Overview
- Unique book which examines advances on isoperimetric problems related with geometric flows and suggests some new directions in the interplay between the two subjects.
- First book to give an introduction to the mean curvature flow with surgery starting with the basis of the existence of solutions and the careful exposition of the ideas in the analysis of singularities.
- The exposition is accessible to graduate students of mathematics with some basic knowledge of Riemannian geometry.
- Includes supplementary material: sn.pub/extras
Part of the book series: Advanced Courses in Mathematics - CRM Barcelona (ACMBIRK)
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Table of contents (15 chapters)
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Front Matter
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Formation of Singularities in the Mean Curvature Flow
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Front Matter
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Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry
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Front Matter
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Back Matter
Keywords
About this book
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.
Authors and Affiliations
Bibliographic Information
Book Title: Mean Curvature Flow and Isoperimetric Inequalities
Authors: Manuel Ritoré, Carlo Sinestrari
Series Title: Advanced Courses in Mathematics - CRM Barcelona
DOI: https://doi.org/10.1007/978-3-0346-0213-6
Publisher: Birkhäuser Basel
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Basel 2010
Softcover ISBN: 978-3-0346-0212-9Published: 19 October 2009
eBook ISBN: 978-3-0346-0213-6Published: 01 January 2010
Series ISSN: 2297-0304
Series E-ISSN: 2297-0312
Edition Number: 1
Number of Pages: VIII, 114
Topics: Differential Geometry, Global Analysis and Analysis on Manifolds