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

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.