An isoperimetric comparison theorem for Schwarzschild space and other manifolds

An Isoperimetric Comparison Theorem for Schwarzschild Space and Other Manifolds

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric (n− 1)-spheres of a spherically symmetric n-manifold are isoperimetric hypersurfaces, meaning that they minimize (n − 1)-dimensional area among hypersurfaces enclosing the same n-volume. This result greatly generalizes the result of Bray (Ph.D. thes...

An isoperimetric comparison theorem

Some Isoperimetric Comparison Theorems for Convex Bodies in Riemannian Manifolds

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M, g) satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Ω. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential po...

An Isoperimetric Theorem in Plane Geometry

An isoperimetric theorem in plane geometry Alan Siegel1 COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY New York Abstract Let be a simple polygon. Let the vertices of be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in . Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal...

Comparison Theorem for Kähler Manifolds and Positivity of Spectrum

The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are CP, C, and CH. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m for a complete, m-dimensional, Kähler manifold with holomorphic bisectional curvature bo...