Acta Mathematica Scientia

, Volume 39, Issue 2, pp 339–356 | Cite as

Rigidity Theorems of Complete Kähler-Einstein Manifolds and Complex Space Forms

  • Tian ChongEmail author
  • Yuxin Dong
  • Hezi Lin
  • Yibin Ren


We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kähler manifolds. We derive some elliptic differential inequalities from Weitzenb¨ock formulas for the traceless Ricci tensor of Kähler manifolds with constant scalar curvature and the Bochner tensor of Kähler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature and complex space forms among Kähler-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 Kähler manifolds and noncompact Kähler-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.

Key words

rigidity theorems Kähler-Einstein complex space forms 

2010 MR Subject Classification

32Q15 32Q20 


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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Sciences; College of Arts and SciencesShanghai Polytechnic UniversityShanghaiChina
  2. 2.School of Mathematical ScienceFudan UniversityShanghaiChina
  3. 3.School of Mathematics and Computer ScienceFujian Normal UniversityShanghaiChina
  4. 4.College of Mathematics; Physics and Information EngineeringZhejiang Normal UniversityJinhuaChina

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