The bridge principle for stable minimal surfaces
type: Item
instance of: scholarly article
title: The bridge principle for stable minimal surfaces
zbMATH Open document ID: 0830.49025
DOI: 10.1007/BF01192091
author: Brian White
published in: Calculus of Variations and Partial Differential Equations
publication date: 1996-01-18T00:00:00Z
review text: The author proves a rather general version of the famous bridge principle for minimal surfaces. This principle asserts that if one attaches the ends of a narrow band \(P\) at two disjoint boundary segments of a connected or disconnected minimal surface \(S\) then the boundary \(\partial(S\cup P)\) spans a minimal surface which almost looks like \(S\cup P\) provided that the width of the band \(P\) is sufficiently small. Beginning with Paul Lévy several authors have set up different versions of this principle; at least the earlier versions are considered to be not rigorous or incomplete. The main interest in this principle stems from the fact that it allows to construct contours spanning uncountably many minimal surfaces. In the present paper, the author establishes the bridge principle for strictly stable smooth minimal submanifolds \(S\) of \(\mathbb{R}^N\) of arbitrary dimension and codimension and for suitable bridges \(P\) that meet \(S\) under an angle strictly between 0 and \(2\pi\). The hypotheses allow for constructing bridges that simultaneously fit several surfaces \(S\) with a common boundary. Using this fact the author is able to construct a Jordan curve in the 2-sphere in \(\mathbb{R}^3\) which is smooth except at one point and which, for given numbers \(g\in \mathbb{N}\cup \{0, \infty\}\) and \(A\in [a, +\infty]\) for some \(a> 0\), spans uncountably many stable embedded minimal surfaces of genus \(g\) and area \(A\). The author also indicates possible generalizations of his results, e.g., to minimal surfaces in Riemannian manifolds or to stationary surfaces for certain parametric elliptic integrals. The guiding principle of the proofs is to take blow up limits and to analyze the limit situation.
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 720497
zbMATH Keywords: Plateau's problem
zbMATH Keywords: bridge principle
zbMATH Keywords: minimal surfaces
zbMATH Keywords: stable smooth minimal submanifolds
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: parametric elliptic integrals
zbMATH Keywords: blow up limits
reviewed by: F. Tomi
MaRDI profile type: Publication
Wikidata QID: Q125772555
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description: scientific article; zbMATH DE number 720497