A note on harmonic forms on complete manifolds
Author:
Luenfai Tam
Journal:
Proc. Amer. Math. Soc. 126 (1998), 30973108
MSC (1991):
Primary 58E20
MathSciNet review:
1459152
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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 ColdingMinicozzi. 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.
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 [CM 3]
 T. Colding and W. Minicozzi, Large scale behavior of kernels of Schrödinger operators, preprint.
 [CM 4]
 T. Colding and W. Minicozzi, Generalized Liouville properties for manifolds, Math. Res. Lett. 3 (1996), 723729. MR 97h:53039
 [CM 5]
 T. Colding and W. Minicozzi, Harmonic functions on manifolds, preprint.
 [CM 6]
 T. Colding and W. Minicozzi, Liouville theorems for harmonic sections and applications manifolds, preprint.
 [CM 7]
 T. Colding and W. Minicozzi, Weyl type bounds for harmonic functions, preprint.
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 A. Grigor'yan, The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik. 72 (1992), 4777.
 [Ha]
 P. Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705727. MR 30:3435
 [Hu]
 A. Huber, On subharmonic functions and differential geometry in the large, Commentarii Math. Helv. 32 (1957), 1372. MR 20:970
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 [L 2]
 P. Li, Curvature and function theory of Riemannian manifolds, preprint.
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 P. Li and R. Schoen, and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), 279301. MR 86j:58147
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Additional Information
Luenfai Tam
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email:
lftam@math.cuhk.edu.hk
DOI:
http://dx.doi.org/10.1090/S0002993998044748
PII:
S 00029939(98)044748
Received by editor(s):
February 19, 1997
Additional Notes:
Research partially supported an Earmarked grant of Hong Kong.
Communicated by:
Peter Li
Article copyright:
© Copyright 1998
American Mathematical Society
