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

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Abstract

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.

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Authors and Affiliations

  1. School of Sciences; College of Arts and Sciences, Shanghai Polytechnic University, Shanghai, 201209, China

    Tian Chong

  2. School of Mathematical Science, Fudan University, Shanghai, 200433, China

    Yuxin Dong

  3. School of Mathematics and Computer Science, Fujian Normal University, Shanghai, 200433, China

    Hezi Lin

  4. College of Mathematics; Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China

    Yibin Ren

Authors
  1. Tian Chong
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  2. Yuxin Dong
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  3. Hezi Lin
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  4. Yibin Ren
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Corresponding author

Correspondence to Tian Chong.

Additional information

The first author is supported by the Foundation for training Young Teachers in University of Shanghai (ZZegd16003). The second author is supported by National Natural Science Foundation of China (11271071, 11771087) and LMNS, Fudan University.

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Chong, T., Dong, Y., Lin, H. et al. Rigidity Theorems of Complete Kähler-Einstein Manifolds and Complex Space Forms. Acta Math Sci 39, 339–356 (2019). https://doi.org/10.1007/s10473-019-0201-y

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  • DOI: https://doi.org/10.1007/s10473-019-0201-y

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