A monotonicity formula on complete K\"ahler manifolds with nonnegative bisectional curvature

TL;DR

This paper introduces a new monotonicity formula for plurisubharmonic functions on complete Kähler manifolds with nonnegative bisectional curvature, leading to sharp dimension estimates for holomorphic functions and partially resolving Yau's conjecture.

Contribution

It presents a novel monotonicity formula and applies it to obtain sharp dimension bounds, advancing understanding of holomorphic functions on such manifolds.

Findings

Derived a new monotonicity formula for plurisubharmonic functions.

Established sharp estimates for the dimension of holomorphic function spaces.

Partially solved a conjecture of Yau regarding holomorphic functions.

Abstract

In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete K\"ahler 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.