Two-point functions and constant mean curvature surfaces in ℝ3

McGrath, Peter; Meekins, Everett

doi:10.2140/involve.2023.16.467

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Two-point functions and constant mean curvature surfaces in $\mathbb{R}^3$

Peter McGrath and Everett Meekins

Vol. 16 (2023), No. 3, 467–482

DOI: 10.2140/involve.2023.16.467

Abstract

Using a two-point maximum principle technique inspired by work of Brendle and Andrews and Li, we give a new proof of a special case of Alexandrov’s theorem: there are no embedded constant mean curvature tori in Euclidean three-space.

Keywords

constant mean curvature, Hopf, Alexandrov, differential geometry, geometric analysis, surface

Mathematical Subject Classification

Primary: 53A10

Secondary: 35J93

Milestones

Received: 2 February 2022

Revised: 21 June 2022

Accepted: 25 June 2022

Published: 10 August 2023

Communicated by Michael Dorff

Authors

Peter McGrath
Department of Mathematics North Carolina State University Raleigh, NC United States

Everett Meekins
North Carolina State University Raleigh, NC United States

Vol. 16, No. 3, 2023