The trapping property of totally geodesic hyperplanes in Hadamard spaces
type: Item
instance of: scholarly article
title: The trapping property of totally geodesic hyperplanes in Hadamard spaces
zbMATH Open document ID: 0865.53028
DOI: 10.1007/BF02247117
author: Yanyan Li
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-07-10T00:00:00Z
full work available at URL: https://eudml.org/doc/58244
review text: The main result of this paper is: For all \(Q\), \(a\geq 1\) and integers \(n>k\geq 2\) there exists a (computable) constant \(D\leq 0\) such that the following holds. Let \(X=X'\times\mathbb{R}^{n-k}\), where \(X\) is either hyperbolic \(k\)-space \(\mathbb{H}^k\) or a Hadamard manifold of dimension \(k=2\) with curvature \(-a^2\leq K\leq -1\). Let \(G=G'\times\mathbb{R}^{n-k}\subset X\) for some totally geodesic \((k-1)\)-plane \(G'\subset X'\). If \(S\in R^{\text{loc}}_{n-1}(X)\) (=space of locally rectifiable \((n-1)\)-currents) is \(Q\)-minimizing in \(X\), \(\text{spt }S\) lies within finite distance from \(G\), and \(\text{spt }\partial S\subset G\), then \(\text{spt }S\) lies within distance \(D\) from \(G\). In addition to this, an analogue trapping property is proved for sufficiently small quasiminimality constant \(Q\) in the case \(X=X'\times Y\), \(Y\) with cocompact isometry group. The results rely on [\textit{V. Bangert} and the author, Comment. Math. Helv. 71, No. 1, 122-143 (1996; Zbl 0856.53047)].
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 934960
zbMATH Keywords: trapping property
zbMATH Keywords: totally geodesic hyperplanes in Hadamard spaces
zbMATH Keywords: quasiminimizing submanifolds
zbMATH Keywords: locally rectifiable currents
reviewed by: Eberhard Teufel
MaRDI profile type: Publication
cites work: Trapping quasiminimizing submanifolds in spaces of negative curvature
cites work: Q5565773
cites work: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension
cites work: Foliated Plateau problem. I: Minimal varieties
cites work: Geodätischer Fluß auf Mannigfaltigkeiten vom hyperbolischen Typ
cites work: Quasi-minimizing surfaces in hyperbolic space
cites work: THE EXISTENCE OF COMPLETE MINIMIZING HYPERSURFACES IN HYPERBOLIC MANIFOLDS
cites work: Q3994358
cites work: A stability theorem for minimal foliations on a torus
cites work: Q3336268
cites work: A strict maximum principle for area minimizing hypersurfaces
description: scientific article; zbMATH DE number 934960