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Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

Yuang Shi

TL;DR

The paper analyzes complete noncompact Kähler manifolds with nonnegative bisectional curvature and maximal volume growth, linking geometric growth, holomorphic function theory, and Kähler-Ricci flow dynamics. It proves a precise relation among refined minimal degrees, AVR, and ASCD, and shows that the KR-flow Lyapunov exponents correspond to these minimal degrees, unifying prior proofs of Yau’s uniformization conjecture. It also establishes that the link of any tangent cone is a sphere, providing a Kähler analogue of Kapovitch’s result, and discusses the structure and regularity of tangent cones via gradient expanding Kähler-Ricci solitons and Sasaki geometry. Together, these results connect asymptotic holomorphic data to the metric geometry at infinity and offer a bridge between different approaches to uniformization in the Kähler setting.

Abstract

We consider noncompact complete Kähler 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 Kähler-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. 3. The link of any tangent cone is homeomorphic to a sphere, providing a Kähler analogue of a result of Kapovitch in the Riemannian case.

Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

TL;DR

The paper analyzes complete noncompact Kähler manifolds with nonnegative bisectional curvature and maximal volume growth, linking geometric growth, holomorphic function theory, and Kähler-Ricci flow dynamics. It proves a precise relation among refined minimal degrees, AVR, and ASCD, and shows that the KR-flow Lyapunov exponents correspond to these minimal degrees, unifying prior proofs of Yau’s uniformization conjecture. It also establishes that the link of any tangent cone is a sphere, providing a Kähler analogue of Kapovitch’s result, and discusses the structure and regularity of tangent cones via gradient expanding Kähler-Ricci solitons and Sasaki geometry. Together, these results connect asymptotic holomorphic data to the metric geometry at infinity and offer a bridge between different approaches to uniformization in the Kähler setting.

Abstract

We consider noncompact complete Kähler 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, (asymptotic volume ratio) and (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the Kähler-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. 3. The link of any tangent cone is homeomorphic to a sphere, providing a Kähler analogue of a result of Kapovitch in the Riemannian case.

Paper Structure

This paper contains 7 sections, 12 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Structure of Kähler manifolds with maximal volume growth and tangent cones
  4. Gradient expanding Kähler-Ricci solitons with nonnegative Ricci curvature
  5. Proof of Theorem \ref{['thm:B']}
  6. Proof of Theorem \ref{['thm:A']}
  7. Tangent cones and further questions

Key Result

Theorem 1

Let $(M^n,g)$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature. Assume that the universal cover of $M$ does not split. Then the following conditions are equivalent: (1) $M$ is of maximal volume growth, i.e. Here $\omega_{2n}$ is the volume of the unit ball in $\mathbb{C}^n$. (2) There exists a nonconstant polynomial growth holomorphic function, i.e. $\mathcal{O}_P(M

Theorems & Definitions (34)

  • Conjecture 1: Yau's Uniformization Conjecture, Yau91
  • Theorem 1: Corollary 3.2 in Ni04, Theorem 1.2 in NT13, Theorem 2 in Liu16b, Theorem 1.4 in Liu16a, Corollary 2.16 in Liu21
  • Remark
  • Definition 1
  • Conjecture 2: Conjecture 2.5.6, Conjecture 2.5.8 in Yang13
  • Theorem 2: Corollary 1 in Ni05, Theorem 1.2, Proposition 3.2 in CT06, Theorem 6.1 in CT11, Theorem 1.5 in LT20, Theorem 1.1 in Lee25
  • Remark
  • Conjecture 3: Conjecture 2.5.16 in Yang13
  • Theorem 3: Theorem 1.4 in Yang22
  • Corollary 1
  • ...and 24 more