## Sharp dimension estimates of holomorphic functions and rigidity

HTML articles powered by AMS MathViewer

- by Bing-Long Chen, Xiao-Yong Fu, Le Yin and Xi-Ping Zhu PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1435-1454 Request permission

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

- Huai-Dong Cao,
*Limits of solutions to the Kähler-Ricci flow*, J. Differential Geom.**45**(1997), no. 2, 257–272. MR**1449972** - 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**, DOI 10.1007/BF01902216 - Tobias H. Colding and William P. Minicozzi II,
*Harmonic functions on manifolds*, Ann. of Math. (2)**146**(1997), no. 3, 725–747. MR**1491451**, DOI 10.2307/2952459 - 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**385749**, DOI 10.1002/cpa.3160280303 - Bing-Long Chen and Xi-Ping Zhu,
*Complete Riemannian manifolds with pointwise pinched curvature*, Invent. Math.**140**(2000), no. 2, 423–452. MR**1757002**, DOI 10.1007/s002220000061 - Chen, B. L. and Zhu, X. P., Volume growth and curvature decay of positively curved Kähler manifolds, Preprint, arXiv: math.DG/0211374.
- Jean-Pierre Demailly,
*$L^2$ 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**, DOI 10.1007/BFb0094302 - Harold Donnelly,
*Harmonic functions on manifolds of nonnegative Ricci curvature*, Internat. Math. Res. Notices**8**(2001), 429–434. MR**1827086**, DOI 10.1155/S1073792801000216 - Richard S. Hamilton,
*Eternal solutions to the Ricci flow*, J. Differential Geom.**38**(1993), no. 1, 1–11. MR**1231700** - 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**, DOI 10.2969/aspm/01810283 - 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**, DOI 10.4310/MRL.1995.v2.n1.a8 - Peter Li,
*Harmonic sections of polynomial growth*, Math. Res. Lett.**4**(1997), no. 1, 35–44. MR**1432808**, DOI 10.4310/MRL.1997.v4.n1.a4 - Peter Li,
*Curvature and function theory on Riemannian manifolds*, Surveys in differential geometry, Surv. Differ. Geom., vol. 7, Int. Press, Somerville, MA, 2000, pp. 375–432. MR**1919432**, DOI 10.4310/SDG.2002.v7.n1.a13 - Peter Li and Luen-Fai Tam,
*Complete surfaces with finite total curvature*, J. Differential Geom.**33**(1991), no. 1, 139–168. MR**1085138** - 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** - 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**, DOI 10.1007/BF02399203 - Ni. L., Monotonicity and Kähler-Ricci flow, Preprint, arXiv: math.DG/0211214.
- Ni, L., A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature, Priprint, arXiv: math.DG/0307275.
- 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** - 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**437797**, DOI 10.2307/1970998 - 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**417452**, DOI 10.1512/iumj.1976.25.25051 - Shing Tung Yau,
*Harmonic functions on complete Riemannian manifolds*, Comm. Pure Appl. Math.**28**(1975), 201–228. MR**431040**, DOI 10.1002/cpa.3160280203 - Yau, S. T., Open problems in geometry, Lectures on Differential Geometry, by Schoen and Yau, International Press (1994), 365-404.

## 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**- 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.
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 1435-1454 - MSC (2000): Primary 32Q30; Secondary 32Q10, 32Q15
- DOI: https://doi.org/10.1090/S0002-9947-05-04105-X
- MathSciNet review: 2186981