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arXiv · 1603.07153

Uniqueness of immersed spheres in three-manifolds

Jose A. Galvez·Pablo Mira
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Abstract

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 $\mathbb{R}^3$.

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Jose A. Galvez, Pablo Mira. 2016-03-23. Uniqueness of immersed spheres in three-manifolds. https://arxiv.org/abs/1603.07153

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