Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere
type: Item
instance of: scholarly article
title: Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere
zbMATH Open document ID: 0682.53056
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: By the famous theorem of Lusternik-Schnirelman there exist at least three closed geodesics without selfintersections on \((S^ 2,g)\) where g is an arbitrary Riemannian metric. Here, the author establishes analogues of this result for minimal surfaces. Specifically, he is able to prove the following beautiful theorems. Theorem 1. Let A be a compact body in \({\mathbb{R}}^ 3\), homeomorphic to the closed unit ball, with boundary \(\partial A\) of class \(C^ 4\), and of positive mean curvature with respect to the interior normal. Then A contains at least three embedded minimal disks meeting \(\partial A\) orthogonally along their boundary, i.e. solving a free boundary problem with boundary \(\partial A\). Theorem 2. Let M be a compact three- dimensional Riemannian manifold, diffeomorphic to the three-sphere \(S^ 3\). Then M contains at least four embedded minimal two-spheres. The numbers three resp. four are what one would expect from the results of Lusternik and Schnirelman. As in the author's previous paper [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 401-426 (1986; Zbl 0669.49024)] and in the joint paper with the reviewer [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 345-390 (1986; Zbl 0617.49017)] the minimal surfaces are obtained by a minimax-procedure. This saddle-point construction in the context of geometric measure theory was first introduced by Pitts. Here, one has to use higher order saddle-point arguments to get more than one minimal surface. However, to exclude multiple coverings one has to construct new classes in which new saddle- point constructions are then performed. For a number of technical details the reader is referred to the author's previous paper mentioned above.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 4117402
zbMATH Keywords: theorem of Lusternik-Schnirelman
zbMATH Keywords: minimal disks
zbMATH Keywords: minimax-procedure
zbMATH Keywords: saddle-point construction
zbMATH Keywords: minimal surface
author: Jürgen Jost
Wikidata QID: Q115182499
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214443602
OpenAlex ID: W1512196425
DOI: 10.4310/JDG/1214443602
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description: scientific article; zbMATH DE number 4117402