## A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

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## Abstract:

In this paper, we derive a new monotonicity formula for the plurisubharmonic functions/positive (1,1) currents on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which, in particular, partially solve a conjecture of Yau. The methods used in this paper, without the assumption of maximum volume of growth, as observed recently by Chen, Fu, Yin, and Zhu, provide a complete solution to Yau’s conjecture.## References

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## Additional Information

**Lei Ni**- Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, Californiz 92093
- MR Author ID: 640255
- Email: lni@math.ucsd.edu
- Received by editor(s): July 22, 2003
- Published electronically: August 27, 2004
- Additional Notes: The author’s research was partially supported by NSF grant DMS-0328624, USA
- © Copyright 2004 by the author. All rights reserved.
- Journal: J. Amer. Math. Soc.
**17**(2004), 909-946 - MSC (2000): Primary 58J35, 53C55
- DOI: https://doi.org/10.1090/S0894-0347-04-00465-5
- MathSciNet review: 2083471