Titlebook: Mean Curvature Flow and Isoperimetric Inequalities; Manuel Ritoré,Carlo Sinestrari Textbook 2010 Birkh?user Basel 2010 Mean curvature.Mini

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Singular behaviour of convex surfacesIn the next two sections we shall see some results showing that, roughly speaking, the convexity properties of a surface evolving by mean curvature flow improve when a singularity is formed. We begin with the case of convex surfaces.

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Mean curvature flow with surgeriesIn this section we describe the mean curvature flow with surgeries which has been defined in [48] for two-convex surfaces of dimension . ≥ 3. Such a construction is inspired by the one which was introduced by Hamilton [37] for the Ricci flow and which enabled Perelman [56] to prove the geometrization conjecture for three-dimensional manifolds.

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Higher dimensionsUnlike surfaces, the use of the mean curvature flow in higher dimensions to prove isoperimetric inequalities is severely limited by the possibility of development of singularities. The reader is referred to Sinestrari’s course in this volume [113] for an updated discussion on these topics.

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Mean Curvature Flow and Isoperimetric Inequalities978-3-0346-0213-6Series ISSN 2297-0304 Series E-ISSN 2297-0312

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