RIGIDITY THEOREMS OF COMPLETE K?HLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS
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摘要
We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L~p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-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 K?hler manifolds and noncompact K?hler-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.
We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L~p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-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 K?hler manifolds and noncompact K?hler-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.
We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L~p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-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 K?hler manifolds and noncompact K?hler-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.
引文
[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]BandoS, Mabuchi T. Uniqueness of Einstein K?hler 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 K?hler geometry(ENG)//Proceedings of the International Congress of Mathematicians. Vol. II. Beijing:Higher Education Press, 2002:273-282
[5]Donaldson S K. Conjectures in K?hler 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]Kuhnel M. Complete K?hler-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 K?hler 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 K?hler manifolds with zero Ricci curvature. I. J Amer Math Soc, 1990, 3(3):579-610
[27]Tian G, Yau S T. Complete K?hler manifolds with zero Ricci curvature. II. Invent Math, 1991, 106:27-60
[28]Zhang Y. A note on conical K?hler-Ricci flow on minimal elliptic K?hler surface. Acta Math Sci, 2018,38B(1):169-176
[2]BandoS, Mabuchi T. Uniqueness of Einstein K?hler 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 K?hler geometry(ENG)//Proceedings of the International Congress of Mathematicians. Vol. II. Beijing:Higher Education Press, 2002:273-282
[5]Donaldson S K. Conjectures in K?hler 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]Kuhnel M. Complete K?hler-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 K?hler 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 K?hler manifolds with zero Ricci curvature. I. J Amer Math Soc, 1990, 3(3):579-610
[27]Tian G, Yau S T. Complete K?hler manifolds with zero Ricci curvature. II. Invent Math, 1991, 106:27-60
[28]Zhang Y. A note on conical K?hler-Ricci flow on minimal elliptic K?hler surface. Acta Math Sci, 2018,38B(1):169-176