Boundary convexity on manifolds with nonnegative Ricci curvature.
type: Item
instance of: scholarly article
title: Boundary convexity on manifolds with nonnegative Ricci curvature.
zbMATH Open document ID: 1038.53038
author: Hui-Hsien Wang
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: Let \(M^n\) be an \(n\)-dimensional compact connected Riemannian manifold with non-empty boundary \(\partial M\), and let \(S_{\partial M}: X\to \nabla_X\mu\) be the second fundamental form on \(\partial M\) with respect to the unit outward normal \(\mu\). For \(\lambda=\inf\) of eigenvalues of \(S_{\partial M}\) over \(\partial M\), if \(\lambda> 0\) then \(\partial M\) is convex, and convexity or strict convexity imply conditions on \(\lambda\) (with \(\lambda\geq 0\)). If \(n= 3\), a Ricci-positively curved \(M^3\) with \(\lambda> 0\) is diffeomorphic to the 3-dimensional ball, as the author shows in the Appendix. For \(n\geq 4\), there are manifolds \(M\) with positive Ricci curvature and convex \(\partial M\) that are not contractible. So a stronger convexity condition is needed for contractibility. Let \(\Lambda(M^n)= \lambda({\text{vol}(\partial M)\over \omega_{n-1}})^{{1\over n-1}}\), \(\omega_{n-1}=\text{volume}\) of the unit \((n-1)\)-sphere, \(\text{Ric}_M>0\), \(K_M\geq 0\) at \(\partial M\), and assume \(\exists \delta_n: 1> \delta_n> 0\) with \(\Lambda(M)> 1-\delta_n\), then \(M\) is contractible.
reviewed by: Alfred Aeppli
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1463737
zbMATH Keywords: boundary convexity
zbMATH Keywords: nonnegative Ricci curvature
MaRDI profile type: Publication
full work available at URL: pjm.1999.191.393
OpenAlex ID: W1972130113
Wikidata QID: Q125941114
DOI: 10.2140/PJM.1999.191.393
description: scientific article; zbMATH DE number 1463737