Let $\mathcal{ A}$ be a class of immersed surfaces in a three-manifold $M$, and assume that $\mathcal{ A}$ is modeled by an elliptic PDE over each tangent plane. In this paper we solve the so-called Hopf uniqueness problem for the class $\mathcal{ A}$ under the only mild assumption of the existence of a transitive family of candidate surfaces $\mathcal{ S}\subset \mathcal{ A}$. Specifically, we prove that any compact immersed surface of genus zero in the class $\mathcal{ A}$ is a candidate sphere. This theorem unifies and extends many previous uniqueness results of different contexts. As an application, we settle in the affirmative a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres with prescribed curvatures in $\mathbb{ R}^3$.
Categories: math.DG math.AP