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

 

On the first Hodge eigenvalue of isometric immersions


Author: Alessandro Savo
Journal: Proc. Amer. Math. Soc. 133 (2005), 587-594
MSC (2000): Primary 58J50; Secondary 53C42
Published electronically: August 25, 2004
MathSciNet review: 2093083
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Abstract: We give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on $p$-forms on a compact manifold without boundary isometrically immersed in $\mathbf R^n$or $\mathbf S^n$. The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature.


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Additional Information

Alessandro Savo
Affiliation: Dipartimento di Metodi e Modelli Matematici, Università di Roma, La Sapienza, Via Antonio Scarpa 16, 00161 Roma, Italy
Email: savo@dmmm.uniroma1.it

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07702-0
PII: S 0002-9939(04)07702-0
Keywords: Laplacian on $p$-forms, first eigenvalue, isometric immersions, minimal immersions
Received by editor(s): January 22, 2003
Published electronically: August 25, 2004
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.