Applications of compactness results for harmonic maps to stable constant mean curvature surfaces
type: Item
instance of: scholarly article
title: Applications of compactness results for harmonic maps to stable constant mean curvature surfaces
zbMATH Open document ID: 0889.53045
DOI: 10.1007/PL00004351
author: Manuel Ritoré
published in: Mathematische Zeitschrift
publication date: 1998-06-18T00:00:00Z
review text: We prove the compactness of the space of compact volume-preserving stable surfaces with genus \(g\geq 2\) immersed in quotients of \(\mathbb{R}^3\) by discrete subgroups of translations. The main tool in the proof is a compactness result for harmonic maps due to \textit{J. Sacks} and \textit{K. Uhlenbeck} [Trans. Am. Math. Soc. 271, 639-652 (1982; Zbl 0527.58008)] which can be applied to the harmonic Gauss maps of the immersions. Our results allow us to describe the isoperimetric domains in \(\mathbb{T}^2 \times \mathbb{R}\), where \(\mathbb{T}^2\) is a flat two-dimensional torus with injectivity radius equal to 1 and large area.
reviewed by: Manuel Ritoré
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1121727
zbMATH Keywords: stable periodic surfaces
zbMATH Keywords: compactness
zbMATH Keywords: isoperimetric domains
MaRDI profile type: Publication
full work available at URL: pl00004351
OpenAlex ID: W1978319452
description: scientific article; zbMATH DE number 1121727