by Shu-yu Hsu. / "July 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 40-42). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Mean curvature flow evolves isometrically immersed base Riemannian manifolds M in the direction of their mean curvature in an ambient manifold M. We consider the classical solutions to the mean curvature flow. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete noncompact isometrically immersed hypersurfaces M (uniformly local lipschitz) in Euclidean space, the short time existence was established by Ecker and Huisken in [10]. The short time existence and the uniqueness of the solutions to the mean curvature flow of complete isometrically immersed manifolds of arbitrary codimensions in the Euclidean space are still open questions. In this thesis, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. More precisely, let (M, g) be a complete Riemannian manifold of dimension n such that the curvature and its covariant derivatives up to order 2 are bounded and the injectivity radius is bounded from below by a positive constant, we prove that the solution of the mean curvature flow with bounded second fundamental form on an isometrically immersed manifold M (may be of high codimension) is unique. In the second part of the thesis, inspired by the Ricci flow, we prove the pseudolocality theorem of mean curvature flow. As a consequence, we obtain the strong uniqueness theorem, which removes the boundedness assumption of the second fundamental form of the solution in the uniqueness theorem (only assume the second fundamental form of the initial submanifold is bounded). / Yin, Le. / "July 2007." / Adviser: Leung Nai-Chung. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 65-68). / 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.

Thesis (Ph. D.)--Lehigh University, 2006. / Includes vita. Includes bibliographical references (leaves 72-76).

Wo, Weifeng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 99-102). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Guan-xin Li. / "January 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 223-225). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Ma, Man Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 75-76). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Basic notions in Riemannian geometry --- p.10 / Chapter 2.1 --- Basic manifold theory --- p.11 / Chapter 2.2 --- "Connection, curvature" --- p.19 / Chapter 2.3 --- Submanifold theory --- p.29 / Chapter 3 --- Basic facts in symplectic and complex geometry --- p.33 / Chapter 3.1 --- "Symplectic manifolds, Lagrangian submanifolds" --- p.34 / Chapter 3.2 --- Kahler and Calabi-Yau manifolds --- p.39 / Chapter 3.3 --- Calibration --- p.49 / Chapter 4 --- Mean curvature flow --- p.52 / Chapter 4.1 --- Basic equations in Lagrangian immersions --- p.53 / Chapter 4.2 --- Evolution equation for --- p.57 / Chapter 4.3 --- Evolution equations for H and θ --- p.62 / Chapter 5 --- Lagrangian angle of a foliation --- p.67 / Chapter 5.1 --- "Proof of equation (5.1), (5.2)" --- p.68 / Chapter 5.2 --- Main theorem --- p.70 / Chapter 5.3 --- Examples of invariant solution --- p.73 / Bibliography --- p.75

by Yiu Chun Chit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 81-82). / Chapter 1 --- Introduction --- p.7 / Chapter 1.1 --- Riemannian Manifolds --- p.7 / Chapter 1.1.1 --- Completeness --- p.8 / Chapter 1.1.2 --- Curvature tensor --- p.9 / Chapter 1.1.3 --- Holonomy --- p.11 / Chapter 1.2 --- Simply-connected Manifold of Nonpositive Sectional Curvature --- p.11 / Chapter 1.2.1 --- Topological structure --- p.12 / Chapter 1.2.2 --- Basic geometric properties --- p.13 / Chapter 1.2.3 --- Examples of nonpositively curved manifold --- p.20 / Chapter 1.2.4 --- Convexity properties --- p.23 / Chapter 1.2.5 --- Points at infinity for M --- p.27 / Chapter 2 --- Symmetric Spaces --- p.36 / Chapter 2.1 --- Symmetric Spaces of Noncompact Type --- p.36 / Chapter 2.1.1 --- Symmetric diffeomorphisms --- p.36 / Chapter 2.1.2 --- Transvections in I(M) --- p.38 / Chapter 2.1.3 --- Symmetric spaces as coset manifolds G/K --- p.39 / Chapter 2.1.4 --- Metric on TpM and the adjoint representation of Lie group --- p.41 / Chapter 2.1.5 --- Curvature tensor of M --- p.43 / Chapter 2.1.6 --- Killing form and classification of symmetric spaces --- p.44 / Chapter 2.1.7 --- Holonomy of M at p --- p.44 / Chapter 2.1.8 --- Rank of a symmetric space M --- p.45 / Chapter 2.1.9 --- Regular and singular points at infinity --- p.46 / Chapter 2.2 --- "The Symmetric Space Mn = SL(n,R)/SO(n,R)" --- p.46 / Chapter 2.2.1 --- Metric on TIMn --- p.47 / Chapter 2.2.2 --- Geodesic and symmetries of Mn --- p.48 / Chapter 2.2.3 --- Curvature of Mn --- p.48 / Chapter 2.2.4 --- Rank and flats in Mn --- p.49 / Chapter 2.2.5 --- Holonomy of Mn at I --- p.49 / Chapter 2.2.6 --- Eigenvalue-flag pair for a point in Mn(∞ ) --- p.50 / Chapter 2.2.7 --- Action of I0(Mn) on Mn(∞ ) --- p.52 / Chapter 2.2.8 --- Flags in opposition --- p.53 / Chapter 2.2.9 --- Joining points at infinity --- p.53 / Chapter 3 --- Group Action --- p.56 / Chapter 3.1 --- Action of Isometries on M(oo) --- p.56 / Chapter 3.1.1 --- Fundamental group as a group of isometries --- p.56 / Chapter 3.1.2 --- Lattices --- p.58 / Chapter 3.1.3 --- Duality condition --- p.59 / Chapter 3.1.4 --- Geodesic flows --- p.61 / Chapter 3.2 --- Action of Geodesic Symmetries on M(oo) --- p.62 / Chapter 3.3 --- Rank --- p.66 / Chapter 3.3.1 --- Rank of a manifold of nonpositive curvature --- p.66 / Chapter 3.3.2 --- Rank of the fundamental group --- p.68 / Chapter 3.4 --- Rigidity Theorems of Locally Symmetric Spaces --- p.69

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