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November 2020 Uniqueness of immersed spheres in three-manifolds
José A. Gálvez, Pablo Mira
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J. Differential Geom. 116(3): 459-480 (November 2020). DOI: 10.4310/jdg/1606964415

Abstract

In this paper we solve two open problems of classical surface theory; we give an affirmative answer to a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres in $\mathbb{R}^3$ that satisfy a general elliptic prescribed curvature equation, and we prove as a consequence that round spheres are the only elliptic Weingarten spheres immersed in $\mathbb{R}^3$. For this, we first extend Hopf’s famous classification of constant mean curvature spheres in $\mathbb{R}^3$ to the general situation of surfaces modeled by elliptic PDEs in arbitrary three-manifolds that admit families of candidate examples.

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José A. Gálvez. Pablo Mira. "Uniqueness of immersed spheres in three-manifolds." J. Differential Geom. 116 (3) 459 - 480, November 2020. https://doi.org/10.4310/jdg/1606964415

Information

Received: 7 July 2017; Published: November 2020
First available in Project Euclid: 3 December 2020

zbMATH: 07282207
MathSciNet: MR4182894
Digital Object Identifier: 10.4310/jdg/1606964415

Rights: Copyright © 2020 Lehigh University

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Vol.116 • No. 3 • November 2020
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