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.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 63-64). Also available on the World Wide Web.

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Li, Yong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 58-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- "Entropy, Pressure and Variational Principle" --- p.4 / Chapter 2.1 --- Entropy --- p.5 / Chapter 2.1.1 --- Entropy of a Partition --- p.5 / Chapter 2.1.2 --- Conditional Entropy --- p.6 / Chapter 2.1.3 --- Entropy of a Measure-Preserving Transformation --- p.7 / Chapter 2.2 --- Topological Pressure --- p.9 / Chapter 3 --- Ruelle Operator Theorem --- p.11 / Chapter 3.1 --- Ruelle Operator Theorem --- p.11 / Chapter 3.2 --- Subshifts of Finite Type --- p.26 / Chapter 4 --- Applications in Dimension Theory --- p.35 / Chapter 4.1 --- Hausdorff Dimension of Cookie-Cutter Sets --- p.35 / Chapter 4.2 --- General Case --- p.44 / Bibliography --- p.58

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