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

doi:10.1007/s10473-019-0201-y

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

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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. 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. 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/Abstract

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 L^p 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 L^p 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

种田, 东瑜昕, 林和子, 任益斌. RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS[J]. 数学物理学报(英文版), 2019, 39(2): 339-356.

Tian CHONG, Yuxin DONG, Hezi LIN, Yibin REN. RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS[J]. Acta mathematica scientia,Series B, 2019, 39(2): 339-356.

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

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

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