The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds

Abstract

We prove that a region D outside a convex set C with smooth boundary in a three-dimensional Cartan-Hadamard manifold M satisfies the relative isoperimetric inequality area, with equality if and only if D is isometric to a Euclidean half ball. We also prove similar sharp inequalities when the sectional curvature of M is bounded above by a negative constant.