The barrier principle for minimal submanifolds of arbitrary codimension
type: Item
instance of: scholarly article
title: The barrier principle for minimal submanifolds of arbitrary codimension
zbMATH Open document ID: 1027.53007
DOI: 10.1023/A:1024791501324
published in: Annals of Global Analysis and Geometry
publication date: 2003-09-01T00:00:00Z
review text: The barrier principle for minimal hypersurfaces is a very useful tool in minimal surface theory. In this paper a corresponding result for minimal submanifolds of any codimension is proved. Recall that a hypersurface in a Riemannian manifold is \(k\)-mean convex if the sum of the \(k\) smallest principal curvatures is nonnegative. The main result of the paper is the following: Let \(M\) be a \(k\)-dimensional minimal submanifold of a Riemannian manifold \(N\), and \(B\subset N\) a \(k\)-mean convex hypersurface with unit normal \(\nu\). Assume that \(M\) and \(B\) are tangent at \(p\) and that \(M\) lies locally near \(p\) in the side of \(B\) where \(\nu\) points. Then \(M\subset B\). The authors construct topologically nontrivial domains in \(\mathbb{R}^n\) with \(2\)-mean convex boundaries and use their main result to prove a nonexistence condition for minimal surfaces with boundary in two parallel hyperplanes of \(\mathbb{R}^n\).
reviewed by: Manuel Ritoré
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 1973349
zbMATH Keywords: minimal surfaces
zbMATH Keywords: barrier principle
zbMATH Keywords: Plateau's problem
author: Luquesio P. de M. Jorge
author: F. Tomi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1973349