Rigidity theorems of complete K\"ahler-Einstein manifolds and complex space forms

TL;DR

This paper establishes new rigidity theorems for complete K"ahler-Einstein manifolds and complex space forms by deriving elliptic inequalities and applying maximum principles to characterize these manifolds.

Contribution

It introduces novel elliptic inequalities and pinching conditions that characterize K"ahler-Einstein manifolds and complex space forms among K"ahler manifolds with constant scalar curvature.

Findings

Characterization of K"ahler-Einstein manifolds via $L^p$ and $L^a0$ pinching results.

Identification of complex space forms among K"ahler-Einstein manifolds.

Unified criteria for complex space forms among K"ahler manifolds with constant scalar curvature.

Abstract

We derive some elliptic differential inequalities from the Weitzenb\"ock formulas for the traceless Ricci tensor of a K\"ahler manifold with constant scalar curvature and the Bochner tensor of a K\"ahler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize K\"ahler-Einstein manifolds among K\"ahler manifolds with constant scalar curvature, and others are given to characterize complex space forms among K\"ahler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among K\"ahler manifolds with constant scalar curvature.