Ricci flow and a sphere theorem.

• Other Authors: Huang, Shaochuang.
• Format: Others
• Language: English; Chinese
• Published: 2013
• Subjects: Ricci flow; Sphere
• Online Access: http://library.cuhk.edu.hk/record=b5549311; http://repository.lib.cuhk.edu.hk/en/item/cuhk-328767

在這篇畢業論文里，我們將闡述微分球面定理。這是由Brelldle和孫理察在2007年用里奇流所證明的。球面定理的研究在微分幾何中有很長的歷史。人們研究一種稱為δ-pillched的截面曲率條件(δ在0和1 之間) ，使得一個緊致單連通的黎曼流形滿足這個曲率條件就會同胚或者微分同胚于一個球面。里奇流是由哈密爾頓在1982年所引進的，當時，他證明了任意一個閉單連通三維黎曼流形只要滿足正的里奇曲率條件就微分同胚于一個球面。 === 在這里，我們會闡述關於里奇流的一些基本結果，包括曲率在里奇流下變化的方式、短時間存在性、唯一性以及曲率和一般張量的高階導數估計。同時，我們也會講述一般的在里奇流下的極大值原理以及收斂性定理。非負迷向曲率條件能夠在里奇流下得到保持的結果也會被詳細證明。最後，我們通過由Böhm 和Wilkillg斤引進的一簇不變圓錐來完成整個微分球面定涅的證明。 === In this thesis, we present the proof of the differentiable sphere theorem which was proved by Brendle and Schoen in 2007 using the Ricci flow. The sphere theorem in differential geometry has a long history. People studied the δ-pinched sectional curvature condition showing that for various cases of δ ∈ (0, 1), any simply-connected, compact Riemannian manifold with this curvature condition is homeomorphic or diffeomorphic to a sphere. The Ricci flow was introduced by Hamilton in his seminal paper in 1982 which proved that every simply-connected, compact three-manifold with positive Ricci curvature is diffeomorphic to a sphere. === In this text, we present some background materials for the Ricci flow, including curvature evolution under the Ricci flow, short-time existence, uniqueness and higher derivatives estimate for curvature and tensor. We also focus on the maximum principle and convergence criterion for the Ricci flow. The fact thatnonnegative isotropic curvature is preserved under the Ricci flow will be showed.Finally, we complete the proof of the differentiable sphere theorem using a family of invariant cones which was constructed by Böhm and Wilking. === Detailed summary in vernacular field only. === Detailed summary in vernacular field only. === Huang, Shaochuang. === Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. === Includes bibliographical references (leaves 96-98). === Abstracts also in Chinese. === Abstract --- p.i === Acknowledgements --- p.iii === Chapter 1 --- Introduction --- p.1 === Chapter 2 --- Background Materials for The Ricci Flow --- p.6 === Chapter 2.1 --- Evolution of curvature under the Ricci flow --- p.6 === Chapter 2.2 --- Short-time existence --- p.13 === Chapter 2.2.1 --- The Ricci tensor is not elliptic --- p.13 === Chapter 2.2.2 --- DeTurck’s trick --- p.17 === Chapter 2.3 --- Uniqueness --- p.20 === Chapter 2.4 --- Estimates for derivatives of curvature --- p.22 === Chapter 2.5 --- Estimates for derivatives of tensor --- p.25 === Chapter 3 --- Hamilton’s Maximum Principle for The Ricci Flow --- p.30 === Chapter 4 --- Hamilton’s Convergence Criterion for The Ricci Flow --- p.39 === Chapter 5 --- Nonnegative Isotropic Curvature --- p.54 === Chapter 5.1 --- Nonnegative isotropic curvature is preserved --- p.54 === Chapter 5.2 --- The Cone Ĉ --- p.67 === Chapter 6 --- Proof of The Differentiable Sphere Theorem --- p.73 === Chapter 6.1 --- An algebraic identity for curvature tensors --- p.73 === Chapter 6.2 --- Construction of a family of invariant cones --- p.80 === Chapter 6.3 --- Proof of The Differentiable Sphere Theorem --- p.87 === Bibliography --- p.96