Abstract
We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kahler manifolds. We derive some elliptic differential inequalities from Weitzenb¨ock formulas for the traceless Ricci tensor of Kahler manifolds with constant scalar curvature and the Bochner tensor of Kahler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L∞ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature and complex space forms among Kahler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kahler manifolds and noncompact Kahler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.
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