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

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 50-51). Also available in electronic version.

In the first part of this thesis, we proved a pseudo-locality theorem for a coupled Ricci flow, extending Perelman’s work on Ricci flow to the Ricci flow coupled with heat equation. By use of the reduced distance and the pseudo-locality theorem, we showed that the parabolic rescaling of a Type I coupled Ricci flow with respect to a Type I singular point converges to a non-trivial Ricci soliton. In the second part of the thesis, we prove the existence of infinitely many solutions to the Hull- Strominger system on generalized Calabi-Gray manifolds, more specifically compact non-K \"ahler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers. We also studied the behavior of the anomaly flow on the generalized Calabi-Gray manifolds, and reduced it to a scalar flow on a Riemann surface. We obtained the long-time existence and convergence after rescaling in the case when the curvature of initial metric is small.

In this thesis, we will first extend the pseudo-locality of the Ricci flow on compact Riemannian manifolds discovered by Perelman to complete noncompact Riemannian manifolds. Then, we will apply it to study the long time existence of the Kahler-Ricci flow on complete noncompact Kahler manifolds with reasonable geometric assumptions. Finally, we will give some examples of complete flat Kahler metrics on the complex projective space with some divisor deleted. / by Yu, Chengjie. / Advisers: Yau-Heng Wan; Luen-Fai Tam. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 113-116). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In 2002, Feldman, Ilmanen, and Knopf constructed the first example of a non-trivial (i.e. non-constant curvature) complete non-compact shrinking soliton, and conjectured that it models a Ricci flow singularity forming on a closed four-manifold. In this thesis, we confirm their conjecture and, as a consequence, show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature. / text

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In this thesis, we explore short time existence and uniqueness of solutions to the Ricci flow on manifolds with boundary, as well as the preservation of natural curvature positivity conditions along the flow. In chapter 2, we establish the existence and uniqueness for linear parabolic systems on vector bundles for Hölder continuous initial data. We introduce appropriate weighted parabolic Hölder spaces to study the existence and uniqueness problem. Having developed the linear theory, we apply it to establish the existence and uniqueness for the Ricci-DeTurck flow, the harmonic map heat flow, and the Ricci flow with Hölder continuous initial data in Chapter 3. In chapter 4, we discuss a general preservation result concerning the preservation of various curvature conditions during boundary deformation. Using a perturbation argument, we construct a family of metrics which interpolate between two metrics that agree on the boundary, and such family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. The results from chapters 2 through 4 will be utilized in proving the Main Theorems in chapter 5. In particular, we construct canonical solutions to the Ricci flow on manifolds with boundary from canonical solutions to the Ricci flow on closed manifolds with Hölder continuous initial data via doubling.

Let (M,g) be a quasi-Sasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain non-negativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g). Naturally associated to a quasi-Sasaki metric g is a transverse Kahler metric g^T. The transverse Kahler-Ricci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in Kahler-Ricci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting. We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasi-Sasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is eta-Einstein.

One of the most fundamental problems in classical general relativity is the measure of e↵ective mass of a pure gravitational field. The principle of equivalence prohibits a purely local measure of this mass. This thesis critically examines the most recent quasi-local measure by Wang and Yau for a maximally rotating black hole spacetime. In particular, it examines a family of spacelike 2-surfaces with constant radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which, the Wang and Yau quasi-local energy has yet to be explored. In this region, the results of this thesis indicate that the Wang and Yau quasi-local energy yields complex values and is essentially equivalent to the previously defined Brown and York quasi-local energy. However, an application of their quasi-local mass is suggested in a dynamical setting, which can potentially give new and meaningful measures. In supporting this thesis, the development of a novel adiabatic isometric mapping algorithm is included. Its purpose is to provide the isometric embedding of convex 2-surfaces with spherical topology into Euclidean 3-space necessary for completing the calculation of quasilocal energy in numerical relativity codes. The innovation of this algorithm is the guided adiabatic pull- back routine. This uses Ricci flow and Newtons method to give isometric embeddings of piecewise simplicial 2-manifolds, which allows the algorithm to provide accuracy of the edge lengths up to a user set tolerance. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection

Let (Mn, g) be a Riemannian manifold. Say K ! E ! M is a principal K-bundle with connection A. We define a natural evolution equation for the pair (g,A) combining the Ricci flow for g and the Yang-Mills flow for A which we dub Ricci Yang-Mills flow. We show that these equations are, up to di eomorphism equivalence, the gradient flow equations for a Riemannian functional on M. Associated to this energy functional is an entropy functional which is monotonically increasing in areas close to a developing singularity. This entropy functional is used to prove a non-collapsing theorem for certain solutions to Ricci Yang-Mills flow. We show that these equations, after an appropriate change of gauge, are equivalent to a strictly parabolic system, and hence prove general unique short-time existence of solutions. Furthermore we prove derivative estimates of Bernstein-Shi type. These can be used to find a complete obstruction to long-time existence, as well as to prove a compactness theorem for Ricci Yang Mills flow solutions. Our main result is a fairly general long-time existence and convergence theorem for volume-normalized solutions to Ricci Yang-Mills flow. The limiting pair (g,A) satisfies equations coupling the Einstein and Yang-Mills conditions on g and A respectively. Roughly these conditions are that the associated curvature FA must be large, and satisfy a certain “stability” condition determined by a quadratic action of FA on symmetric two-tensors.