Uniqueness results for minimal surfaces and constant mean curvature surfaces in Riemannian manifolds

Abstract

A classical theorem due to Alexandrov asserts that the only embedded surfaces in Euclidean space with constant mean curvature are the round spheres. Alexandrov's theorem is closely related to the classical isoperimetric inequality, which asserts that round spheres have minimal surface area among all surfaces that enclose a given amount of volume. It is an interesting question whether Alexandrov's theorem can be generalized to surfaces of constant mean curvatures in a Riemannian manifold. In particular, if the ambient Riemannian manifold is rotationally symmetric, can one conclude that every embedded surface of constant mean curvature is a sphere of symmetry? This question turns out to have interesting links with general relativity. We also discuss minimal surfaces in the sphere $S^3$ and our recent solution of Lawson's conjecture.