Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives

This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject. It discusses inequalities in Euclidean and Riemannian geometry, methods of classical differential geometry and elementary modern geometric measure theory, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on Riemannian manifolds.

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.