A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov

Abstract

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A. D. Alexandrov about immersed spheres of prescribed Weingarten curvature in $$\mathbb {R}^3$$ for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in $$\mathbb {R}^3$$ .