On the isometric minimal immersion into a Euclidean space
type: Item
instance of: scholarly article
title: On the isometric minimal immersion into a Euclidean space
zbMATH Open document ID: 0942.53045
author: Qing Chen
publication date: 2000-08-16T00:00:00Z
review text: Let \(M ^n\) be a complete \(n\)-dimensional Riemannian manifold and \(p \in M ^n\). Denote by \(B _p (t)\) the geodesic ball of \(M ^n\) with center \(p\) and radius \(t\). The author gives a necessary condition for the existence of a minimal isometric immersion of \(M ^n\) into a Euclidean space. More precisely, the author proves: if there exists a minimal isometric immersion of \(M ^n\) into a Euclidean space, then for each \(p \in M ^n\) the function \(\text{vol} (B _p (t)) / t ^n \) is monotone non-decreasing with respect to \(t\). A classification of non-positively curved minimal hypersurfaces of Euclidean spaces is given in the third paragraph of the paper.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1385072
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: minimal immersion
zbMATH Keywords: geodesic ball
zbMATH Keywords: volume growth
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
full work available at URL: s10114-999-0087-z
OpenAlex ID: W2354424232
cites work: Some pinching and classification theorems for minimal submanifolds
cites work: Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
cites work: Riemannian geometry
cites work: Q3336268
cites work: Isometric embedding of Kähler manifolds with nonpositive sectional curvature
DOI: 10.1007/S10114-999-0087-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1385072