Mean Curvature Flow and Isoperimetric Inequalities

Name: Mean Curvature Flow and Isoperimetric Inequalities
Price: 40.9 USD
Availability: InStock
ISBN: 9783034602129

ISBN-10: 303460212X

ISBN-13: 9783034602129

Edition: 2010

Authors: Manuel Ritorï¿½, Carlo Sinestrari, Vicente Miquel, Joan Porti

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Description:

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…

Book details

List price: $26.99
Copyright year: 2010
Publisher: Birkhauser Verlag GmbH
Publication date: 10/19/2009
Binding: Paperback
Pages: 114
Size: 6.50" wide x 9.25" long x 0.50" tall
Weight: 0.682
Language: English



Foreword



Formation of Singularities in the Mean Curvature Flow




Introduction



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


Bibliography



Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry



Preface



The classical isoperimetric inequality in Euclidean space



Introduction



Preliminaries



Area and volume



Variational formulas



The isoperimetric profile



Isoperimetric regions



The isoperimetric inequality in Euclidean space



A calibration argument



Schwarz symmetrization



Another proof of the isoperimetric inequality



Surfaces



Introduction



The Gauss-Bonnet Theorem



Curve shortening flow



Basic results



The avoidance principle



Applications of curve shortening flow to isoperimetric inequalities



Planes



Spheres



Higher dimensions



Introduction



H<sup>k</sup>-ftows and isoperimetric inequalities



Estimates on the Willmore functional and isoperimetric inequalities



Euclidean spaces



3-dimensional Hadamard manifolds



Singularities in the volume-preserving mean curvature flow



Some applications to hyperbolic geometry



Introduction



Bounds on the Heegaard genus of a hyperbolic manifold



The isoperimetric profile for small volumes


Bibliography