Mean Curvature Flow and Isoperimetric Inequalities

Geometric flows have many applications in physics and geometry, while isoperimetric inequalities can help in treating several aspects of convergence of these flows. Based on a series of lectures given by the authors, the material here deals with both subjects.

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.

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

Contenu
Formation of Singularities in the Mean Curvature Flow.- Geometry of hypersurfaces.- Examples.- Local existence and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Huisken's monotonicity formula.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.- Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry.- The classical isoperimetric inequality in Euclidean space.- Surfaces.- Higher dimensions.- Some applications to hyperbolic geometry.