Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T14:00:34.836Z Has data issue: false hasContentIssue false
  • English
  • Français

REILLY INEQUALITIES OF ELLIPTIC OPERATORS ON CLOSED SUBMANIFOLDS

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  29 June 2009

RUSHAN WANG
RUSHAN WANG*
Affiliation:
College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

Keywords

closed submanifoldselliptic operatorr-mean curvature

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 335 - 346
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

The author was partially supported by grant number 60804044 of the NSFC.

References

[1] Alencar, H., do Carmo, M. and Marques, F., ‘Upper bounds for the first eigenvalue of the operator L r and some applications’, Illinois J Math. 45 (2001), 851–863.CrossRefGoogle Scholar
[2] Alias, L. J. and Malacarne, J. M., ‘On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms’, Illinois J. Math. 48 (2004), 219–240.CrossRefGoogle Scholar
[3] Barbosa, J. L. and Colares, A. G., ‘Stablity of hypersurfaces of constant r-mean curvature’, Ann. Global Anal. Geom. 15 (1997), 277–297.CrossRefGoogle Scholar
[4] Chavel, I., ‘On A. Hurwitz’ method in isoperimetric inequalities’, Proc. Amer. Math. Soc. 71 (1978), 275–279.Google Scholar
[5] Grosjean, J. F., ‘Extrinsic upper bounds for the first eigenvalue of elliptic operators’, Hokkaido Math. J. 33(2) (2004), 341–355.CrossRefGoogle Scholar
[6] Reilly, R. C., ‘On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space’, Comment. Math. Helv. 52 (1977), 525–553.CrossRefGoogle Scholar
[7] Rosenberg, H., ‘Hypersurfaces of constant curvature in space forms’, Bull. Sci. Math. 117 (1993), 211–239.Google Scholar
[8] Rushan, W., ‘Sharp upper bounds for λ 1L r of immersed hypersurfaces and their r-stability in space forms’, Acta. Math. Sinica. (Engl. Ser.) 24(5) (2008), 749–760.Google Scholar
[9] Sakai, T., Riemannian Geometry, Translations of Mathematical Monographs, 149 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar