Sharp dimension estimates of holomorphic functions and rigidity
Authors:
BingLong Chen, XiaoYong Fu, Le Yin and XiPing Zhu
Journal:
Trans. Amer. Math. Soc. 358 (2006), 14351454
MSC (2000):
Primary 32Q30; Secondary 32Q10, 32Q15
Published electronically:
November 18, 2005
MathSciNet review:
2186981
Fulltext PDF Free Access
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Abstract: Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that for all , with equality for some positive integer if and only if is holomorphically isometric to . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.
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 2.
 Cheeger, J., Colding, T. H. and Minicozzi, W. P., Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature, Geom. Funct. Anal. 5 (1995), no. 6, 948954. MR 1361516 (96j:53038)
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 4.
 Cheng, S. Y. and Yau, S. T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333354. MR 0385749 (52:6608)
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 Chen, B. L. and Zhu, X. P., Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math. 140 (2000), 423452. MR 1757002 (2001g:53118)
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 Chen, B. L. and Zhu, X. P., Volume growth and curvature decay of positively curved Kähler manifolds, Preprint, arXiv: math.DG/0211374.
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 Demailly, J. P., vanishing theorems for positive line bundles and adjunction theory, Transcendental Methods in Algebraic Geometry, CIME, Cetrro, 1994, Lecture Notes in Math. 1646, SpringerVerlag, 1996. MR 1603616 (99k:32051)
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 Donnelly, H., Harmonic functions on manifolds of nonnegative Ricci curvature, Internat. Math. Res. Notices, 2001, no. 8, 429434. MR 1827086 (2002k:53062)
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 Hamilton, R., Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993), 111. MR 1231700 (94g:58043)
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 Kasue, A., Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature II, Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., vol. 18, NorthHolland, 1989. MR 1145260 (93e:53043)
 11.
 Li, P., Linear growth harmonic functions on Kähler manifolds with nonnegative Ricci curvature, Math. Res. Lett. 2 (1995), 7994. MR 1312979 (95m:53057)
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 Li, P., Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), 3544. MR 1432808 (98i:53054)
 13.
 Li, P., Curvature and function theory on Riemannian manifolds, Survey in Differential Geometry vol. VII, International Press, Cambridge, 2000, 71111. MR 1919432 (2003g:53047)
 14.
 Li, P. and Tam, L. F., Complete surfaces with finite total curvature, J. Differential Geom. 33 (1991), 139168. MR 1085138 (92e:53051)
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 Li, P. and Tam, L. F., Linear growth harmonic functions on a complete manifold, J. Differential Geom. 29 (1989), 421425. MR 0982183 (90a:58202)
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 Li, P. and Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 139168. MR 0834612 (87f:58156)
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 Ni. L., Monotonicity and KählerRicci flow, Preprint, arXiv: math.DG/0211214.
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 Ni, L., A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature, Priprint, arXiv: math.DG/0307275.
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 Ni, L. and Tam, L. F., Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 (2003), no. 3, 457524. MR 2032112 (2005a:32023)
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 Siu, Y. T. and Yau, S. T., Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Ann. Math. 105 (1977), 225264. MR 0437797 (55:10719)
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 Yau, S. T., Some functiontheoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659670. MR 0417452 (54:5502)
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Additional Information
BingLong Chen
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
XiaoYong Fu
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Le Yin
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
XiPing Zhu
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
DOI:
http://dx.doi.org/10.1090/S000299470504105X
PII:
S 00029947(05)04105X
Received by editor(s):
October 1, 2003
Published electronically:
November 18, 2005
Additional Notes:
The first author was partially supported by NSFC 10401042 and FANEDD 200216. The second author was partially supported by NSFC 10171114. The last author was partially supported by NSFC 10428102.
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
