一类Hartogs域的完备Einstein-Khler度量和比较定理(英文)
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摘要
本文研究的是一类赋予完备Einstein-K(a|¨)hler度量的非齐性Hartogs域Ω_(μ,v).首先得到了该度量的K(a|¨)hler势函数隐函数形式的表达式;其次当参数满足一定条件时,将隐函数转化为了显函数;最后还得到了该类域上Einstein-K(a|¨)hler度量和Kobayashi度量的比较定理.
In this paper,we consider a class of non-homogeneous Hartogs domains Ω_(μ,v)endowed with a complete Einstein-Kahler metric.First,we get the Kahler potential function which is an implicit function.Second,we get the explicit function from the implicit function when the parameters satisfy certain conditions.Finally,we obtain the comparison theorem between Einstein-Kahler metric and Kobayashi metric on those domains.
In this paper,we consider a class of non-homogeneous Hartogs domains Ω_(μ,v)endowed with a complete Einstein-Kahler metric.First,we get the Kahler potential function which is an implicit function.Second,we get the explicit function from the implicit function when the parameters satisfy certain conditions.Finally,we obtain the comparison theorem between Einstein-Kahler metric and Kobayashi metric on those domains.
引文
[1]Aubin,T.,Metriques riemanniennes et courbure,J.Diff.Geom.,1970,4:383-424(in French).
[2]Aubin,T.,Equations du type Monge-Ampere sur les varietes Khleriennes compactes,C.R.Acad.Sci.Paris Sir.A-B,1976,283:119-121(in French).
[3]Aubin,T.,Nonlinear Analysis on Manifolds,Monge-Ampere Equations,New York:Springer-Verlag,1982.
[4]Calabi,E.,The space of Khler metrics,In:Proc.Internat.Congress Math.(Amsterdam),1954,2:206-207.
[5]Calabi,E.,On Khler manifolds with vanishing canonical class,In:Algebraic Geometry and Topology,A Symposium in Honor of S.Lefschetz,Princeton:Princeton University Press,1957,78-89.
[6]Cheng,S.Y.and Yau,S.Y.,On the existence of a complete Kahler metric on non-compact complex manifolds and the regularity of Fefferman's equation,Commun,Pure Appl.Math.,1980,33(4):507-544.
[7]Hahn,K.T.,Inequality between the Bergman metric and Caratheodory differential metric,Proc.Amer.Math.Soc.,1978,68(2):193-194.
[8]Hahn,K.T.and Pflug,P.,The Kobayashi and Bergman metrics on generalized Thullen domains,Proc.Amer.Math.Soc.,1988,104(1):207-214.
[9]Heins,M.,On a class of conformal metrics,Nagoya Math.J.,1962,21:1-60.
[10]Kobayashi,S.,Intrinsic distances,measures and geometric function theory,Bull.Amer.Math.Soc,1976,82(3):357-416.
[11]Lempert,L.,Holomorphic retracts and intrinsic metrics in convex domains,Anal.Math.,1982,8(4):257-261.
[12]Lin P.and Yin W.P.,The comparison theorem on Cartan-Hartogs domain of the fourth type,Advances in Mathematics(China),2003,32(1):124-126.
[13]Liu K.F.,Sun X.F.and Yau,S.T.,Geometric aspects of the moduli space of Riemann surfaces,Sci.China,Ser.A,2005,48(S1):97-122.
[14]Look,K.H.,Schwarz lemma and analytic invariants,Scientia Sinica,1958,7(5):453-504.
[15]Look,K.H.,Classical Manifolds and Classical Domains,Beijing:Science Press,2011(in Chinese).
[16]Loos,O.,Jordan Pairs,Lecture Notes in Mathematics,Vol.460,New York:Springer-Verlag,1975.
[17]Mok,N.and Yau,S.T.,Completeness,of the Kahler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions,Proc.Symposia Pure Math.,1983,39:41-49.
[18]Schoen,R.and Yau,S.T.,Lectures on Differential Geometry,Cambridge,MA:International Press,1994.
[19]Shi J.H.,The Function Theory of Several Complex Variables,Beijing:Higher Education Press,1996(in Chinese).
[20]Tian G.,On Calabi's conjecture for complex surfaces with positive first Chern class,Invent.Math.,1990,101(1):101-172.
[21]Wang A.,The Einstein-Kahler metrics on Cartan-Hartogs domain of the first type,Sci.China,Ser.A,2004,47(2):220-235.
[22]Wang A.,Wang M.J.and Zhang L.Y.,Kahler-Einstein metric on Cartan-Hartogs domain,Advances in Mathematics(China),2009,38(4):493-502.
[23]Wang A.and Yin W.P.,The Einstein-Khler metric on Hua construction of the first type,Sci.China,Ser.A,2005,48(5):711-719.
[24]Wang A.and Yin W.P.,Einstein-Khler metric with explicit formula on super-Cartan domain of the fourth type,Acta Math.Sinica,Engl.Series,2006,22(2):367-376.
[25]Wang A.,Yin W.P.,Zhang L.Y.and Roos,G.,The Khler-Einstein metric for some Hartogs domains over symmetric domains,Sci.China,Ser.A,2006,49(9):1175-1210.
[26]Wang A.,Yin W.P.,Zhang L.Y.and Zhang W.J.,The Einstein-Khler metric with explicit formulas on some non-homogeneous domains,Asian J.Math.,2004,8(1):39-50.
[27]Wang A.,Yin W.P.and Zhang W.J.,Einstein-Khler metric on super-Cartan domain of the third type,Advances in Mathematics(China),2004,33(2):215-228(in Chinese).
[28]Wu,H.,Old and new invariant metrics on complex manifolds,several complex variables:In:Proceedings of the Mittag-LefBer Institute,1987-1988(J.E.,Fornaess ed.),Math.Notes,Vol.38,Princeton:Princeton Univ.Press,1993,640-682.
[29]Yau,S.T.,Calabi's conjecture and some new results in algebraic geometry,Proc.Nat.Acad.Sci.USA,1977,74(5):1798-1799.
[30]Yau,S.T.,On the Ricci curvature of a compact Khler manifold and the complex Monge-Ampere equation,Comm.Pure Appl.Math.,1978,31(3):339-411.
[31]Yin W.P.,Curvatures on a class of Reinhardt domains,Sci.China,Ser.A,1992,35(11):1281-1293.
[32]Yin W.P.,The comparison theorem for the Bergman and Kobayashi metrics on certain pseudoconvex domains,Complex Variables,1997,34(4):351-373.
[33]Yin W.P.,The Bergman kernels on super-Cartan domains of the first type,Sci China,Ser.A,2000,43(1):13-21.
[34]Yin W.P.and Wang A.,The equivalence on classical metrics,Sci.China,Ser.A,2007,50(2):183-200.
[35]Yin W.P.,Wang A.and Zhao X.X.,Comparison theorem on Cartan-Hartogs domain of the first type,Sci.China,Ser.A,2001,44(5):587-598.
[36]Yin W.P.and Yin X.L.,A research into the numerical method of Dirichlet's problem of complex MongeAmpere equation on Cartan-Hartogs domain of the third type,J.Math.Anal.Appl.,2008,339(1):295-302.
[37]Yin W.P.and Yin X.L.,On the solution of Dirichlet's problem of the complex Monge-Ampere equation for the Cartan-Hartogs domain of the first type,Nonlinear Analysis,2008,69(7):2077-2085.
[38]Yin W.P.and Zhao X.X.,The Comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the third type,Complex Variables,2002,47(3):183-201.
[39]Yin X.L.and Zhao X.X.,The computations of Einstein-Kahler metric of Cartan-Hartogs domain,Sci.China,Ser.A,2005,48(S1):365-376.
[40]Zhao X.X.,Zhang L.Y.and Yin W.P.,Einstein-Kahler metric on Cartan-Hartogs domain of the second type,Progress in Natural Science,2004,14(3):201-212.
[2]Aubin,T.,Equations du type Monge-Ampere sur les varietes Khleriennes compactes,C.R.Acad.Sci.Paris Sir.A-B,1976,283:119-121(in French).
[3]Aubin,T.,Nonlinear Analysis on Manifolds,Monge-Ampere Equations,New York:Springer-Verlag,1982.
[4]Calabi,E.,The space of Khler metrics,In:Proc.Internat.Congress Math.(Amsterdam),1954,2:206-207.
[5]Calabi,E.,On Khler manifolds with vanishing canonical class,In:Algebraic Geometry and Topology,A Symposium in Honor of S.Lefschetz,Princeton:Princeton University Press,1957,78-89.
[6]Cheng,S.Y.and Yau,S.Y.,On the existence of a complete Kahler metric on non-compact complex manifolds and the regularity of Fefferman's equation,Commun,Pure Appl.Math.,1980,33(4):507-544.
[7]Hahn,K.T.,Inequality between the Bergman metric and Caratheodory differential metric,Proc.Amer.Math.Soc.,1978,68(2):193-194.
[8]Hahn,K.T.and Pflug,P.,The Kobayashi and Bergman metrics on generalized Thullen domains,Proc.Amer.Math.Soc.,1988,104(1):207-214.
[9]Heins,M.,On a class of conformal metrics,Nagoya Math.J.,1962,21:1-60.
[10]Kobayashi,S.,Intrinsic distances,measures and geometric function theory,Bull.Amer.Math.Soc,1976,82(3):357-416.
[11]Lempert,L.,Holomorphic retracts and intrinsic metrics in convex domains,Anal.Math.,1982,8(4):257-261.
[12]Lin P.and Yin W.P.,The comparison theorem on Cartan-Hartogs domain of the fourth type,Advances in Mathematics(China),2003,32(1):124-126.
[13]Liu K.F.,Sun X.F.and Yau,S.T.,Geometric aspects of the moduli space of Riemann surfaces,Sci.China,Ser.A,2005,48(S1):97-122.
[14]Look,K.H.,Schwarz lemma and analytic invariants,Scientia Sinica,1958,7(5):453-504.
[15]Look,K.H.,Classical Manifolds and Classical Domains,Beijing:Science Press,2011(in Chinese).
[16]Loos,O.,Jordan Pairs,Lecture Notes in Mathematics,Vol.460,New York:Springer-Verlag,1975.
[17]Mok,N.and Yau,S.T.,Completeness,of the Kahler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions,Proc.Symposia Pure Math.,1983,39:41-49.
[18]Schoen,R.and Yau,S.T.,Lectures on Differential Geometry,Cambridge,MA:International Press,1994.
[19]Shi J.H.,The Function Theory of Several Complex Variables,Beijing:Higher Education Press,1996(in Chinese).
[20]Tian G.,On Calabi's conjecture for complex surfaces with positive first Chern class,Invent.Math.,1990,101(1):101-172.
[21]Wang A.,The Einstein-Kahler metrics on Cartan-Hartogs domain of the first type,Sci.China,Ser.A,2004,47(2):220-235.
[22]Wang A.,Wang M.J.and Zhang L.Y.,Kahler-Einstein metric on Cartan-Hartogs domain,Advances in Mathematics(China),2009,38(4):493-502.
[23]Wang A.and Yin W.P.,The Einstein-Khler metric on Hua construction of the first type,Sci.China,Ser.A,2005,48(5):711-719.
[24]Wang A.and Yin W.P.,Einstein-Khler metric with explicit formula on super-Cartan domain of the fourth type,Acta Math.Sinica,Engl.Series,2006,22(2):367-376.
[25]Wang A.,Yin W.P.,Zhang L.Y.and Roos,G.,The Khler-Einstein metric for some Hartogs domains over symmetric domains,Sci.China,Ser.A,2006,49(9):1175-1210.
[26]Wang A.,Yin W.P.,Zhang L.Y.and Zhang W.J.,The Einstein-Khler metric with explicit formulas on some non-homogeneous domains,Asian J.Math.,2004,8(1):39-50.
[27]Wang A.,Yin W.P.and Zhang W.J.,Einstein-Khler metric on super-Cartan domain of the third type,Advances in Mathematics(China),2004,33(2):215-228(in Chinese).
[28]Wu,H.,Old and new invariant metrics on complex manifolds,several complex variables:In:Proceedings of the Mittag-LefBer Institute,1987-1988(J.E.,Fornaess ed.),Math.Notes,Vol.38,Princeton:Princeton Univ.Press,1993,640-682.
[29]Yau,S.T.,Calabi's conjecture and some new results in algebraic geometry,Proc.Nat.Acad.Sci.USA,1977,74(5):1798-1799.
[30]Yau,S.T.,On the Ricci curvature of a compact Khler manifold and the complex Monge-Ampere equation,Comm.Pure Appl.Math.,1978,31(3):339-411.
[31]Yin W.P.,Curvatures on a class of Reinhardt domains,Sci.China,Ser.A,1992,35(11):1281-1293.
[32]Yin W.P.,The comparison theorem for the Bergman and Kobayashi metrics on certain pseudoconvex domains,Complex Variables,1997,34(4):351-373.
[33]Yin W.P.,The Bergman kernels on super-Cartan domains of the first type,Sci China,Ser.A,2000,43(1):13-21.
[34]Yin W.P.and Wang A.,The equivalence on classical metrics,Sci.China,Ser.A,2007,50(2):183-200.
[35]Yin W.P.,Wang A.and Zhao X.X.,Comparison theorem on Cartan-Hartogs domain of the first type,Sci.China,Ser.A,2001,44(5):587-598.
[36]Yin W.P.and Yin X.L.,A research into the numerical method of Dirichlet's problem of complex MongeAmpere equation on Cartan-Hartogs domain of the third type,J.Math.Anal.Appl.,2008,339(1):295-302.
[37]Yin W.P.and Yin X.L.,On the solution of Dirichlet's problem of the complex Monge-Ampere equation for the Cartan-Hartogs domain of the first type,Nonlinear Analysis,2008,69(7):2077-2085.
[38]Yin W.P.and Zhao X.X.,The Comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the third type,Complex Variables,2002,47(3):183-201.
[39]Yin X.L.and Zhao X.X.,The computations of Einstein-Kahler metric of Cartan-Hartogs domain,Sci.China,Ser.A,2005,48(S1):365-376.
[40]Zhao X.X.,Zhang L.Y.and Yin W.P.,Einstein-Kahler metric on Cartan-Hartogs domain of the second type,Progress in Natural Science,2004,14(3):201-212.