Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives [Hardcover]

This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject.Classical isoperimetric inequalities in the plane relate the area of a domain to the length of its boundary, and, in space, the volume to the area of its boundary. This advan ced introduction emphasizes the subject's variety of ideas, techniques, and applications. The author discusses inequalities in Euclidean and Riemannian geometry, methods of classical differential geometry and elementary modern geometric measure, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on Riemannian manifolds.Requiring only of a basic course in differential geometry, this book will appeal to graduate students and researchers alike in differential geometry, analysis, and related subjects.Classical isoperimetric inequalities in the plane relate the area of a domain to the length of its boundary, and, in space, the volume to the area of its boundary. This advan ced introduction emphasizes the subject's variety of ideas, techniques, and applications. The author discusses inequalities in Euclidean and Riemannian geometry, methods of classical differential geometry and elementary modern geometric measure, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on Riemannian manifolds.Requiring only of a basic course in differential geometry, this book will appeal to graduate students and researchers alike in differential geometry, analysis, and related subjects.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 Rl³$