On stable minimal surfaces in manifolds of positive bi-Ricci curvatures
type: Item
instance of: scholarly article
title: On stable minimal surfaces in manifolds of positive bi-Ricci curvatures
zbMATH Open document ID: 0874.53049
DOI: 10.1215/S0012-7094-96-08505-1
author: Ying Shen
author: Rugang Ye
published in: Duke Mathematical Journal
publication date: 1997-11-11T00:00:00Z
review text: The authors introduce a new and interesting curvature invariant for a Riemannian manifold \(M^m\) as follows. Let \(u,v\in T_pM\) be orthonormal tangent vectors. Then the bi-Ricci curvature is defined to be \[ BRc(u,v)=\text{Ric}(u)+\text{Ric}(v)-K(u,v) \] where \(K(u,v)\) is the sectional curvature of the plane spanned by \(u\) and \(v\). Positive sectional curvature implies positive bi-Ricci curvature and an example is given that shows that the converse is not true. The main result of the paper under review is motivated by the work of \textit{R. Schoen} and \textit{S.-T. Yau} [Commun. Math. Phys. 90, 575-579 (1983; Zbl 0541.53054)] and is the following: Assume \(BRc\geq\kappa > 0\) and \(m\leq 5\). Then any closed stable minimal hypersurface \(S\) in \(M^m\) satisfies \[ \text{diam }S\leq \sqrt{c(m)} \frac{\pi}{\sqrt{\kappa}} \] where \(c(3)= 4/3\), \(c(4)=2\) and \(c(5)=4\). Moreover, let \(\Gamma_1\) and \(\Gamma_2\) be submanifolds with codimension 2. If \(\text{dist}(\Gamma_1,\Gamma_2)>\sqrt{c(m)} (\pi/\sqrt{\kappa}),\) then \(\Gamma_1\) and \(\Gamma_2\) cannot be connected by a stable minimal hypersurface.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 949887
zbMATH Keywords: bi-Ricci curvature
zbMATH Keywords: closed stable minimal hypersurface
reviewed by: Gudlaugur Thorbergsson
MaRDI profile type: Publication
cites work: Q4067705
cites work: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
cites work: The classification of simply connected manifolds of positive scalar curvature
cites work: Q3312869
cites work: On the structure of manifolds with positive scalar curvature
cites work: The existence of a black hole due to condensation of matter
cites work: Q3834643
cites work: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces
full work available at URL: s0012-7094-96-08505-1
OpenAlex ID: W1495083299
Recommended article: Q4783133
Recommended article: \(L^2\) harmonic forms on a complete stable hypersurfaces with constant mean curvature
Recommended article: Positive biorthogonal curvature on $S^2\times S^2$
Recommended article: Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature
Recommended article: Q4219020
Recommended article: Properly immersed submanifolds in complete Riemannian manifolds
Recommended article: Biharmonic hypersurfaces with constant scalar curvature in space forms
Recommended article: Stability and the index of biharmonic hypersurfaces in a Riemannian manifold
Recommended article: Biharmonic submanifolds in a Riemannian manifold with non-positive curvature
Recommended article: Q3834643
description: scientific article; zbMATH DE number 949887