Laplacian comparison theorems on complete Kähler manifolds and applications

TL;DR

The paper introduces Laplacian comparison and rigidity results for complete Kähler manifolds under curvature notions that interpolate between Ricci curvature and holomorphic bisectional curvature, via the symmetrized curvature operator $\mathcal{R}$. It develops a new index form and Hessian/Laplacian comparison framework, combined with curvature-synthesis arguments and a novel index theorem, to obtain global and local bounds on $\Delta r$ and to derive diameter and volume rigidity statements. The main contributions include global and local Laplacian comparison theorems for $\mathcal{R}-2c\cdot\mathrm{id}$ with $c<0$ or $c>0$, a sharp diameter bound with topological consequences, and volume comparison results that force isometric biholomorphism to model spaces under equality. These results extend classical Riemannian and Kähler comparison theory, providing new rigidity phenomena and sharper geometric/topological implications under interpolated curvature conditions.

Abstract

In this paper, we establish new Laplacian comparison theorems and rigidity theorems for complete Kähler manifolds under new curvature notions that interpolate between Ricci curvature and holomorphic bisectional curvature.

Theorems & Definitions (31)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 2.1
• proof