Berger's isoperimetric problem and minimal immersions of surfaces
type: Item
instance of: scholarly article
title: Berger's isoperimetric problem and minimal immersions of surfaces
zbMATH Open document ID: 0868.58079
DOI: 10.1007/BF02246788
author: S. H. Smith
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-08-17T00:00:00Z
full work available at URL: https://eudml.org/doc/58250
review text: The author establishes inequalities between eigenvalues of the Laplacian of a compact surface and the geometrical characteristics of this surface, and he considers the case of an orientable surface of genus one, \(T^2\). It can be shown that in the class of flat tori of fixed area the first nonzero eigenvalue of the Beltrami-Laplace operator \(\lambda_1\) attains its supremum on the equilateral torus, i.e., on the torus \(\mathbb{R}^2/\Gamma\), where \(\Gamma\) is the lattice generated by (1,0) and \((1/2,\sqrt{3/2})\). The main result of this paper is that the flat equilateral torus is \(\lambda_1\)-maximal. It means that the author gives affirmative answers to Berger's problem.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 954723
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: Laplace operator
zbMATH Keywords: surface of genus one
zbMATH Keywords: Berger's problem
reviewed by: Stana Nikčević
MaRDI profile type: Publication
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OpenAlex ID: W2050092130
Wikidata QID: Q126181243
description: scientific article; zbMATH DE number 954723