|
The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold
Author(s):
Song-Ying
Li;
Hing-Sun
Luk
Journal:
Proc. Amer. Math. Soc.
132
(2004),
789-798.
MSC (2000):
Primary 32V05, 32V20;
Secondary 53C56
Posted:
August 7, 2003
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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  , 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  . For dimension  , the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.
References:
-
- 1.
- M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real submanifolds in complex space and their mappings. Princeton Mathematical Series, 47, Princeton University Press, Princeton, NJ, 1999. MR 2000b:32066
- 2.
- P. H. Bérard, From vanishing theorems to estimating theorems: the Bochner technique revisited, Bull. Amer. Math. Soc., 19 (1988), 371-406. MR 89i:58152
- 3.
- J. P. Bourguignon, The ``magic'' of Weitzenböck formulas, Variational methods (Paris, 1988), 251-271, Progr. Nonlinear Differential Equations Appl., 4, Birkhäuser, Boston, MA, 1990. MR 94a:58181
- 4.
- I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Inc., New York, London, Toronto, Tokyo, 1984. MR 86g:58140
- 5.
- Allan Greenleaf, The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold, Comm. Partial Differential Equations, 10(2) (1985), 191-217. MR 86f:58157
- 6.
- P. Li, Lecture Notes on Geometric Analysis, Lecture Notes Series, No. 6, RIM, Global Analysis Research Center, Seoul National Univ., Korea (1993).
- 7.
- N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book-Store, Tokyo, 1975. MR 53:3361
- 8.
- D.-C. Chang and S.-Y. Li, A Riemann zeta function associated to the sub-Laplacian on the unit sphere in
 , J. Anal. Math., 86 (2002), 25-48. - 9.
- S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., 13 (1978), 25-41. MR 80e:32015
- 10.
- S. Webster, A remark on the Chern-Moser tensor, Special issue for S. S. Chern, Houston J. Math., 28 (2002), 433-435. MR 2003d:32047
- 11.
- S. Webster, On the pseudo-conformal geometry of a Kähler manifold, Math. Z., 157 (1977), 265-270. MR 57:16666
- 12.
- Wu, Hung Hsi, The Bochner technique in differential geometry, Math. Rep., 3 (1988), No. 2, i-xii, 289-538. MR 91h:58031
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
32V05, 32V20,
53C56
Retrieve articles in all Journals with MSC
(2000):
32V05, 32V20,
53C56
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:
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
Posted:
August 7, 2003
Communicated by:
Mei-Chi Shaw
Copyright of article:
Copyright
2003,
American Mathematical Society
|
Comments: webmaster@ams.org