Mean Curvature Flow and Isoperimetric Inequalities

• Main Authors: Ritoré, Manuel (Author), Sinestrari, Carlo (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Basel : Birkhäuser Basel, 2010.
• Series: Advanced Courses in Mathematics — CRM Barcelona, Centre de Recerca Matemàtica
• Subjects: Mathematics.; Global analysis (Mathematics).; Manifolds (Mathematics).; Differential geometry.; Differential Geometry.; Global Analysis and Analysis on Manifolds.

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 pro...

Table of Contents:

• 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.