Uniqueness of immersed spheres in three-manifolds

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Let A be a class of immersed surfaces in a three-manifold M, and assume that 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 A under the only mild assumption of the existence of a transitive family of candidate surfaces S⊂ A. Specifically, we prove that any compact immersed surface of genus zero in the class 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 R³.