Uniqueness of immersed spheres in three-manifolds

Abstract

In this paper we solve two open problems of classical surface theory; we give an affirmative answer to a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres in $\mathbb{R}^3$ that satisfy a general elliptic prescribed curvature equation, and we prove as a consequence that round spheres are the only elliptic Weingarten spheres immersed in $\mathbb{R}^3$. For this, we first extend Hopf’s famous classification of constant mean curvature spheres in $\mathbb{R}^3$ to the general situation of surfaces modeled by elliptic PDEs in arbitrary three-manifolds that admit families of candidate examples.