Mean Curvature Flow and Isoperimetric Inequalities (Advanced Courses in Mathematics - CRM Barcelona)

De (autor) Editat de Vicente Miquel De (autor) Editat de Joan Porti
Notă GoodReads:
en Limba Engleză Paperback – 19 Oct 2009
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.
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ISBN-13: 9783034602129
ISBN-10: 303460212X
Pagini: 114
Dimensiuni: 170 x 244 x 10 mm
Greutate: 0.23 kg
Ediția: 2010
Editura: Birkhäuser Basel
Colecția Birkhäuser
Seria Advanced Courses in Mathematics - CRM Barcelona

Locul publicării: Basel, Switzerland

Public țintă



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.


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.