|
A note on harmonic forms on complete manifolds
Author(s):
Luen-fai
Tam
Journal:
Proc. Amer. Math. Soc.
126
(1998),
3097-3108.
MSC (1991):
Primary 58E20
MathSciNet review:
1459152
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this note, we will prove that under certain conditions, the space of polynomial growth harmonic functions and harmonic forms with a fixed growth rate on manifolds which are asymptotically nonnegatively curved is finite dimensional. This is a partial generalization of the works of Li and Colding-Minicozzi. We will also give an explicit estimate for the dimension in case the manifold is a complete surface of finite total curvature. This is a generalization to harmonic forms of the work of Li and the author.
References:
- [B-C]
- R. Bishop and R. Crittenden, Geometry of Manifolds, Academic Press: New York, 1964. MR 29:6401
- [Cg-Y]
- S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333-354. MR 52:6608
- [C-M 1]
- T. Colding and W. Minicozzi, On function theory on spaces with a lower Ricci curvature bound, Math. Research Letters 3 (1996), 241-246. MR 97e:53073
- [C-M 2]
- T. Colding and W. Minicozzi, Harmonic functions with polynomial growth, preprint.
- [C-M 3]
- T. Colding and W. Minicozzi, Large scale behavior of kernels of Schrödinger operators, preprint.
- [C-M 4]
- T. Colding and W. Minicozzi, Generalized Liouville properties for manifolds, Math. Res. Lett. 3 (1996), 723-729. MR 97h:53039
- [C-M 5]
- T. Colding and W. Minicozzi, Harmonic functions on manifolds, preprint.
- [C-M 6]
- T. Colding and W. Minicozzi, Liouville theorems for harmonic sections and applications manifolds, preprint.
- [C-M 7]
- T. Colding and W. Minicozzi, Weyl type bounds for harmonic functions, preprint.
- [G]
- A. Grigor'yan, The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik. 72 (1992), 47-77.
- [Ha]
- P. Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705-727. MR 30:3435
- [Hu]
- A. Huber, On subharmonic functions and differential geometry in the large, Commentarii Math. Helv. 32 (1957), 13-72. MR 20:970
- [L 1]
- P. Li, Polynomial growth harmonic sections, to appear in Math. Res. Letters.
- [L 2]
- P. Li, Curvature and function theory of Riemannian manifolds, preprint.
- [L-S]
- P. Li and R. Schoen,
 and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), 279-301. MR 86j:58147 - [L-T 1]
- P. Li and L. F. Tam, Linear growth harmonic functions on a complete manifold, J. Diff. Geom. 29 (1989), 421-425. MR 90a:58202
- [L-T 2]
- P. Li and L. F. Tam, Complete surfaces with finite total curvature, J. Diff. Geom. 33 (1991), 139-168. MR 92e:53051
- [L-T 3]
- P. Li and L. F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1-46. MR 93e:58039
- [L-T 4]
- P. Li and L.-F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), 359-383. MR 93b:53033
- [L-T 5]
- P. Li and L.-F. Tam, Green's functions, harmonic functions and volume comparison, J. Differential Geom. 41 (1995), 277-318. MR 96f:53054
- [SC]
- L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Inter. Math. Research Notices 2 (1992), 27-38. MR 93d:58158
- [W]
- J. Wang, Linear growth harmonic functions on complete manifolds, Comm. Anal. Geom. 4 (1995), 683-698. MR 96m:53044
- [Y]
- S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. MR 55:4042
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
58E20
Retrieve articles in all Journals with
MSC (1991):
58E20
Additional Information:
Luen-fai
Tam
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email:
lftam@math.cuhk.edu.hk
DOI:
10.1090/S0002-9939-98-04474-8
PII:
S 0002-9939(98)04474-8
Received by editor(s):
February 19, 1997
Additional Notes:
Research partially supported an Earmarked grant of Hong Kong.
Communicated by:
Peter Li
Copyright of article:
Copyright
1998,
American Mathematical Society
|
