On the regularity of minimal surfaces with free boundaries in Riemannian manifolds
type: Item
instance of: scholarly article
title: On the regularity of minimal surfaces with free boundaries in Riemannian manifolds
zbMATH Open document ID: 0601.49028
DOI: 10.1007/BF01180769
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155171
review text: The author proves regularity for stationary minimal surfaces with a free boundary in a Riemannian manifold by the way extending earlier results obtained for example by \textit{M. Grüter}, \textit{S. Hildebrandt} and \textit{J. C. Nitsche} [Manuscr. Math. 35, 387-410 (1981; Zbl 0483.49037)] for supporting surfaces \(\Sigma\) located in Euclidean space \({\mathbb{R}}^ 3\). Now \({\mathbb{R}}^ 3\) is replaced by an n-dimensional Riemannian manifold X with bounded sectional curvature and with injectivity radius bounded away from zero, and \(\Sigma\) denotes a \(C^ 2\)-hypersurface in X with bounded second fundamental form. Moreover, \(\Sigma\) has a uniform tubular neighborhood on which a smooth nearest point retraction is defined. In this case a stationary solution of the minimal surface problem with free boundary \(\Sigma\) is of class \(C^{1,\alpha}\) up to the boundary. The proof of this theorem uses methods developed by \textit{M. Grüter} and \textit{J. Jost} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, No.1, 129-169 (1986)] and exploits geometric constructions from \textit{J. Jost} and \textit{H. Karcher} [Manuscr. Math. 40, 27-77 (1982; Zbl 0502.53036)].
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3969498
zbMATH Keywords: regularity
zbMATH Keywords: stationary minimal surfaces
zbMATH Keywords: free boundary
zbMATH Keywords: Riemannian manifold
author: Jürgen Jost
MaRDI profile type: Publication
Wikidata QID: Q115393952
cites work: Q3909666
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cites work: Q4722728
cites work: Harmonic maps between surfaces (with a special chapter on conformal mappings)
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OpenAlex ID: W2170395545
description: scientific article; zbMATH DE number 3969498