off with what is topologically either a punctured \"\\Bbb{RP}^3\" or punctured \"S^3\" (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as \"C\"-capped strong \"\\varepsilon\"-tubes (we will define this term precisely later). One can also consider doubly \"C\"-capped strong \"\\varepsilon\"-tubes, in which an approximately round finite cylinder \"S^2 is capped off at both ends by either a punctured \"\\Bbb{RP}^3\" or punctured \"S^3\"; such manifolds then become homeomorphic to either \"S^3\" or \"{\\Bbb. (Note we need to cap off any ends that show up in order to keep the manifold M complete.) \"\\diamond\"

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An important theorem of Perelman shows that these examples of \"\\kappa\"-solutions are in fact the only ones:

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Theorem 1. (Perelman classification theorem, imprecise version) Every 3-dimensional \"\\kappa\"-solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):

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  1. A shrinking round 3-sphere \"S^3\" (or shrinking round spherical space form \"S^3/\\Gamma\");
    2. \n
  2. A shrinking round 3-cylinder \"S^2, the quotient \"\\Bbb{RP}^2, or one of its quotients (either oriented or unoriented);
    3. \n
  3. A C-component;
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  4. A C-capped strong \"\\varepsilon\"-tube;
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  5. A doubly C-capped strong \"\\varepsilon\"-tube.
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We will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).

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Remark 3. At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong \"\\varepsilon\"-tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong \"\\varepsilon\"-tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional \"\\kappa\"-solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as \"S^1, or as the product of the cigar soliton and a line) or 3-dimensional (as in \"{\\Bbb) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not \"\\kappa\"-solutions. The unoriented quotiented cylinder \"\\Bbb{RP}^2 also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of \"\\Bbb{RP}^2 (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur. \"\\diamond\"

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Remark 4. In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round \"S^2\" with the trivial plane \"{\\Bbb. This is a \"\\kappa\"-solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).

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The classification theorem lets one understand the geometry of neighbourhoods of any given point in a \"\\kappa\"-solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):

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Definition 2. (Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.

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  1. We say that U is an \"\\varepsilon\"-neck if it is close (in a smooth topology) to a round cylinder \"S^2, with x well in the middle of of this cylinder;
    2. \n
  2. We say that U is a C-component if U is diffeomorphic to \"S^3\" or \"\\Bbb{RP}^3\" (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.
    3. \n
  3. We say that U is \"\\varepsilon\"-round if it is close (in a smooth topology) to a round sphere \"S^3\" or spherical space form \"S^3/\\Gamma\" (i.e. it is close to a constant curvature manifold).
    4. \n
  4. We say that U is a \"(C,\\varepsilon)\"-cap if it consists of an \"\\varepsilon\"-neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured \"{\\Bbb and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)
    5. \n
  5. We say that U is a canonical neighbourhood of x if it is one of the above four types.
    6. \n
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When the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.

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Using Theorem 1 (and defining all terms precisely), one can easily show the following important statement:

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Corollary 1 (Canonical neighbourhood theorem for \"\\kappa\"-solitons, informal version) Every point in a 3-dimensional \"\\kappa\"-solution that does not contain an embedded copy of \"\\Bbb{RP}^2\" with trivial normal bundle is contained in a canonical neighbourhood.

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The next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.

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— High curvature regions of Ricci flows —

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Corollary 1 is an assertion about \"\\kappa\"-solutions only, but it implies an important property about more general Ricci flows:

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Theorem 2. (Canonical neighbourhood for Ricci flows, informal version) Let \"t be a Ricci flow of compact 3-manifolds on a time interval \"{}[0,T)\", without any embedded copy of \"\\Bbb{RP}^2\" with trivial normal bundle. Then every point \"(t,x) with sufficiently large scalar curvature is contained in a canonical neighbourhood.

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(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)

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The importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)

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Theorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:

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  1. Suppose for contradiction that Theorem 2 failed. Then one could find a sequence \"(t_n,x_n) of points with \"R(t_n,x_n) which were not contained in canonical neighbourhoods.
    2. \n
  2. M, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.
    3. \n
  3. On any compact time interval \"{}[0,t], the scalar curvature is necessarily bounded, and thus \"t_n. As a consequence, if we define the rescaled Ricci flows \"g^{(n)}(t), where \"L_n is the natural length scale associated to the scalar curvature at \"(t_n,x_n)\", then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds \"(M,g^{(n)}(t))\" may cease to be compact, but will remain complete.
    4. \n
  4. Because of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3), we expect the rescaled flows \"t to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).
    5. \n
  5. If we can pick the points \"(t_n,x_n)\" suitably (so that the scalar curvature \"R(t_n,x_n)\" is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows \"t have bounded curvature in the limit.
    6. \n
  6. Since \"\\kappa\"-noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7) should ensure that the rescaled flows remain \"\\kappa\"-noncollapsed in the limit.
    7. \n
  7. Since the rescaled scalar curvature at the base point \"x_n\" of \"(M,g^{(n)})\" is equal to 1 by construction, any limiting flow will be non-flat.
    8. \n
  8. Various compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow \"(M^{(\\infty)},. This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.
    9. \n
  9. Applying 2-8, we see that the limiting flow \"(M^{(\\infty)}, is a \"\\kappa\"-solution.
    10. \n
  10. Applying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points \"(t_n,x_n)\" also lie in canonical neighbourhoods for sufficiently large n, a contradiction.
    11. \n
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There are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for \"\\kappa\"-solutions which we will deduce in later lectures. For Step 5, the problem is that the points \"(t_n,x_n)\" that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow \"(M^{(\\infty)}, ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point \"(t_n,x_n)\" (much further away than the natural length scale \"L_n\", in particular), so that it does not show up in the limiting flow \"(M^{(\\infty)},. We will discuss these issues in more detail in later lectures.

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— Benchmarks in controlling \"\\kappa\"-solutions —

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As mentioned earlier, the next few lectures will be focused on controlling \"\\kappa\"-solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.

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Proposition 1. (Consequences of Hamilton’s Harnack inequality) Let \"t be a \"\\kappa\"-solution. Then \"R(t,x)\" is a non-decreasing function of time. Furthermore, for any \"(t_0,x_0), we have the pointwise inequalities

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\"\\displaystyle (1)

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and

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\"\\displaystyle (2)

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on \"(-\\infty,t_0), where of course \"\\tau is the backwards time variable.

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These inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of \"\\kappa\"-solutions, as well as the Ricci flow equation \"\\dot of course.

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Now one can handle the two-dimensional case:

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Proposition 2. (Classification of 2-dimensional \"\\kappa\"-solutions) The only two-dimensional \"\\kappa\"-solutions are the round shrinking 2-spheres.

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This proposition relies on first studying a certain asymptotic limit of the \"\\kappa\"-solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original \"\\kappa\"-solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.

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Turning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.

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Proposition 3. (Asymptotic curvature) Let \"t be a 3-dimensional \"\\kappa\" solution which is not compact. Then for any time \"t and any base point \"p, we have \"\\limsup_{x.

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The proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.

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The increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:

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Proposition 4. (Asymptotic volume collapse) Let \"t be a 3-dimensional \"\\kappa\" solution which is not compact. Then for any time \"t and any base point \"p, we have \"\\limsup_{r.

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Note that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.

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An important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball \"B_{g(0)}(p,r)\" at time zero on which the volume is large (e.g. larger than \"\\nu for some \"\\nu), one has bounded normalised curvature, thus \"R on this ball. This fact helps us deduce

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Theorem 3. (Perelman compactness theorem, informal version) The space of all pointed \"\\kappa\"-solutions (allowing \"\\kappa to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.

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One corollary of this compactness is that there is in fact a universal \"\\kappa_0 such that every \"\\kappa\"-solution is a \"\\kappa_0\"-solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on \"\\kappa\"-solutions which come from the previous volume analysis.

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The proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.

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9 comments

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16 May, 2008 at 12:40 pm

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Anton Fonarev

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Dear prof. Tao,

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I mentioned a small misprint. There’s no closing bracket in Theorem 3 after \"\\kappa.

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16 May, 2008 at 12:44 pm

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Terence Tao

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Thanks for the correction!

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18 May, 2008 at 7:32 pm

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This past two weeks… « It’s Equal, but It’s Different…

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[…] 285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume, 285G, Lecture 11: κ-noncollapsing via Perelman reduced volume, 285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions […]

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19 May, 2008 at 10:13 am

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285G, Lecture 13: Li-Yau-Hamilton Harnack inequalities and κ-solutions « What’s new

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[…] The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the -geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send ). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the -solutions introduced in the previous lecture. […]

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19 May, 2008 at 12:57 pm

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Sylvain Maillot

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Another minor correction: in Example 2, you should delete the words
\n“shrinking round” before \"S^1\\times : this manifold
\nis flat, so it does not shrink by Ricci flow (and I wouldn’t call it “round”.)

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A more serious point: I am surprised to see five cases in the classification of
\n\"\\kappa\"-solutions (Theorem 1), since in Perelman there are only four.
\nWhat is exactly the difference between case 3 “C-component” and case 5
\n“doubly \"C\"-capped strong \"\\varepsilon\"-tube” ?

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19 May, 2008 at 2:39 pm

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Terence Tao

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Dear Sylvain: Thanks again for the corrections!

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Regarding Theorem 1, I am basing this on Theorem 9.93 of Morgan-Tian (though I am grouping the cases a little differently). As I understand it, a C-component and a doubly C-capped strong epsilon-tube are topologically the same, but geometrically rather different; in the former the sectional curvature is bounded above and below everywhere, whereas in the latter the sectional curvature can be very small in the epsilon-tube component of the manifold (in planes parallel to the direction of the tube, of course). Also, if we normalise the maximal curvature to be 1, then C-components need to have diameter comparable to 1, but doubly C-capped epsilon-tubes can have arbitrarily large diameter.

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Perhaps the key difference here between Morgan-Tian and Perelman is that in Morgan-Tian one tries to classify the global structure of kappa-solutions, even though for applications it is only the local structure (i.e. the canonical neighbourhoods) which is of importance. Thus, for instance, if one takes a doubly C-capped strong epsilon-tube, the canonical neighbourhood of any given point in such an object would be either an epsilon-neck, epsilon-cap, or a closed manifold in Perelman’s notation, depending on whether the tube is long or short and whether the point lies in the middle of the manifold or near one of the ends.

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[In Morgan-Tian, the relevant canonical neighbourhood theorem is Corollary 9.94; the definition of canonical neighbourhoods on page 232 (and reproduced here as Definition 2) has exactly four cases and matches up with the corresponding notion in Perelman’s second paper.]

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20 May, 2008 at 10:12 am

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Sylvain Maillot

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OK. In fact, if you take a \"\\kappa\"-solution which is a \"C\"-component at time zero and look into its past, then you will see a doubly capped neck, where the neck part becomes longer and longer as time goes to negative infinity.

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This means that you can merge those two cases if you add time-shifting in your equivalence relation (in addition to isometry and rescaling). I would say that your statement is more a classification of time-zero slices of \"\\kappa\"-solutions than a classification of \"\\kappa\"-solutions strictly speaking.

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Well, all this is irrelevant to the Poincare conjecture anyway…

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2 June, 2008 at 4:12 pm

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285G, Lecture 17: The structure of κ-solutions « What’s new

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[…] 2 June, 2008 in 285G – poincare conjecture, math.DG by Terence Tao Tags: compactness and contradiction, geometric limits, gradient shrinking solitons, kappa-solutions, necks Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the previous lecture, we now use this classification, combined with extensive use of compactness and contradiction arguments, as well as the comparison geometry of complete Riemannian manifolds of non-negative curvature, to understand the structure of -solutions in these dimensions, with the aim being to state and prove precise versions of Theorem 1 and Corollary 1 from Lecture 12. […]

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4 June, 2008 at 8:37 am

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285G, Lecture 18: The structure of high-curvature regions of Ricci flow « What’s new

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[…] now use them to describe the structure of high curvature regions of Ricci flow, as promised back in Lecture 12, in particular controlling their geometry and topology to the extent that surgery will be applied, […]

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\n\t \n\n\n\n", + "m_content": "In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a $\\kappa$ -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the $\\kappa$ -solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the $\\kappa$ -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of $\\kappa$ -solutions. A secondary compactness argument lets us obtain that control of $\\kappa$ -solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.\n\n[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, $\\kappa$ -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]\n\nThe next few lectures will be devoted to the analysis of $\\kappa$ -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a $\\kappa$ -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about $\\kappa$ -solutions.\n\nOur treatment here is based primarily on the book of Morgan and Tian.\n\n— Definition of a $\\kappa$ -solution —\n\nWe fix a small number $\\kappa > 0$ (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a $\\kappa$ -solution:\n\nDefinition 1. ( $\\kappa$ -solutions) A $\\kappa$ -solution is a Ricci flow $t \\mapsto (M,g(t))$ which is\n\n1. Ancient , in the sense that t ranges on the interval $(-\\infty,0]$ ;\n2. Complete and connected (i.e. (M,g(t)) is complete and connected for every t);\n3. Non-negative Riemann curvature , i.e. $\\hbox{Riem}: \\bigwedge^2 TM \\to \\bigwedge^2 TM$ is positive semidefinite at all points in spacetime;\n4. Bounded curvature , thus $\\sup_{(t,x) \\in (-\\infty,0] \\times M} |\\hbox{Riem}|_g < +\\infty$ ;\n5. $\\kappa$ -noncollapsed (see Definition 1 of Lecture 7 ) at every point $(t_0,x_0)$ in spacetime and at every scale $r_0 > 0$ ;\n6. Non-flat , i.e. the curvature is non-zero at at least one point in spacetime.\n\nThis laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.\n\nRemark 1. If a d-dimensional Riemann manifold is both flat (thus $\\hbox{Riem}=0$ ) and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space ${\\Bbb R}^d$ . Thus the non-flat condition is only excluding the trivial Ricci flow $M = {\\Bbb R}^d$ with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere. $\\diamond$\n\nRemark 2. In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and $\\hbox{Ric}$ are non-negative, and that $|\\hbox{Riem}|_g, |\\hbox{Ric}|_g = O(R)$ and $R = O_d(1)$ . Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of $\\kappa$ -solutions are controlled by the scalar curvature. $\\diamond$\n\nNow we discuss examples (and non-examples) of $\\kappa$ -solutions.\n\nExample 1. Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a $\\kappa$ -solution for sufficiently small $\\kappa$ if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere $S^d$ with the standard metric is a gradient shrinking solution and will generate a $\\kappa$ -solution for any $d \\geq 2$ and sufficiently small $\\kappa > 0$ , which we will refer to as the shrinking round sphere $\\kappa$ -solution. $\\diamond$\n\nExercse 1. Show that the Cartesian product of two $\\kappa$ -solutions is again a $\\kappa$ -solution (with a smaller value of $\\kappa$ ), as is the Cartesian product of a $\\kappa$ -solution. Thus for instance the product $S^2 \\times {\\Bbb R}$ of the shrinking round 2-sphere and the Euclidean line is a $\\kappa$ -solution, which we refer to as the shrinking round 3-cylinder $S^2 \\times {\\Bbb R}$ . $\\diamond$\n\nExample 2. In one dimension, there are no $\\kappa$ -solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is not a $\\kappa$ -solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere $S^2$ is $\\kappa$ -solution, as discussed above. We can quotient this by the obvious ${\\Bbb Z}/2$ action to also get a shrinking round projective plane $\\Bbb{RP}^2$ as a $\\kappa$ -solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional $\\kappa$ -solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder $S^1 \\times {\\Bbb R}$ is not a $\\kappa$ -solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a $\\kappa$ -solution due to it being collapsed at large scales. $\\diamond$\n\nExample 3. In three dimensions, we begin to get significantly more variety amongst $\\kappa$ -solutions. We have the round shrinking 3-sphere $S^3$ , but also all the quotients $S^3/\\Gamma$ of such round spheres by free finite group actions (including the projective space ${\\Bbb RP}^3$ , but with many other examples. We refer to these examples as round shrinking 3-spherical space forms. We have also seen the shrinking round cylinder $S^2 \\times {\\Bbb R}$ ; there are also finite quotients of this example such as shrinking round projective cylinder $\\Bbb{RP}^2 \\times {\\Bbb R}$ , or the quotient of the cylinder by the orientation-preserving free involution $(\\omega,z) \\mapsto (-\\omega,-z)$ . We refer to these examples as the unoriented and oriented quotients of the shrinking round 3-cylinder respectively. The oriented quotient can be viewed as a half-cylinder $S^2 \\times [1,+\\infty)$ capped off with a punctured $\\Bbb{RP}^3$ (and the whole manifold is in fact homeomorphic to a punctured $\\Bbb{RP}^3$ ). $\\diamond$\n\nExample 4. One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking $S^2$ or shrinking ${\\Bbb RP}^3$ example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as C-components (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders $S^2 \\times [1,+\\infty)$ capped off with what is topologically either a punctured $\\Bbb{RP}^3$ or punctured $S^3$ (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as $C$ -capped strong $\\varepsilon$ -tubes (we will define this term precisely later). One can also consider doubly $C$ -capped strong $\\varepsilon$ -tubes, in which an approximately round finite cylinder $S^2 \\times [-T,T]$ is capped off at both ends by either a punctured $\\Bbb{RP}^3$ or punctured $S^3$ ; such manifolds then become homeomorphic to either $S^3$ or ${\\Bbb RP}^3$ . (Note we need to cap off any ends that show up in order to keep the manifold M complete.) $\\diamond$\n\nAn important theorem of Perelman shows that these examples of $\\kappa$ -solutions are in fact the only ones:\n\nTheorem 1. (Perelman classification theorem, imprecise version) Every 3-dimensional $\\kappa$ -solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):\n\n1. A shrinking round 3-sphere $S^3$ (or shrinking round spherical space form $S^3/\\Gamma$ );\n2. A shrinking round 3-cylinder $S^2 \\times {\\Bbb R}$ , the quotient $\\Bbb{RP}^2 \\times {\\Bbb R}$ , or one of its quotients (either oriented or unoriented);\n3. A C-component;\n4. A C-capped strong $\\varepsilon$ -tube;\n5. A doubly C-capped strong $\\varepsilon$ -tube.\n\nWe will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).\n\nRemark 3. At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong $\\varepsilon$ -tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong $\\varepsilon$ -tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional $\\kappa$ -solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as $S^1 \\times {\\Bbb R}^2$ , or as the product of the cigar soliton and a line) or 3-dimensional (as in ${\\Bbb R}^3$ ) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not $\\kappa$ -solutions. The unoriented quotiented cylinder $\\Bbb{RP}^2 \\times {\\Bbb R}$ also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of $\\Bbb{RP}^2 \\times {\\Bbb R}$ (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur. $\\diamond$\n\nRemark 4. In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round $S^2$ with the trivial plane ${\\Bbb R}^2$ . This is a $\\kappa$ -solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).\n\nThe classification theorem lets one understand the geometry of neighbourhoods of any given point in a $\\kappa$ -solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):\n\nDefinition 2. (Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.\n\n1. We say that U is an $\\varepsilon$ -neck if it is close (in a smooth topology) to a round cylinder $S^2 \\times (-R,R)$ , with x well in the middle of of this cylinder;\n2. We say that U is a C-component if U is diffeomorphic to $S^3$ or $\\Bbb{RP}^3$ (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.\n3. We say that U is $\\varepsilon$ -round if it is close (in a smooth topology) to a round sphere $S^3$ or spherical space form $S^3/\\Gamma$ (i.e. it is close to a constant curvature manifold).\n4. We say that U is a $(C,\\varepsilon)$ -cap if it consists of an $\\varepsilon$ -neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured ${\\Bbb RP}^3$ and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)\n5. We say that U is a canonical neighbourhood of x if it is one of the above four types.\n\nWhen the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.\n\nUsing Theorem 1 (and defining all terms precisely), one can easily show the following important statement:\n\nCorollary 1 (Canonical neighbourhood theorem for $\\kappa$ -solitons, informal version) Every point in a 3-dimensional $\\kappa$ -solution that does not contain an embedded copy of $\\Bbb{RP}^2$ with trivial normal bundle is contained in a canonical neighbourhood.\n\nThe next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.\n\n— High curvature regions of Ricci flows —\n\nCorollary 1 is an assertion about $\\kappa$ -solutions only, but it implies an important property about more general Ricci flows:\n\nTheorem 2. (Canonical neighbourhood for Ricci flows, informal version) Let $t \\mapsto (M,g)$ be a Ricci flow of compact 3-manifolds on a time interval ${}[0,T)$ , without any embedded copy of $\\Bbb{RP}^2$ with trivial normal bundle. Then every point $(t,x) \\in [0,T) \\times M$ with sufficiently large scalar curvature is contained in a canonical neighbourhood.\n\n(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)\n\nThe importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)\n\nTheorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:\n\n1. Suppose for contradiction that Theorem 2 failed. Then one could find a sequence $(t_n,x_n) \\in [0,T) \\times M$ of points with $R(t_n,x_n) \\to +\\infty$ which were not contained in canonical neighbourhoods.\n2. M, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.\n3. On any compact time interval ${}[0,t] \\times M$ , the scalar curvature is necessarily bounded, and thus $t_n \\to T$ . As a consequence, if we define the rescaled Ricci flows $g^{(n)}(t) = \\frac{1}{L_n^2} g( t_n + L_n^2 t )$ , where $L_n := R(t_n,x_n)^{-1/2}$ is the natural length scale associated to the scalar curvature at $(t_n,x_n)$ , then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds $(M,g^{(n)}(t))$ may cease to be compact, but will remain complete.\n4. Because of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3 ), we expect the rescaled flows $t \\mapsto (M, g^{(n)}(t))$ to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).\n5. If we can pick the points $(t_n,x_n)$ suitably (so that the scalar curvature $R(t_n,x_n)$ is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows $t \\mapsto (M, g^{(n)}(t))$ have bounded curvature in the limit.\n6. Since $\\kappa$ -noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7 ) should ensure that the rescaled flows remain $\\kappa$ -noncollapsed in the limit.\n7. Since the rescaled scalar curvature at the base point $x_n$ of $(M,g^{(n)})$ is equal to 1 by construction, any limiting flow will be non-flat.\n8. Various compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ . This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.\n9. Applying 2-8, we see that the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ is a $\\kappa$ -solution.\n10. Applying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points $(t_n,x_n)$ also lie in canonical neighbourhoods for sufficiently large n, a contradiction.\n\nThere are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for $\\kappa$ -solutions which we will deduce in later lectures. For Step 5, the problem is that the points $(t_n,x_n)$ that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point $(t_n,x_n)$ (much further away than the natural length scale $L_n$ , in particular), so that it does not show up in the limiting flow $(M^{(\\infty)}, g^{(\\infty)})$ . We will discuss these issues in more detail in later lectures.\n\n— Benchmarks in controlling $\\kappa$ -solutions —\n\nAs mentioned earlier, the next few lectures will be focused on controlling $\\kappa$ -solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.\n\nProposition 1. (Consequences of Hamilton’s Harnack inequality) Let $t \\mapsto (M,g(t))$ be a $\\kappa$ -solution. Then $R(t,x)$ is a non-decreasing function of time. Furthermore, for any $(t_0,x_0) \\in (-\\infty,0] \\times M$ , we have the pointwise inequalities\n\n$\\displaystyle |\\nabla l_{(t_0,x_0)}|^2 + R \\leq \\frac{3 l_{(t_0,x_0)}}{\\tau}$ (1)\n\nand\n\n$\\displaystyle -2 \\frac{l_{(t_0,x_0)}}{\\tau} \\leq \\frac{\\partial l_{(t_0,x_0)}}{\\partial \\tau} \\leq \\frac{l_{(t_0,x_0)}}{\\tau}$ (2)\n\non $(-\\infty,t_0) \\times M$ , where of course $\\tau := t_0 - t$ is the backwards time variable.\n\nThese inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of $\\kappa$ -solutions, as well as the Ricci flow equation $\\dot g = -2 \\hbox{Ric}$ of course.\n\nNow one can handle the two-dimensional case:\n\nProposition 2.(Classification of 2-dimensional $\\kappa$ -solutions) The only two-dimensional $\\kappa$ -solutions are the round shrinking 2-spheres.\n\nThis proposition relies on first studying a certain asymptotic limit of the $\\kappa$ -solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original $\\kappa$ -solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.\n\nTurning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.\n\nProposition 3. (Asymptotic curvature) Let $t \\mapsto (M,g(t))$ be a 3-dimensional $\\kappa$ solution which is not compact. Then for any time $t \\in (-\\infty,0)$ and any base point $p \\in M$ , we have $\\limsup_{x \\to \\infty} R(t,x) d_{g(t)}(x,p)^2 = +\\infty$ .\n\nThe proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.\n\nThe increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:\n\nProposition 4. (Asymptotic volume collapse) Let $t \\mapsto (M,g(t))$ be a 3-dimensional $\\kappa$ solution which is not compact. Then for any time $t \\in (-\\infty,0)$ and any base point $p \\in M$ , we have $\\limsup_{r \\to +\\infty} \\hbox{Vol}_{g(t)}( B_{g(t)})(p,r) ) / r^3 = 0$ .\n\nNote that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.\n\nAn important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball $B_{g(0)}(p,r)$ at time zero on which the volume is large (e.g. larger than $\\nu r^3$ for some $\\nu > 0$ ), one has bounded normalised curvature, thus $R = O_\\nu( 1 / r^2 )$ on this ball. This fact helps us deduce\n\nTheorem 3. (Perelman compactness theorem, informal version) The space of all pointed $\\kappa$ -solutions (allowing $\\kappa > 0$ to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.\n\nOne corollary of this compactness is that there is in fact a universal $\\kappa_0 > 0$ such that every $\\kappa$ -solution is a $\\kappa_0$ -solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on $\\kappa$ -solutions which come from the previous volume analysis.\n\nThe proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.", + "t_content": "In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the *-solutions* to the Ricci flow equation; we will be able to use compactness arguments (as well as the -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of -solutions. A secondary compactness argument lets us obtain that control of -solutions from control of an even more special type of solution, the *gradient shrinking solitons* that we already encountered in Lecture 8.\n\n[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]\n\nThe next few lectures will be devoted to the analysis of -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the *canonical neighbourhood theorem* for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about -solutions.\n\nOur treatment here is based primarily on the book of Morgan and Tian.\n\n\n— Definition of a -solution —\n\nWe fix a small number (basically the parameter that comes out of the non-collapsing theorem). Here is the formal definition of a -solution:\n\n\nDefinition 1.(-solutions) A-solutionis a Ricci flow which is\n-\nAncient, in the sense that t ranges on the interval ;\n-\nComplete and connected(i.e. (M,g(t)) is complete and connected for every t);\n-\nNon-negative Riemann curvature, i.e. is positive semidefinite at all points in spacetime;\n-\nBounded curvature, thus ;\n-\n-noncollapsed(see Definition 1 of Lecture 7) at every point in spacetime and at every scale ;\n-\nNon-flat, i.e. the curvature is non-zero at at least one point in spacetime.\nThis laundry list of properties arises because they are the properties that we are able to directly establish on limits of rescaled Ricci flows; see below.\n\n**Remark 1. ** If a d-dimensional Riemann manifold is both flat (thus ) and non-collapsed at every scale, then (by Cheeger’s lemma, Theorem 1 from Lecture 7) its injectivity radius is infinite, and by normal coordinates the manifold is isometric to Euclidean space . Thus the non-flat condition is only excluding the *trivial Ricci flow* with the standard (and static) metric. The non-flat condition tells us that the (scalar, say) curvature is positive in at least one point of spacetime, but we will shortly be able to use the strong maximum principle to conclude in fact that the curvature is positive everywhere.\n**Remark 2.** In three dimensions, the condition of non-negative RIemann curvature is equivalent to that of non-negative sectional curvature; see the discussion in Lecture 0. In any dimension, the conditions of non-negative bounded Riemann curvature imply that R and are non-negative, and that and . Thus as far as magnitude is concerned, the Riemann and Ricci curvatures of -solutions are controlled by the scalar curvature.\nNow we discuss examples (and non-examples) of -solutions.\n\n**Example 1.** Every gradient shrinking soliton or gradient steady soliton (M,g) (see Lecture 8) gives an ancient flow. This flow will be a -solution for sufficiently small if the Einstein manifold (M,g) is complete, connected, non-collapsed at every scale, and is not Euclidean space. For instance, the round sphere with the standard metric is a gradient shrinking solution and will generate a -solution for any and sufficiently small , which we will refer to as the *shrinking round sphere* -solution.\n**Exercse 1.** Show that the Cartesian product of two -solutions is again a -solution (with a smaller value of ), as is the Cartesian product of a -solution. Thus for instance the product of the shrinking round 2-sphere and the Euclidean line is a -solution, which we refer to as the *shrinking round 3-cylinder* .\n**Example 2.** In one dimension, there are no -solutions, as every manifold is flat; in particular, the 1-sphere (i.e. a circle) is *not* a -solution (it is flat and also collapsed at large scales). In two dimensions, the shrinking round 2-sphere is -solution, as discussed above. We can quotient this by the obvious action to also get a shrinking round projective plane as a -solution. But we shall show in later lectures that if we restrict attention to oriented manifolds, then the shrinking round 2-sphere is the only 2-dimensional -solutions; this result is due to Hamilton, see e.g. Chapter 5 of Chow-Knopf. For instance, the 2-cylinder is not a -solution (it is both flat and collapsed at large scales). The cigar soliton (Example 3 from Lecture 8) also fails to be a -solution due to it being collapsed at large scales.\n**Example 3.** In three dimensions, we begin to get significantly more variety amongst -solutions. We have the round shrinking 3-sphere , but also all the quotients of such round spheres by free finite group actions (including the projective space , but with many other examples. We refer to these examples as *round shrinking 3-spherical space forms*. We have also seen the shrinking round cylinder ; there are also finite quotients of this example such as shrinking round projective cylinder , or the quotient of the cylinder by the orientation-preserving free involution . We refer to these examples as the *unoriented and oriented quotients of the shrinking round 3-cylinder* respectively. The oriented quotient can be viewed as a half-cylinder capped off with a punctured (and the whole manifold is in fact homeomorphic to a punctured ).\n**Example 4. **One can also imagine perturbations of the shrinking solutions mentioned above. For instance, one could imagine non-round versions of the shrinking or shrinking example, in which the manifold has sectional curvature which is still positive but not constant. We shall informally refer to such solutions as *C-components* (we will define this term formally later, and explain the role of the parameter C). Similarly one could imagine variants of the oriented quotient of the shrinking round cylinder, which are approximately round half-cylinders capped off with what is topologically either a punctured or punctured (i.e. with something homeomorphic to a ball); a 3-dimensional variant of a cigar soliton would fall into this category (such solitons have been constructed by Bryant and by Cao). We informally refer to such solutions as -capped strong -tubes (we will define this term precisely later). One can also consider *doubly -capped strong -tubes*, in which an approximately round finite cylinder is capped off at both ends by either a punctured or punctured ; such manifolds then become homeomorphic to either or . (Note we need to cap off any ends that show up in order to keep the manifold M complete.)\nAn important theorem of Perelman shows that these examples of -solutions are in fact the only ones:\n\n\nTheorem 1.(Perelman classification theorem, imprecise version) Every 3-dimensional -solution takes on one of the following forms at time zero (after isometry and rescaling, if necessary):\nA shrinking round 3-sphere (or shrinking round spherical space form );\n-\nA shrinking round 3-cylinder , the quotient , or one of its quotients (either oriented or unoriented);\n-\nA C-component;\n-\nA C-capped strong -tube;\n-\nA doubly C-capped strong -tube.\n-\nWe will make this theorem more precise in later lectures (or if you are impatient, you can read Chapter 9 of Morgan-Tian).\n\n**Remark 3.** At very large scales, Theorem 1 implies that an ancient solution at time zero either looks 0-dimensional (because the manifold was compact, as in the case of a sphere, spherical space form, C-component, or doubly C-capped strong -tube) or 1-dimensional, resembling a line (in the case of the cylinder) or half-line (for C-capped strong -tube). Oversimplifying somewhat, this 0- or 1-dimensionality of the three-dimensional -solutions is the main reason why surgery is even possible; if Ricci flow singularities could look 2-dimensional (such as , or as the product of the cigar soliton and a line) or 3-dimensional (as in ) then it is not clear at all how to define a surgery procedure to excise the singularity. The point is that all the potential candidates for singularity that look 2-dimensional or 3-dimensional at large scales (after rescaling) are either flat or collapsed (or do not have bounded nonnegative curvature), and so are not -solutions. The unoriented quotiented cylinder also causes difficulties with surgery (despite being only one-dimensional at large scales), because it is hard to cap off such a cylinder in a manner which is well-behaved with respect to Ricci flow; however if we assume that the original manifold M contains no embedded copy of (which is for instance the case if the manifold is oriented, and in particular if it is simply connected) then this case does not occur.\n**Remark 4.** In four and higher dimensions, things look much worse; consider for instance the product of a shrinking round with the trivial plane . This is a -solution but has a two-dimensional large-scale structure, and so there is no obvious way to remove singularities of this shape by surgery. It may be that in order to have analogues of Perelman’s theory in higher dimensions one has to make much stronger topological or geometric assumptions on the manifold. Note however that four-dimensional Ricci flows with surgery were already considered by Hamilton (with a rather different definition of surgery, however).\nThe classification theorem lets one understand the geometry of neighbourhoods of any given point in a -solution. Let us make the following imprecise definitions (which, again, will be made precise in later lectures):\n\n\nDefinition 2.(Canonical neighbourhoods, informal version) Let (M,g) be a complete connected 3-manifold, let x be a point in M, and let U be an open neighbourhood of x. We normalise the scalar curvature at x to be 1.\nWe say that U is an\n-\n-neckif it is close (in a smooth topology) to a round cylinder , with x well in the middle of of this cylinder;We say that U is a\n-\nC-componentif U is diffeomorphic to or (in particular, it must be all of M) with sectional curvatures bounded above and below by positive constants, and with diameter comparable to 1.We say that U is\n-\n-roundif it is close (in a smooth topology) to a round sphere or spherical space form (i.e. it is close to a constant curvature manifold).We say that U is a\n-\n-capif it consists of an -neck together with a cap at one end, where the cap is homeomorphic to either an open 3-ball or a punctured and obeys similar bounds as a C-component, and that x is contained inside the cap. (For technical reasons one also needs some derivative bounds on curvature, but we omit them here.)We say that U is a\n-\ncanonical neighbourhoodof x if it is one of the above four types.When the scalar curvature is some other positive number than 1, we can generalise the above definition by rescaling the metric to have curvature 1.\n\nUsing Theorem 1 (and defining all terms precisely), one can easily show the following important statement:\n\nCorollary 1(Canonical neighbourhood theorem for -solitons, informal version) Every point in a 3-dimensional -solution that does not contain an embedded copy of with trivial normal bundle is contained in a canonical neighbourhood.\nThe next few lectures will be devoted to establishing precise versions of Theorem 1, Definition 2, and Corollary 1.\n\n— High curvature regions of Ricci flows —\n\nCorollary 1 is an assertion about -solutions only, but it implies an important property about more general Ricci flows:\n\nTheorem 2.(Canonical neighbourhood for Ricci flows, informal version) Let be a Ricci flow of compact 3-manifolds on a time interval , without any embedded copy of with trivial normal bundle. Then every point with sufficiently large scalar curvature is contained in a canonical neighbourhood.\n(Actually, as with many other components of this proof, we actually need a generalisation of this result for Ricci flow with surgery, but we will address this (non-trivial) complication later.)\n\nThe importance of this theorem lies in the fact that all the singular regions that need surgery will have large scalar curvature, and Theorem 2 provides the crucial topological and geometric control in order to perform surgery on these regions. (This is a significant oversimplification, as one has to also study certain “horns” that appear at the singular time in order to find a particularly good place to perform surgery, but we will postpone discussion of this major additional issue later in this course.)\n\nTheorem 2 is deduced from Corollary 1 and a significant number of additional arguments. The strategy is to use a compactness-and-contradiction argument. As a very crude first approximation, the proof goes as follows:\n\nSuppose for contradiction that Theorem 2 failed. Then one could find a sequence of points with which were not contained in canonical neighbourhoods.\n-\nM, being compact, has finitely many components; by restricting attention to a subsequence of points if necessary, we can take M to be connected.\n-\nOn any compact time interval , the scalar curvature is necessarily bounded, and thus . As a consequence, if we define the rescaled Ricci flows , where is the natural length scale associated to the scalar curvature at , then these flows will become increasingly ancient. Note that in the limit (which we will not define rigorously yet, but think of a pointed Gromov-Hausdorff limit for now), the increasingly large manifolds may cease to be compact, but will remain complete.\n-\nBecause of the Hamilton-Ivey pinching phenomenon (Theorem 1 from Lecture 3), we expect the rescaled flows to have non-negative Ricci curvature in the limit (and hence non-negative Riemann curvature also, as we are in three dimensions).\n-\nIf we can pick the points suitably (so that the scalar curvature is larger than or comparable to the scalar curvatures at other nearby points), then we should be able to ensure that the rescaled flows have bounded curvature in the limit.\n-\nSince -noncollapsing is invariant under rescaling, the non-collapsing theorem (Theorem 2 of Lecture 7) should ensure that the rescaled flows remain -noncollapsed in the limit.\n-\nSince the rescaled scalar curvature at the base point of is equal to 1 by construction, any limiting flow will be non-flat.\n-\nVarious compactness theorems (of Gromov, Hamilton, and Perelman) exploiting the non-collapsed, bounded curvature, and parabolic nature of the rescaled Ricci flows now allows one to extract a limiting flow . This limit is initially in a fairly weak sense, but one can use parabolic theory to upgrade the convergence to quite a strong (and smooth) convergence. In particular, the limit of the Ricci flows will remain a Ricci flow.\n-\nApplying 2-8, we see that the limiting flow is a -solution.\n-\nApplying Corollary 1, we conclude that every point in the limiting flow lies inside a canonical neighbourhood. Using the strong nature of the convergence (and the scale-invariant nature of canonical neighbourhoods), we deduce that the points also lie in canonical neighbourhoods for sufficiently large n, a contradiction.\n-\nThere are some non-trivial technical difficulties in executing the above scheme, especially in Step 5 and Step 8. Step 8 will require some compactness theorems for -solutions which we will deduce in later lectures. For Step 5, the problem is that the points that we are trying to place inside canonical neighbourhoods have large curvature, but they may be adjacent to other points of significantly higher curvature, so that the limiting flow ends up having unbounded curvature. To get around this, Perelman established Theorem 2 by a downwards induction argument on the curvature, first establishing the result for extremely high curvature, then for slightly less extreme curvature, and so forth. The point is that with such an induction hypothesis, any potentially bad adjacent points of really high curvature will be safely tucked away in a canonical neighbourhood of high curvature, which in turn is connected to another canonical neighbourhood of high curvature, and so forth; some basic topological and geometric analysis then eventually lets us conclude that this bad point must in fact be quite far from the base point (much further away than the natural length scale , in particular), so that it does not show up in the limiting flow . We will discuss these issues in more detail in later lectures.\n\n— Benchmarks in controlling -solutions —\n\nAs mentioned earlier, the next few lectures will be focused on controlling -solutions. It turns out that the various properties in Definition 1 interact very well with each other, and give remarkably precise control on these solutions. In this section we state (without proofs) some of the results we will establish concerning such solutions.\n\n\nProposition 1.(Consequences of Hamilton’s Harnack inequality) Let be a -solution. Then is a non-decreasing function of time. Furthermore, for any , we have the pointwise inequalities(1)\n\nand\n\n(2)\n\non , where of course is the backwards time variable.\n\nThese inequalities follow from an important Harnack inequality of Hamilton (also related to earlier work of Li and Yau) that we will discuss in the next lecture. These results rely primarily on the ancient and non-negatively curved nature of -solutions, as well as the Ricci flow equation of course.\n\nNow one can handle the two-dimensional case:\n\nProposition 2.(Classification of 2-dimensional -solutions) The only two-dimensional -solutions are the round shrinking 2-spheres.\nThis proposition relies on first studying a certain asymptotic limit of the -solution, known as the asymptotic soliton, to be defined later. One shows that this asymptotic limit is a round shrinking 2-sphere, which implies that the original -solution is asymptotically a round shrinking 2-sphere. One can then invoke Hamilton’s rounding theorem to finish the claim.\n\nTurning now to three dimensions, the first important result that the curvature R decays slower at infinity than what scaling naively predicts.\n\nProposition 3.(Asymptotic curvature) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .\nThe proof of Proposition 3 is based on another compactness-and-contradiction argument which also heavily exploits some splitting theorems in Riemannian geometry, as well as the soul theorem.\n\nThe increasing curvature at infinity can be used to show that the volume does not grow as fast at infinity as scaling predicts:\n\n\nProposition 4.(Asymptotic volume collapse) Let be a 3-dimensional solution which is not compact. Then for any time and any base point , we have .\nNote that Proposition 4 does not contradict the non-collapsed nature of the flow, since one does not expect bounded normalised curvature at extremely large scales. Proposition 4 morally follows from Bishop-Gromov comparison geometry theory, but its proof in fact uses yet another compactness-and-contradiction argument combined with splitting theory.\n\nAn important variant of Proposition 4 and Proposition 3 (and yet another compactness-and-contradiction argument) states that on any ball at time zero on which the volume is large (e.g. larger than for some ), one has bounded normalised curvature, thus on this ball. This fact helps us deduce\n\nTheorem 3.(Perelman compactness theorem, informal version) The space of all pointed -solutions (allowing to range over the positive real numbers) is compact (in a suitable topology) after normalising the scalar curvature at the base point to be 1.\nOne corollary of this compactness is that there is in fact a universal such that every -solution is a -solution. (Indeed, the proof of this universality is one of the key steps in the proof of the above theorem.) This theorem is proven by establishing some uniform curvature bounds on -solutions which come from the previous volume analysis.\n\nThe proof of Theorem 1 (and thus Corollary 1) follows from this compactness once one can classify the asymptotic solitons mentioned earlier. This task in turn requires many of the techniques already mentioned, together with some variational analysis of the gradient curves of the potential function f that controls the geometry of the soliton.\n" +} diff --git a/docs/mineru_xunlianying/table.json b/docs/mineru_xunlianying/table.json new file mode 100644 index 0000000..4fd2e14 --- /dev/null +++ b/docs/mineru_xunlianying/table.json @@ -0,0 +1,7 @@ +{ + "id": "table_01", + "url": "https://nauticalstudies.org/category/psychico/", + "html": "\n\n\n\n\n\npsychico – Institute of International Nautical Studies\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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Category: psychico

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Κολλεγιο Αθηνων-Ψυχικου – Ναυτιλιακά Προγραμματα Ενηλικων

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Το Κολλέγιο Ψυχικού-Αθηνών πρωτοπορεί και δημιουργεί 15 ναυτιλιακά προγράμματα Ενηλίκων για επαγγελματίες που ενδιαφέρονται να πιστοποιηθούν στον χώρο της Ναυτιλίας. Τα προγράμματα προσφέρονται τόσο εξ αποστάσεως όσο και στο campus στο Π.Ψυχικό και παρέχουν πιστοποίηση στους εξής τομείς:

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MARINE (Ship Orientated)MARITIME (Shore Orientated)BUSINESS SKILLS
Regulatory Framework of ShippingShipping BusinessCommunication & Resilience
Ship OperationsTransportation LawMaritime Leadership
Vetting InspectionsPort & Terminal ManagementProject Management
Ship Design & TechnologyLogistics & Supply Chain ManagementTrain the Trainer
Safety & Security CultureShip CharteringBusiness Policy & Innovation
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Κάθε πρόγραμμα προσφέρεται σε 4 συναντήσεις στο campus του Ψυχικού ή εξ αποστάσεως, με δίδακτρα 150 ευρώ. Δείτε το Πρόγραμμα μαθημάτων

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Επίσης προσφέρονται όλα τα μαθήματα ως Ολοκληρωμένο Πρόγραμμα Ειδίκευσης στις Ναυτιλιακές Σπουδές με δίδακτρα 1050 ευρώ.

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Για αιτήσεις συμμετοχής έως 10 Οκτωβρίου ΠΑΤΗΣΤΕ ΕΔΩ

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\n\n\n\n", + "m_content": "Το Κολλέγιο Ψυχικού-Αθηνών πρωτοπορεί και δημιουργεί 15 ναυτιλιακά προγράμματα Ενηλίκων για επαγγελματίες που ενδιαφέρονται να πιστοποιηθούν στον χώρο της Ναυτιλίας. Τα προγράμματα προσφέρονται τόσο εξ αποστάσεως όσο και στο campus στο Π.Ψυχικό και παρέχουν πιστοποίηση στους εξής τομείς:\n\n| MARINE (Ship Orientated) | MARITIME (Shore Orientated) | BUSINESS SKILLS |\n|---|---|---|\n| Regulatory Framework of Shipping | Shipping Business | Communication & Resilience |\n| Ship Operations | Transportation Law | Maritime Leadership |\n| Vetting Inspections | Port & Terminal Management | Project Management |\n| Ship Design & Technology | Logistics & Supply Chain Management | Train the Trainer |\n| Safety & Security Culture | Ship Chartering | Business Policy & Innovation |\n\nΚάθε πρόγραμμα προσφέρεται σε 4 συναντήσεις στο campus του Ψυχικού ή εξ αποστάσεως, με δίδακτρα 150 ευρώ. Δείτε το Πρόγραμμα μαθημάτων\n\nΕπίσης προσφέρονται όλα τα μαθήματα ως Ολοκληρωμένο Πρόγραμμα Ειδίκευσης στις Ναυτιλιακές Σπουδές με δίδακτρα 1050 ευρώ.\n\nΓια αιτήσεις συμμετοχής έως 10 Οκτωβρίου ΠΑΤΗΣΤΕ ΕΔΩ", + "t_content": "Το Κολλέγιο Ψυχικού-Αθηνών πρωτοπορεί και δημιουργεί 15 ναυτιλιακά προγράμματα Ενηλίκων για επαγγελματίες που ενδιαφέρονται να πιστοποιηθούν στον χώρο της Ναυτιλίας. Τα προγράμματα προσφέρονται τόσο εξ αποστάσεως όσο και στο campus στο Π.Ψυχικό και παρέχουν πιστοποίηση στους εξής τομείς:\n\n**MARINE (Ship Orientated)** |\n**MARITIME (Shore Orientated)** |\n**BUSINESS SKILLS** |\n| Regulatory Framework of Shipping |\nShipping Business |\nCommunication & Resilience |\n| Ship Operations |\nTransportation Law |\nMaritime Leadership |\n| Vetting Inspections |\nPort & Terminal Management |\nProject Management |\n| Ship Design & Technology |\nLogistics & Supply Chain Management |\nTrain the Trainer |\n| Safety & Security Culture |\nShip Chartering |\nBusiness Policy & Innovation |\n\nΚάθε πρόγραμμα προσφέρεται σε 4 συναντήσεις στο campus του Ψυχικού ή εξ αποστάσεως, με δίδακτρα 150 ευρώ. Δείτε το Πρόγραμμα μαθημάτων\n\nΕπίσης προσφέρονται όλα τα μαθήματα ως Ολοκληρωμένο Πρόγραμμα Ειδίκευσης στις Ναυτιλιακές Σπουδές με δίδακτρα 1050 ευρώ.\n\nΓια αιτήσεις συμμετοχής έως 10 Οκτωβρίου ΠΑΤΗΣΤΕ ΕΔΩ\n\n\n### Like this:\n\nLike Loading..." +}