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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
DOI: https://doi.org/10.1090/S0894-0347-04-00465-5
Published electronically: August 27, 2004
MathSciNet review: 2083471
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Abstract | References | Similar Articles | Additional Information

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: https://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.

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