Minimal Degrees, Volume Growth, and Curvature Decay on Complete K\"ahler Manifolds

Abstract

We consider noncompact complete K\"ahler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms, $\operatorname{AVR}$ (asymptotic volume ratio) and $\operatorname{ASCD}$ (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the K\"ahler-Ricci flow can be described in terms of polynomial growth holomorphic functions. This provides a unifying perspective that bridges the two distinct proofs of Yau's uniformization conjecture by Liu and Chau-Lee-Tam. These resolve two conjectures made by Yang.