A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

Author:
Lei Ni

Journal:
J. Amer. Math. Soc. **17** (2004), 909-946

MSC (2000):
Primary 58J35, 53C55

Published electronically:
August 27, 2004

MathSciNet review:
2083471

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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.

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Additional Information

**Lei Ni**

Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, Californiz 92093

Email:
lni@math.ucsd.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-04-00465-5

Keywords:
Monotonicity formula,
holomorphic functions of polynomial growth,
heat equation deformation of plurisubharmonic functions

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

Article copyright:
© Copyright 2004
by the author. All rights reserved.