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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
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