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

 

The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold


Authors: Song-Ying Li and Hing-Sun Luk
Journal: Proc. Amer. Math. Soc. 132 (2004), 789-798
MSC (2000): Primary 32V05, 32V20; Secondary 53C56
Published electronically: August 7, 2003
MathSciNet review: 2019957
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Abstract: This paper studies, using the Bochner technique, a sharp lower bound of the first eigenvalue of a subelliptic Laplace operator on a strongly pseudoconvex CR manifold in terms of its pseudo-Hermitian geometry. For dimensions greater than or equal to $7$, the lower bound under a condition on the Ricci curvature and the torsion was obtained by Greenleaf. We give a proof for all dimensions greater than or equal to $5$. For dimension $3$, the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.


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Additional Information

Song-Ying Li
Affiliation: Department of Mathematics, University of California, Irvine, California 92697–3875
Email: sli@math.uci.edu

Hing-Sun Luk
Affiliation: Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong
Email: hsluk@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07174-0
PII: S 0002-9939(03)07174-0
Keywords: Strongly pseudoconvex CR manifold, pseudo-Hermitian geometry, sub-Laplacian, eigenvalues
Received by editor(s): October 28, 2002
Published electronically: August 7, 2003
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2003 American Mathematical Society