数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (2): 339-356.doi: 10.1007/s10473-019-0201-y

• 论文 •    下一篇

RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS

种田1, 东瑜昕2, 林和子3, 任益斌4   

  1. 1. School of Sciences;College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China;
    2. School of Mathematical Science, Fudan University, Shanghai 200433, China;
    3. School of Mathematics and Computer Science, Fujian Normal University, Shanghai 200433, China;
    4. College of Mathematics;Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
  • 收稿日期:2018-04-08 出版日期:2019-04-25 发布日期:2019-05-06
  • 通讯作者: Tian CHONG E-mail:chongtian@sspu.edu.cn
  • 作者简介:Yuxin DONG,yxdong@fudan.edu.cn;Hezi LIN,lhz1@fjnu.edu.cn;Yibin REN,allenrybqqm@hotmail.com
  • 基金资助:
    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.

RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS

Tian CHONG1, Yuxin DONG2, Hezi LIN3, Yibin REN4   

  1. 1. School of Sciences;College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, China;
    2. School of Mathematical Science, Fudan University, Shanghai 200433, China;
    3. School of Mathematics and Computer Science, Fujian Normal University, Shanghai 200433, China;
    4. College of Mathematics;Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
  • Received:2018-04-08 Online:2019-04-25 Published:2019-05-06
  • Contact: Tian CHONG E-mail:chongtian@sspu.edu.cn
  • Supported by:
    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.

摘要: 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 Weitzenbock 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.

关键词: rigidity theorems, Kähler-Einstein, complex space forms

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 Weitzenbock 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