Overview

Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.

Full Product Details

Author:   Manuel Ritoré ,  Vicente Miquel ,  Carlo Sinestrari ,  Joan Porti
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2010 ed.
Dimensions:   Width: 17.00cm , Height: 1.00cm , Length: 24.40cm
Weight:   0.454kg
ISBN:  

9783034602129

ISBN 10:   303460212
Pages:   114
Publication Date:   19 October 2009
Audience:   College/higher education ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

The classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.- $H^k$-flows and isoperimetric inequalities.- Estimates on the Willmore functional and isoperimetric inequalities.- Singularities in the volume-preserving mean curvature flow.- Bounds on the Heegaard genus of a hyperbolic manifold.- The isoperimetric profile for small volumes.- Local existence of flows driven by the second fundamental form and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.

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