de Sitter空间中类空子流形的拼挤问题
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摘要
本论文主要研究de Sitter空间中具有平行平均曲率向量、常数量曲率或第二基本形式模长平方是常数的三类类空子流形,并通过分别估计三种情形下子流形的第二基本形式模长平方的Laplace,利用Yau的极大值原理和Stokes定理,获得了这些子流形的一些拼挤定理和刚性定理。
论文共分为四节:
第一节是准备工作,介绍de Sitter空间中类空子流形的基本概念和基本公式。
第二节中,我们研究了de Sitter空间中具有平行平均曲率向量的类空子流形M~n,并证明了当M~n的第二基本形式模长平方||h||~2满足适当的上界时,M~n是全脐子流形(见定理2.1),同时获得了当M~n具有平坦法丛和正截面曲率时的一个刚性定理(详见定理2.2)。
第三节中,我们主要讨论了de Sitter空间中具有常数量曲率的类空子流形M~n,获得了关于M~n的第二基本形式模长平方的两个拼挤定理,具体见定理3.1和定理3.2。
第四节中,我们考虑了de Sitter空间中第二基本形式模长平方是常数的类空子流形M~n,并得到了M~n是全脐的三个充分条件,见定理4.1。
In this article,we mainly study the space-like submanifolds with parallel mean curva-ture vector,constant scalar curvature or constant square length of the second fundamental form,respectively.We obtain some pinching theorems and rigidity results by estimating Laplace of the square length of the second fundamental form of the such submanifolds and using yau's maximum principle or Stokes Theorem.
The article divided into four sections.
In section one,some basic concepts and associate formulas on space-like submanifolds in de Sitter space are presented.
In section two,we consider the space-like submanifold M~n with parallel mean curva-ture vector in de Sitter space S_p~(n+P)(c),and shows that M~n is totally umbilical if its square length of the second fundamental form ||h||~2 bounded from above(see Theorem 2.1).At the same time,we give a rigidity theorem of M~n with flat normal bundle and positive sectional curvature(see Theorem 2.2).
In section three,we study the space-like submanifold M~n with constant scalar curva-ture in de Sitter space S_p~(n+P)(c),and get two pinching theorems for M~n about the square length of the second fundamental form(see Theorem 3.1 and 3.2).
In section four,we discuss the space-like submanifold M~n with constant square norm of the second fundamental form in de Sitter space S_p~(n+P)(c),and obtain three sufficient conditions for M~n to be totally umbilical(see Theorem 4.1).
论文共分为四节:
第一节是准备工作,介绍de Sitter空间中类空子流形的基本概念和基本公式。
第二节中,我们研究了de Sitter空间中具有平行平均曲率向量的类空子流形M~n,并证明了当M~n的第二基本形式模长平方||h||~2满足适当的上界时,M~n是全脐子流形(见定理2.1),同时获得了当M~n具有平坦法丛和正截面曲率时的一个刚性定理(详见定理2.2)。
第三节中,我们主要讨论了de Sitter空间中具有常数量曲率的类空子流形M~n,获得了关于M~n的第二基本形式模长平方的两个拼挤定理,具体见定理3.1和定理3.2。
第四节中,我们考虑了de Sitter空间中第二基本形式模长平方是常数的类空子流形M~n,并得到了M~n是全脐的三个充分条件,见定理4.1。
In this article,we mainly study the space-like submanifolds with parallel mean curva-ture vector,constant scalar curvature or constant square length of the second fundamental form,respectively.We obtain some pinching theorems and rigidity results by estimating Laplace of the square length of the second fundamental form of the such submanifolds and using yau's maximum principle or Stokes Theorem.
The article divided into four sections.
In section one,some basic concepts and associate formulas on space-like submanifolds in de Sitter space are presented.
In section two,we consider the space-like submanifold M~n with parallel mean curva-ture vector in de Sitter space S_p~(n+P)(c),and shows that M~n is totally umbilical if its square length of the second fundamental form ||h||~2 bounded from above(see Theorem 2.1).At the same time,we give a rigidity theorem of M~n with flat normal bundle and positive sectional curvature(see Theorem 2.2).
In section three,we study the space-like submanifold M~n with constant scalar curva-ture in de Sitter space S_p~(n+P)(c),and get two pinching theorems for M~n about the square length of the second fundamental form(see Theorem 3.1 and 3.2).
In section four,we discuss the space-like submanifold M~n with constant square norm of the second fundamental form in de Sitter space S_p~(n+P)(c),and obtain three sufficient conditions for M~n to be totally umbilical(see Theorem 4.1).
引文
铩颷1]Akutagawa K.On space-like hypersurfaces with constant mean curvature in the deSitter space[J].Math.Z.,1987,196:13-19.
[2]Alencar H and do Carmo M.Hypersurfaces with constant mean curvature inspheres[J].Proc.Amer.Math.Soc.,1994,120:1223-1229.
[3]Cheng Q M.Complete space-like submanifolds in a de Sitter space with parallel meancurvature vector[J].Math.Z.,1991,206:333-339.
[4]Cheng Q M.Submanifolds with constant scalar curvature[J].Proc.Royal.Soc.Edin-bergh,2002,132:1163-1183.
[5]Cheng Q M and Ishikawa S.Space-like hypersurfaces with constant scalar curva-ture[J].Manuscripta Math.,1998,95:499-505.
[6]Cheng S Y,Yau S T.Uypersurfaces with constant scalar curvature[J].Math.Ann.,1977,225:195-204.
[7]Chern S S,do Carmo M and Koboyashi S.Minimal submanifolds of a sphere withsecond fundamental form of constant length[J].Fun.Ann.Rel.Fie.Conf.Chicago,1970:59-75.
[8]Choquei Bruhai Y,Fischer A E and Marsden J E.Maximal hypersurfaces and pos-itivity of mass[J].Proc.of the Enrico Fermi Summer School of the Italian PhysicalSociety,J.Ehlers,Ed.,North-Holland,1979.
[9]Goddard A J.Some remarks on the existence of space-like hypersurfaces of constantmean curvature[J].Math.Proc.Cambrige Phil.Soc.,1977,82:489-495.
[10]Hang KI U,Kim H J and Nakagawa H.On sapce-like hypersurfaces with constantmean curvature of a Lorentz space form[J].Tokyo J.of Math.,1991,14:205-216.
[11]纪永强.子流形几何[M]_北京:科学出版社,2004.
[12]Jr A B,Colares A G and Palmas O.Complete space-like hypersurfaces with constantmean curvature in the de Sitter spaces:A gap Theorem[J].Illinois J.Math.,2003,47:847-866.
[13]Liu X.Complete space-like hypersurfaces with constant scalar curvature[J].Manu-scripta Math.,2001,105:367-377.
[14]Liu X.Space-like hypersurfaces of constant scalar curvature in the de Sitter space[J].Atti.Sere.Mat.Fis.Univ.Modena.,2000,48:99-106.
[15]Liu X.Space-like submanifolds with constant scalar curvature[J].C.R.Acad.Sci.Paris,t.332,Sénè,Géométrie différentielle,2001,332:729-734.
铩颷16]Lumiste Ulo.Semi-parallel time-like surfaces in Lorentzian spacetime forms[J].Diff.Geom.and its Appl.,1997,7:59-74.
[17]Marsden J E,Tiple F J.Maximal hypersurfaces and foliations of constant meancurvature in general relativity[J].Phys.Rev.,1980,66:109-139.
[18]Montiel S.An integral inequality of compact sapce-like hypersurfaces in de Sitterspace and applications to the case of constant mean curvature[J].Indian Univ.Math.J.,1988,37:909-917.
[19]莫小欢.常曲率空间中具有平行平均曲率向量的子流形[J].数学年刊,1989,9A(5):530-540.
[20]Oliker V.A priori estimates of the principal curvatures of spacelike hypersurfaces inde Sitter space with applications to hypersurfaces in hyperbolic space[J].Amer.J.Math.,1992,114:605-626.
[21]Omori H.Isometric immersion of Riemannian manifolds[J].J.Math.Soc.Japan,1997,19:205-214.
[22]Remanathan J.Complete space-like hypersurfaces of constant mean curvature in thede Sitter space[J].Indian Univ.Math.J.,1987,36:349-359.
[23]Santos W.Submanifolds with parallel mean curvature vector in spheres[J].TohokuMath.J.,1994,46:403-415.
[24]舒世昌,刘三阳.de Sitter空间中具有平行平均曲率向量的完备类空子流形(Ⅱ)[J].应用数学,2002,15(3):76-80.
[25]Wang Q,Xia C.Rigidity theorems for closed hypersurfaces in a unit sphere[J].J.Geom.and Phy.,2005,55:227-240.
[26]吴传喜,李光汉.子流形几何[M].北京:科学出版社,2002.
[27]许志才.de Sitter空间中具有常平均曲率的类空超曲面[J].数学杂志,1998,4:466-468.
[28]许志才.子流形几何[M].合肥:中国科学技术大学出版社,2003.
[29]Yau S T.Harmonic functions on complete Riemannian manifolds[J].Comm.Pure andAppl.Math.,1975,28:201-228.
[30]Yau S T.Submanifolds with constant mean curvature(Ⅰ,Ⅱ)[J].Amer.J.Math.,1974,96:346-366;1975,97:76-100.
[31]张剑锋.关于常数数量曲率的子流形和Finsler流形上的调和函数.博士论文,浙江大学,2005.
[32]Zheng Y.On space-like hypersurfaces in the de Sitter spaces[J].Ann.of Global Anal.and Geom.,1995,13:317-321.
[33]Zheng Y.Space-like hypersurfaces with constant scalar curvature in the de Sitterspaces[J].Diff.Geom.Appl.,1996,6:51-54.
[2]Alencar H and do Carmo M.Hypersurfaces with constant mean curvature inspheres[J].Proc.Amer.Math.Soc.,1994,120:1223-1229.
[3]Cheng Q M.Complete space-like submanifolds in a de Sitter space with parallel meancurvature vector[J].Math.Z.,1991,206:333-339.
[4]Cheng Q M.Submanifolds with constant scalar curvature[J].Proc.Royal.Soc.Edin-bergh,2002,132:1163-1183.
[5]Cheng Q M and Ishikawa S.Space-like hypersurfaces with constant scalar curva-ture[J].Manuscripta Math.,1998,95:499-505.
[6]Cheng S Y,Yau S T.Uypersurfaces with constant scalar curvature[J].Math.Ann.,1977,225:195-204.
[7]Chern S S,do Carmo M and Koboyashi S.Minimal submanifolds of a sphere withsecond fundamental form of constant length[J].Fun.Ann.Rel.Fie.Conf.Chicago,1970:59-75.
[8]Choquei Bruhai Y,Fischer A E and Marsden J E.Maximal hypersurfaces and pos-itivity of mass[J].Proc.of the Enrico Fermi Summer School of the Italian PhysicalSociety,J.Ehlers,Ed.,North-Holland,1979.
[9]Goddard A J.Some remarks on the existence of space-like hypersurfaces of constantmean curvature[J].Math.Proc.Cambrige Phil.Soc.,1977,82:489-495.
[10]Hang KI U,Kim H J and Nakagawa H.On sapce-like hypersurfaces with constantmean curvature of a Lorentz space form[J].Tokyo J.of Math.,1991,14:205-216.
[11]纪永强.子流形几何[M]_北京:科学出版社,2004.
[12]Jr A B,Colares A G and Palmas O.Complete space-like hypersurfaces with constantmean curvature in the de Sitter spaces:A gap Theorem[J].Illinois J.Math.,2003,47:847-866.
[13]Liu X.Complete space-like hypersurfaces with constant scalar curvature[J].Manu-scripta Math.,2001,105:367-377.
[14]Liu X.Space-like hypersurfaces of constant scalar curvature in the de Sitter space[J].Atti.Sere.Mat.Fis.Univ.Modena.,2000,48:99-106.
[15]Liu X.Space-like submanifolds with constant scalar curvature[J].C.R.Acad.Sci.Paris,t.332,Sénè,Géométrie différentielle,2001,332:729-734.
铩颷16]Lumiste Ulo.Semi-parallel time-like surfaces in Lorentzian spacetime forms[J].Diff.Geom.and its Appl.,1997,7:59-74.
[17]Marsden J E,Tiple F J.Maximal hypersurfaces and foliations of constant meancurvature in general relativity[J].Phys.Rev.,1980,66:109-139.
[18]Montiel S.An integral inequality of compact sapce-like hypersurfaces in de Sitterspace and applications to the case of constant mean curvature[J].Indian Univ.Math.J.,1988,37:909-917.
[19]莫小欢.常曲率空间中具有平行平均曲率向量的子流形[J].数学年刊,1989,9A(5):530-540.
[20]Oliker V.A priori estimates of the principal curvatures of spacelike hypersurfaces inde Sitter space with applications to hypersurfaces in hyperbolic space[J].Amer.J.Math.,1992,114:605-626.
[21]Omori H.Isometric immersion of Riemannian manifolds[J].J.Math.Soc.Japan,1997,19:205-214.
[22]Remanathan J.Complete space-like hypersurfaces of constant mean curvature in thede Sitter space[J].Indian Univ.Math.J.,1987,36:349-359.
[23]Santos W.Submanifolds with parallel mean curvature vector in spheres[J].TohokuMath.J.,1994,46:403-415.
[24]舒世昌,刘三阳.de Sitter空间中具有平行平均曲率向量的完备类空子流形(Ⅱ)[J].应用数学,2002,15(3):76-80.
[25]Wang Q,Xia C.Rigidity theorems for closed hypersurfaces in a unit sphere[J].J.Geom.and Phy.,2005,55:227-240.
[26]吴传喜,李光汉.子流形几何[M].北京:科学出版社,2002.
[27]许志才.de Sitter空间中具有常平均曲率的类空超曲面[J].数学杂志,1998,4:466-468.
[28]许志才.子流形几何[M].合肥:中国科学技术大学出版社,2003.
[29]Yau S T.Harmonic functions on complete Riemannian manifolds[J].Comm.Pure andAppl.Math.,1975,28:201-228.
[30]Yau S T.Submanifolds with constant mean curvature(Ⅰ,Ⅱ)[J].Amer.J.Math.,1974,96:346-366;1975,97:76-100.
[31]张剑锋.关于常数数量曲率的子流形和Finsler流形上的调和函数.博士论文,浙江大学,2005.
[32]Zheng Y.On space-like hypersurfaces in the de Sitter spaces[J].Ann.of Global Anal.and Geom.,1995,13:317-321.
[33]Zheng Y.Space-like hypersurfaces with constant scalar curvature in the de Sitterspaces[J].Diff.Geom.Appl.,1996,6:51-54.