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 $\mathbb{R}^3$.

中文翻译：

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三流形中浸入球的唯一性

令 $\mathcal{A}$ 是三流形 $M$ 中的一类浸入曲面，并假设 $\mathcal{A}$ 由每个切平面上的椭圆 PDE 建模。在本文中，我们在候选表面的传递族 $\mathcal{S}\subset\mathcal{A} 存在的唯一温和假设下解决了类 $\mathcal{A}$ 的所谓 Hopf 唯一性问题$. 具体来说，我们证明了类 $\mathcal{A}$ 中任何属 0 的紧凑浸没表面都是候选球体。该定理统一和扩展了不同上下文的许多先前的唯一性结果。作为应用，我们肯定了 AD Alexandrov 1956 年关于 $\mathbb{R}^3$ 中具有规定曲率的浸入球的唯一性的猜想。