A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

Ni, Lei

doi:10.1090/S0894-0347-04-00465-5

A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature
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by Lei Ni

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

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

Published electronically: August 27, 2004

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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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