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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sharp dimension estimates of holomorphic functions and rigidity


Authors: Bing-Long Chen, Xiao-Yong Fu, Le Yin and Xi-Ping Zhu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1435-1454
MSC (2000): Primary 32Q30; Secondary 32Q10, 32Q15
Published electronically: November 18, 2005
MathSciNet review: 2186981
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Abstract: Let $ M^n$ be a complete noncompact Kähler manifold of complex dimension $ n$ with nonnegative holomorphic bisectional curvature. Denote by $ \mathcal{O}_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $ d$ on $ M^n$. In this paper we prove that

$\displaystyle dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$

for all $ d>0$, with equality for some positive integer $ d$ if and only if $ M^n$ is holomorphically isometric to $ \mathbb{C}^n$. 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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Additional Information

Bing-Long Chen
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China

Xiao-Yong 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

Xi-Ping Zhu
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04105-X
PII: S 0002-9947(05)04105-X
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.