A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov

TL;DR

This paper proves a uniqueness theorem for immersed spheres with prescribed non-constant mean curvature in homogeneous three-manifolds, extending classical results and confirming a conjecture by A.D. Alexandrov.

Contribution

It establishes a new uniqueness theorem for spheres with prescribed non-constant mean curvature, generalizing classical results and confirming a longstanding conjecture.

Findings

Proved a uniqueness theorem for immersed spheres with prescribed mean curvature.

Extended Hopf's classical theorem to non-constant mean curvature cases.

Confirmed A.D. Alexandrov's conjecture for prescribed Weingarten curvature in R3.

Abstract

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.