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Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives - Isaac Chavel

Isoperimetric Inequalities

Differential Geometric and Analytic Perspectives

Paperback Published: 21st July 2011
ISBN: 9781107402270
Number Of Pages: 282

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This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond!

Industry Reviews

Review of the hardback: 'The presentation of the book is clear and elegant, and gives expression to the beauty of the subject. It is a great pleasure to read this book, which is a profound source text for both classical and modern methods, and which will be equally valuable to graduate students and researchers in analysis and geometry.' Bulletin of the London Mathematical Society
Review of the hardback: 'The book is very useful in two ways. First, it nicely explains the story of the classical isoperimetric inequality, a result with a big disproportion between the ease of formulation and difficulty of the proof. This second part contains deep results obtained by the author.' European Mathematical Society
Review of the hardback: '... very useful ...' EMS Newsletter
Review of the hardback: '[This book] constitues a valuable addition to the modern theory of inequalities.' Bulletin of the Belgian Mathematical Society

Introduction
The isoperimetric problem
The isoperimetric inequality in the plane
Preliminaries
Bibliographic notes
Differential Geometric Methods
The C2 uniqueness theory
The C1 isoperimetric inequality
Bibliographic notes
Minkowski Area and Perimeter
The Hausdorff metric on compacta
Minkowski area and Steiner symmetrization
Application: the Faber-Krahn inequality
Perimeter
Bibliographic notes
Hausdorff Measure and Perimeter
Hausdorff measure
The area formula for Lipschitz maps
Bibliographic notes
Isoperimetric Constants
Riemannian geometric preliminaries
Isoperimetric constants
Discretizations and isoperimetric inequalities
Bibliographic notes
Analytic Isoperimetric Inequalities
L2-Sobolev inequalities
The compact case
Faber-Kahn inequalities
The Federer-Fleming theorem: the discrete case
Sobolev inequalities and discretizations
Bibliographic notes
Laplace and Heat Operators
Self-adjoint operators and their semigroups
The Laplacian
The heat equation and its kernels
The action of the heat semigroup
Simplest examples
Bibliographic notes
Large-Time Heat Diffusion
The main problem
The Nash approach
The Varopoulos approach
Coulhon's modified Sobolev inequality
The denoument: geometric applications
Epilogue: the Faber-Kahn method
Bibliographic notes
Bibliography
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781107402270
ISBN-10: 1107402271
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 282
Published: 21st July 2011
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 1.91
Weight (kg): 0.43

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