UNIQUENESS OF IMMERSED SPHERES IN THREE-MANIFOLDS

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 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 R-3. For this, we first extend Hopf's famous classification of constant mean curvature spheres in R-3 to the general situation of surfaces modeled by elliptic PDEs in arbitrary three-manifolds that admit families of candidate examples.