Uniqueness of stable minimal surfaces with partially free boundaries
type: Item
instance of: scholarly article
title: Uniqueness of stable minimal surfaces with partially free boundaries
zbMATH Open document ID: 0831.53006
author: S. Hildebrandt
author: Friedrich Sauvigny
published in: Journal of the Mathematical Society of Japan
publication date: 1996-02-12T00:00:00Z
review text: We consider parametric minimal surfaces \(X : B \to \mathbb{R}^3\) of disc- type which are stationary in a boundary configuration \(\langle T, S\rangle\) consisting of a support surface \(S \subset \mathbb{R}^3\) and a Jordan arc \(\Gamma\) with endpoints on \(S\). We introduce the notion of ``freely stable minimal surfaces'', whose second variation of the area functional is nonnegative with respect to displacements of \(X\) keeping the arc \(\Gamma\) fixed and remaining with the support surface \(S\). If \(\Gamma\) is a graph above the \(x,y\)-plane \(E\) with a convex projection curve \(\underline {\Gamma} \subset E\) and \(S\) is a cylindrical surface with a generating curve \(\sum_0 \subset E\) satisfying a certain oscillation condition, we prove: Each freely stable, parametric minimal surface in this configuration \(\langle \Gamma, S\rangle\) is necessarily a graph above \(E\), and its height function solves a mixed boundary value problem for the minimal surface equation. Consequently, these configurations \(\langle \Gamma, S\rangle\) only bound one freely stable, parametric minimal surface.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 800606
zbMATH Keywords: uniqueness
zbMATH Keywords: disc-type minimal surfaces
zbMATH Keywords: freely stable minimal surfaces
zbMATH Keywords: boundary configuration
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04730423
OpenAlex ID: W2077819608
DOI: 10.2969/JMSJ/04730423
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description: scientific article; zbMATH DE number 800606