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 | References | Similar Articles | Additional Information

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

**1.**Huai-Dong Cao,*Limits of solutions to the Kähler-Ricci flow*, J. Differential Geom.**45**(1997), no. 2, 257–272. MR**1449972****2.**J. Cheeger, T. H. Colding, and W. P. Minicozzi II,*Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature*, Geom. Funct. Anal.**5**(1995), no. 6, 948–954. MR**1361516**, 10.1007/BF01902216**3.**Tobias H. Colding and William P. Minicozzi II,*Harmonic functions on manifolds*, Ann. of Math. (2)**146**(1997), no. 3, 725–747. MR**1491451**, 10.2307/2952459**4.**S. Y. Cheng and S. T. Yau,*Differential equations on Riemannian manifolds and their geometric applications*, Comm. Pure Appl. Math.**28**(1975), no. 3, 333–354. MR**0385749****5.**Bing-Long Chen and Xi-Ping Zhu,*Complete Riemannian manifolds with pointwise pinched curvature*, Invent. Math.**140**(2000), no. 2, 423–452. MR**1757002**, 10.1007/s002220000061**6.**Chen, B. L. and Zhu, X. P., Volume growth and curvature decay of positively curved Kähler manifolds, Preprint, arXiv: math.DG/0211374.**7.**Jean-Pierre Demailly,*𝐿² vanishing theorems for positive line bundles and adjunction theory*, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 1–97. MR**1603616**, 10.1007/BFb0094302**8.**Harold Donnelly,*Harmonic functions on manifolds of nonnegative Ricci curvature*, Internat. Math. Res. Notices**8**(2001), 429–434. MR**1827086**, 10.1155/S1073792801000216**9.**Richard S. Hamilton,*Eternal solutions to the Ricci flow*, J. Differential Geom.**38**(1993), no. 1, 1–11. MR**1231700****10.**Atsushi Kasue,*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, Academic Press, Boston, MA, 1990, pp. 283–301. MR**1145260****11.**Peter Li,*Harmonic functions of linear growth on Kähler manifolds with nonnegative Ricci curvature*, Math. Res. Lett.**2**(1995), no. 1, 79–94. MR**1312979**, 10.4310/MRL.1995.v2.n1.a8**12.**Peter Li,*Harmonic sections of polynomial growth*, Math. Res. Lett.**4**(1997), no. 1, 35–44. MR**1432808**, 10.4310/MRL.1997.v4.n1.a4**13.**Peter Li,*Curvature and function theory on Riemannian manifolds*, Surveys in differential geometry, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000, pp. 375–432. MR**1919432**, 10.4310/SDG.2002.v7.n1.a13**14.**Peter Li and Luen-Fai Tam,*Complete surfaces with finite total curvature*, J. Differential Geom.**33**(1991), no. 1, 139–168. MR**1085138****15.**Peter Li and Luen-Fai Tam,*Linear growth harmonic functions on a complete manifold*, J. Differential Geom.**29**(1989), no. 2, 421–425. MR**982183****16.**Peter Li and Shing-Tung Yau,*On the parabolic kernel of the Schrödinger operator*, Acta Math.**156**(1986), no. 3-4, 153–201. MR**834612**, 10.1007/BF02399203**17.**Ni. L., Monotonicity and Kähler-Ricci flow, Preprint, arXiv: math.DG/0211214.**18.**Ni, L., A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature, Priprint, arXiv: math.DG/0307275.**19.**Lei Ni and Luen-Fai Tam,*Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature*, J. Differential Geom.**64**(2003), no. 3, 457–524. MR**2032112****20.**Yum Tong Siu and Shing Tung Yau,*Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay*, Ann. of Math. (2)**105**(1977), no. 2, 225–264. MR**0437797****21.**Shing Tung Yau,*Some function-theoretic properties of complete Riemannian manifold and their applications to geometry*, Indiana Univ. Math. J.**25**(1976), no. 7, 659–670. MR**0417452****22.**Shing Tung Yau,*Harmonic functions on complete Riemannian manifolds*, Comm. Pure Appl. Math.**28**(1975), 201–228. MR**0431040****23.**Yau, S. T., Open problems in geometry, Lectures on Differential Geometry, by Schoen and Yau, International Press (1994), 365-404.

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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

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.