Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Author: Qing-Ming Cheng
Journal: Proc. Amer. Math. Soc. 136 (2008), 3309-3318
MSC (2000): Primary 53C42; Secondary 58J50
Published electronically: May 5, 2008
MathSciNet review: 2407097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be an $ n$-dimensional compact hypersurface with constant scalar curvature $ n(n-1)r$, $ r> 1$, in a unit sphere $ S^{n+1}(1)$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $ \int_MHdM$ of the mean curvature $ H$. In this paper, we first study the eigenvalue of the Jacobi operator $ J_s$ of $ M$. We derive an optimal upper bound for the first eigenvalue of $ J_s$, and this bound is attained if and only if $ M$ is a totally umbilical and non-totally geodesic hypersurface or $ M$ is a Riemannian product $ S^m(c)\times S^{n-m}(\sqrt{1-c^2})$, $ 1\leq m\leq n-1$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 58J50

Retrieve articles in all journals with MSC (2000): 53C42, 58J50

Additional Information

Qing-Ming Cheng
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan

PII: S 0002-9939(08)09304-0
Keywords: Hypersurface with constant scalar curvature, Jacobi operator, mean curvature, first eigenvalue and principal curvatures
Received by editor(s): November 14, 2006
Received by editor(s) in revised form: August 2, 2007
Published electronically: May 5, 2008
Additional Notes: The author’s research was partially supported by a Grant-in-Aid for Scientific Research from JSPS
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia