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

doi:10.1007/s10473-019-0201-y

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (2): 339-356.doi: 10.1007/s10473-019-0201-y

• Articles • Next Articles

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.

Abstract

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

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.

Trendmd

References

[1] Bando S, Kasue A, Nakajima H. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent Math, 1989, 97(2):313-349
[2] Bando S, Mabuchi T. Uniqueness of Einstein Kähler metrics modulo connected group actions//Adv Stud Pure Math, 10. Amsterdam:North-Holland, 1987
[3] Bochner S. Curvature and Betti numbers. 2. Ann Math, 1949, 50(1):77-93
[4] Chen X. Recent progress in Kähler geometry (ENG)//Proceedings of the International Congress of Mathematicians. Vol. Ⅱ. Beijing:Higher Education Press, 2002:273-282
[5] Donaldson S K. Conjectures in Kähler geometry//Douglas M, Gauntlett J, Gross M. Strings and Geometry. American Mathematical Society, 2004:71-78
[6] Hebey E. Variational methods and elliptic equations in Riemannian geometry. Workshop on Recent Trends in Nonlinear Variational Problems. Notes from Lectures at ICTP, 2003
[7] Hebey E. Nonlinear Analysis on Manifolds:Sobolev Spaces and Inequalities. Courant Lect Notes Math, 5. New York:Courant Institute of Mathematical Sciences, 1999
[8] Huisken G. Ricci deformation of the metric on a Riemannian manifold. J Differential Geom, 1985, 21(1985):2438-2443
[9] Hebey E, Vaugon M. Effective L^p pinching for the concircular curvature. J Geom Anal, 1996, 6(4):531-553
[10] Itoh M, Nakagawa T. Variational stability and local rigidity of Einstein metrics. Yokohama Math J, 2005, 51(2):103-115
[11] Itoh M, Satoh H. Isolation of the Weyl conformal tensor for Einstein manifolds. Proc Japan Acad, 2002, 2015(5):1163-1168
[12] Itoh M, Kobayashi D. Isolation theorems of the Bochner curvature type tensors. Tokyo J Math, 2004, 27(2004):227-237
[13] Kim S. Rigidity of noncompact complete manifolds with harmonic curvature. Manuscripta Math, 2009, 135(1/2):107-116
[14] Kobayashi S, Nomizu K. Foundations of Differential Geometry. Vol. Ⅱ. Wiley Classics Library, 1996
[15] Kühnel M. Complete Kähler-Einstein manifolds//Ebeling W, Hulek K, Smoczyk K, eds. Complex and Differential Geometry. Berlin, Heidelberg:Springer, 2011:171-181
[16] Lee J M, Parker T H. The Yamabe problem. Bull Amer Math Soc, 1987, 17(1):37-91
[17] Lebrun C, Simanca S R. Extremal Kähler metrics and complex deformation theory. Geom Funct Anal, 1994, 4(3):298-336
[18] Obata M. The conjectures on conformal transformations of Riemannian manifolds. Bull Amer Math Soc, 1971, 77(1971):247-258
[19] Okumura M. Hypersurfaces and a 'pinching problem on the second fundamental tensor. Amer J Math, 1974, 96(1):207-213
[20] Pigola S, Rigoli M, Setti A. Some characterizations of space-forms. Trans Amer Math Soc, 2007, 359(4):1817-1828
[21] Ren Y, Lin H, Dong Y. Rigidity theorems for complete Sasakian manifolds with constant pseudo-Hermitian scalar curvature. J Geom Anal, 2014:1-29
[22] Shen Z M. Some rigidity phenomena for Einstein metrics. Proc Amer Math Soc, 1990, 108(4):981-987
[23] Shen Z. Rigidity theorems for nonpositive Einstein metrics. Proc Amer Math Soc, 1992, 116(4):1107-1114
[24] Schoen R, Yau S T. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent Math, 1988, 92(1):47-71
[25] Tian G. Extremal metrics and geometric stability. Houston J Math, 2002, 28(2):411-432
[26] Tian G, Yau S T. Complete Kähler manifolds with zero Ricci curvature. I. J Amer Math Soc, 1990, 3(3):579-610
[27] Tian G, Yau S T. Complete Kähler manifolds with zero Ricci curvature. Ⅱ. Invent Math, 1991, 106:27-60
[28] Zhang Y. A note on conical Kähler-Ricci flow on minimal elliptic Kähler surface. Acta Math Sci, 2018, 38B(1):169-176

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

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 0

Metrics

Comments

Recommended 4