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Topics in mean curvature flow of hypersurfaces Hikspoors, Samuel

Abstract

In this thesis we study the possible solutions of the mean curvature flow problem restricted to hypersurface geometries: We give a complete exposition of the theory contained in some of the articles included in the bibliography. The text is divided in three parts. The first part consist of an informal discussion on some useful knowledge in partial differential equations of parabolic type. The second part of the text is the core of the thesis: It contains a detailed exposition (with full proof of all main results) of the now classical work of Huisken and Hamilton on the MCF of compact-convex initial surfaces. The main result state the existence of a long time solution of the mean curvature flow and that this smooth solution converge to a round point. The third part is meant to be an introduction to some further developments under less restrictive initial data. In some situations the flow can generate singular points and then a smooth solution do no longer exist after a given finite time interval: We characterize in a simple way some of these singularities. An informal discussion on some important results of MCF of entire graphs then conclude our work.

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