Parametric surfaces of least \(H\)-energy in a Riemannian manifold
type: Item
instance of: scholarly article
title: Parametric surfaces of least \(H\)-energy in a Riemannian manifold
zbMATH Open document ID: 0953.53007
author: Frank Duzaar
author: Klaus Steffen
published in: Mathematische Annalen
publication date: 2001-01-22T00:00:00Z
review text: The main aim of this paper is to establish existence theorems for the Plateau problem with prescribed mean curvature in Riemannian \(3\)-manifolds, going beyond the results of \textit{R. Gulliver} [J. Differ. Geom. 8, 317-330 (1973; Zbl 0275.53033)] and \textit{St. Hildebrandt} and \textit{H. Kaul} [Commun. Pure Appl. Math. 25, 187-223 (1972; Zbl 0245.53006)], more precisely, by renouncing on some of the conditions that the cited authors put on the boundary. Hence, in the present paper, the considered surfaces are not contained in a given coordinate domain of the manifold and the existence results are proven for all curves bounding a surface of sufficiently small area. The authors first prove a general existence theorem for weak solutions of the Plateau problem. Then, they combine it with conditions in Riemannian manifolds and some results on isoperimetric inequalities [same authors, Calc. Var. Partial Differ. Equ. 1, No. 4, 355-406 (1993; Zbl 0806.49028)] to obtain geometric conditions that the Jordan boundary curve and the mean curvature function must satisfy in order to assure the existence of the solution. The paper ends with some brief comments on proving the regularity of the solutions obtained before.
reviewed by: Liviu Ornea
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1337013
zbMATH Keywords: Plateau problem
zbMATH Keywords: mean curvature
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Sobolev space
zbMATH Keywords: isoperimetric inequality
MaRDI profile type: Publication
full work available at URL: s002080050292
OpenAlex ID: W2082674408
Wikidata QID: Q115388966
DOI: 10.1007/S002080050292
description: scientific article; zbMATH DE number 1337013