Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives

Couverture
Cambridge University Press, 23 juil. 2001 - 268 pages
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!
 

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Table des matières

Differential Geometric Methods
30
Minkowski Area and Perimeter
52
Hausdorff Measure and Perimeter
100
Isoperimetric Constants
117
Analytic Isoperimetric Inequalities
157
Laplace and Heat Operators
185
Large Time Heat Diffusion
225
Bibliography
255
Author Index
263
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