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 qualitative features of the inequality which 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.
Rezensionen / Stimmen
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
Reihe
Sprache
Verlagsort
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Maße
Höhe: 235 mm
Breite: 157 mm
Dicke: 21 mm
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ISBN-13
978-0-521-80267-3 (9780521802673)
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Schweitzer Klassifikation
Autor*in
City University of New York
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.
