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

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 $L^{\mathrm{\infty }}$ 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.