  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 $\mathbb{R}^3$.
