A Lipschitz decomposition of minimal surfaces
type: Item
instance of: scholarly article
title: A Lipschitz decomposition of minimal surfaces
zbMATH Open document ID: 0777.53014
author: John B. Garnett
author: Peter Jones
author: Donald E. Marshall
published in: Journal of Differential Geometry
publication date: 1993-02-18T00:00:00Z
review text: A Jordan curve \(\Gamma\) in \(\mathbb{R}^ n\) satisfies an \(M\)-cord-arc condition if the length \(l(z,w)\) of the shorter subarc of \(\Gamma\) joining the points \(z\), \(w\in\Gamma\) is bounded by \(M| z-w|\). The authors prove the following decomposition theorem for classical minimal surfaces (i.e. conformal, branched, minimal immersions) \(F:D\to\mathbb{R}^ n\), \(D\) the unit disc: there is a universal constant \(M\) such that whenever \(F\) is a minimal surface with boundary a rectifiable Jordan curve \(\Gamma\), then there exists a locally finite partition of \(D\) into non-overlapping simply connected subdomains \(D_ j\) such that (i) \(F\) is an embedding on \(\overline D_ j\), (ii) \(F(\partial D_ j)\) is an \(M\)- chord-arc curve, (iii) \(\sum l(F(\partial D_ j))\leq Ml(\Gamma)\).
reviewed by: F. Tomi
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 120217
zbMATH Keywords: \(M\)-chord-arc condition
zbMATH Keywords: minimal surfaces
zbMATH Keywords: rectifiable Jordan curve
Wikidata QID: Q115181027
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448261
OpenAlex ID: W1584013604
DOI: 10.4310/JDG/1214448261
description: scientific article; zbMATH DE number 120217