Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives

by Isaac Chavel

Overview

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!

Product Details

ISBN-13:	9780521802673
Publisher:	Cambridge University Press
Publication date:	07/23/2001
Series:	Cambridge Tracts in Mathematics , #145
Pages:	282
Product dimensions:	5.98(w) x 9.02(h) x 0.75(d)

Table of Contents

Part I. Introduction: 1. The isoperimetric problem; 2. The isoperimetric inequality in the plane; 3. Preliminaries; 4. Bibliographic notes; Part II. Differential Geometric Methods: 1. The C2 uniqueness theory; 2. The C1 isoperimetric inequality; 3. Bibliographic notes; Part III. Minkowski Area and Perimeter: 1. The Hausdorff metric on compacta; 2. Minkowski area and Steiner symmetrization; 3. Application: the Faber-Krahn inequality; 4. Perimeter; 5. Bibliographic notes; Part IV. Hausdorff Measure and Perimeter: 1. Hausdorff measure; 2. The area formula for Lipschitz maps; 3. Bibliographic notes; Part V. Isoperimetric Constants: 1. Riemannian geometric preliminaries; 2. Isoperimetric constants; 3. Discretizations and isoperimetric inequalities; 4. Bibliographic notes; Part VI. Analytic Isoperimetric Inequalities: 1. L2-Sobolev inequalities; 2. The compact case; 3. Faber-Kahn inequalities; 4. The Federer-Fleming theorem: the discrete case; 5. Sobolev inequalities and discretizations; 6. Bibliographic notes; Part VII. Laplace and Heat Operators: 1. Self-adjoint operators and their semigroups; 2. The Laplacian; 3. The heat equation and its kernels; 4. The action of the heat semigroup; 5. Simplest examples; 6. Bibliographic notes; Part VIII. Large-Time Heat Diffusion: 1. The main problem; 2. The Nash approach; 3. The Varopoulos approach; 4. Coulhon's modified Sobolev inequality; 5. The denoument: geometric applications; 6. Epilogue: the Faber–Kahn method; 7. Bibliographic notes; Bibliography.

Editorial 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

From the Publisher

As part of a series whose volumes pursue a narrow path in a broad subject, this introductory tract follows two venues of the isoperimetric inequality which span the evolution from the Euclidian space of classical differential geometry to the inequalities of modern measure theory. The treatment of Riemann manifolds is guided by the dichotomy between their local Euclidian character and global geometric properties, and covers discretization techniques and application to large time heat diffusion problems. Chapters include proofs. Chavel (mathematics, City U. of New York) is the author of and . Annotation c. Book News, Inc., Portland, OR (booknews.com)