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

TL;DR

This paper explores the relationships between polynomial growth holomorphic functions, volume, and curvature decay on noncompact K"ahler manifolds, linking K"ahler-Ricci flow behavior with Yau's uniformization conjecture.

Contribution

It establishes precise relations among minimal degree, volume growth, and curvature decay, and connects K"ahler-Ricci flow asymptotics with polynomial growth functions, resolving two conjectures by Yang.

Findings

Relations among minimal degree, volume ratio, and curvature decay are established.

K"ahler-Ricci flow behavior is described via polynomial growth holomorphic functions.

Provides a unifying perspective on proofs of Yau's uniformization conjecture.

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.