On the nonexistence of stable currents in submanifolds of a Euclidean space
type: Item
instance of: scholarly article
title: On the nonexistence of stable currents in submanifolds of a Euclidean space
zbMATH Open document ID: 1075.53078
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-04-15T00:00:00Z
review text: Let \(M^m\) be a compact Riemannian manifold immersed in Euclidean space \(\mathbb R^{m+k}\). The main result of the paper is that there are no stable currents in \(M^m\) if one of the following pinching conditions holds: 1) The sectional curvature \(K\) satisfies \(K > (k/4)(\lambda_0 - \mu_0)^2\) where \(\lambda_0\) and \(\mu_0\) are the maximum and the minimum of the principal curvatures. 2) \(\lambda\mu > (1/4) (\lambda-\mu)^2\) where \(\lambda\) and \(\mu\) are any two principal curvatures. The author also recognizes \(M^m\) as sphere if a pinching condition in terms of the principal curvatures is satisfied and shows that the nonexistence of stable currents can be extended to suitable compact submanifolds of \(M^m\).
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 2157240
zbMATH Keywords: stable currents
zbMATH Keywords: pinching
zbMATH Keywords: submanifolds
author: Xueshan Zhang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246746
OpenAlex ID: W1973444153
cites work: Normal and integral currents
cites work: The nonexistence of stable submanifolds, varifolds, and harmonic maps in sufficiently pinched simply connected Riemannian manifolds
cites work: On stable currents and their application to global problems in real and complex geometry
cites work: Q3217184
cites work: Nonexistence of stable currents in submanifolds of a product of two spheres
cites work: Geometry and topology of submanifolds immersed in space forms and ellipsoids
DOI: 10.2748/TMJ/1113246746
description: scientific article; zbMATH DE number 2157240