Uniqueness of immersed spheres in three-manifolds

TL;DR

This paper proves a general uniqueness theorem for immersed spheres in three-manifolds, extending previous results and confirming a 1956 conjecture on the uniqueness of spheres with prescribed curvature.

Contribution

It establishes a unified framework for the uniqueness of immersed spheres in three-manifolds under elliptic PDE conditions, solving a longstanding conjecture.

Findings

Any genus-zero immersed surface in the class is a candidate sphere.

Unified and extended previous uniqueness results.

Confirmed Alexandrov's conjecture on sphere uniqueness.

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 $\mathbb{R}^3$.