The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces
type: Item
instance of: scholarly article
title: The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces
zbMATH Open document ID: 0945.53037
DOI: 10.1016/S0926-2245(99)00021-2
author: Yoe Itokawa
author: Katsuhiro Shiohama
published in: Differential Geometry and its Applications
publication date: 2000-09-20T00:00:00Z
review text: We prove that a minimal immersion of a complete Riemannian manifold \(M\) into another complete noncompact Riemannian manifold \(N\) of positive curvature must have an unbounded image provided that \(M\) has scalar curvature bounded away from \(- \infty\). This extends the unboundedness theorems of \textit{D. Gromoll} and \textit{W. Meyer} [Ann. Math., II. Ser. 90, 75-90 (1969; Zbl 0191.19904)] for complete geodesics and of \textit{G. J. Galloway} and \textit{L. Rodriguez} [Geom. Dedicata 39, 29-42 (1991; Zbl 0724.53037)] for parabolic minimal surfaces. Furthermore, we prove that in case \(M\) is of codimension 1, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space \(N\) needs to be positive in order for the same conclusion to hold.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1367707
zbMATH Keywords: minimal submanifold
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: sectional curvature
zbMATH Keywords: Ricci curvature
zbMATH Keywords: scalar curvature
Wikidata QID: Q115337418
MaRDI profile type: Publication
cites work: Q5341449
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full work available at URL: s0926-2245(99)00021-2
OpenAlex ID: W2082045644
description: scientific article; zbMATH DE number 1367707