关于常曲率空间形式中子流形的一些结果
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摘要
本文主要讨论常曲率空间形式中的子流形,得到以下结果:
1.对于单位球面中的紧致子流形,得到了第二基本形式模长平方S关于其上Laplacian算子第一非零特征值的SimonS型不等式;进而,在S为常数的假设下,得到S的下界估计式。
2.对于欧氏空间中的紧致子流形,得到了某些类子流形稳定流的非存在性结果。关于超曲面,我们在假设这些类子流形的主曲率,截面曲率或者第二基本形式模长平方分别满足某种条件下,证明了相应的非存在性定理。关于余维数大于1的情形,我们也证明了,如果子流形的第二基本形式模长平方满足某一拼挤条件,那么该子流形中不存在稳定流,而且这类子流形与欧氏球同胚。
3.对于双曲空间中的子流形,讨论了双曲空间中具有非正Ricci曲率超曲面的性质,得到了超曲面第二基本形式模长平方的一个下界。进而,得到了超曲面主曲率乘积的一个上界。
The aim of this paper is to deal with the submanifolds immersed into the space form with constant curvature and obtain the following results:
1. For the compact submanifolds of the standard Euclidean sphere, we shall give a Simons-type inequality involving the squared norm S of the second fundamental form h of M~n in terms of the first nonzero eigenvalueλ_1 of the Laplacian of M~n. Furthermore, if, in addition, S is a constant, we give the lower bound for S.
2. For the compact submanifolds of the Euclidean space, we shall prove the non-existence of stable currents for certain classes of them. For hypersurfaces, we prove the non-existence theorems under the assumptions about the principal curvature, sectional curvature or the square length of the second fundamental form respectively. For high-codimension, we also prove that there are no stable currents in submanifolds of the Euclidean space when the square length of the second fundamental form satisfies a pinching condition. As a result, such submanifolds are homeomorphic to the Euclidean sphere.
3. For the submanifolds of the hyperbolic space, we consider the hypersurfaces with non-positive Ricci curvature and give a lower bound for the square length of the second fundamental form of the hypersurfaces. Further, we obtain an upper bound for the product of the principal curvatures of the hypersurfaces.
1.对于单位球面中的紧致子流形,得到了第二基本形式模长平方S关于其上Laplacian算子第一非零特征值的SimonS型不等式;进而,在S为常数的假设下,得到S的下界估计式。
2.对于欧氏空间中的紧致子流形,得到了某些类子流形稳定流的非存在性结果。关于超曲面,我们在假设这些类子流形的主曲率,截面曲率或者第二基本形式模长平方分别满足某种条件下,证明了相应的非存在性定理。关于余维数大于1的情形,我们也证明了,如果子流形的第二基本形式模长平方满足某一拼挤条件,那么该子流形中不存在稳定流,而且这类子流形与欧氏球同胚。
3.对于双曲空间中的子流形,讨论了双曲空间中具有非正Ricci曲率超曲面的性质,得到了超曲面第二基本形式模长平方的一个下界。进而,得到了超曲面主曲率乘积的一个上界。
The aim of this paper is to deal with the submanifolds immersed into the space form with constant curvature and obtain the following results:
1. For the compact submanifolds of the standard Euclidean sphere, we shall give a Simons-type inequality involving the squared norm S of the second fundamental form h of M~n in terms of the first nonzero eigenvalueλ_1 of the Laplacian of M~n. Furthermore, if, in addition, S is a constant, we give the lower bound for S.
2. For the compact submanifolds of the Euclidean space, we shall prove the non-existence of stable currents for certain classes of them. For hypersurfaces, we prove the non-existence theorems under the assumptions about the principal curvature, sectional curvature or the square length of the second fundamental form respectively. For high-codimension, we also prove that there are no stable currents in submanifolds of the Euclidean space when the square length of the second fundamental form satisfies a pinching condition. As a result, such submanifolds are homeomorphic to the Euclidean sphere.
3. For the submanifolds of the hyperbolic space, we consider the hypersurfaces with non-positive Ricci curvature and give a lower bound for the square length of the second fundamental form of the hypersurfaces. Further, we obtain an upper bound for the product of the principal curvatures of the hypersurfaces.
引文
[1]Barbosa J.N.,Barros A.A lower bound for the norm of the second fundamental form of rninimal hypersurfaces of S~(n+1)[J].Arch.Math.,2003,81:478-484.
[2]Chen B.Y.,Total mean curvature and submanifolds of finite type[M].Singapore:World Scientific,1984.
[3]Chen B.Y.,Okuraura M.Scalar curvature,inequalities and submanifolds[J].Proc.Amer.Math.Soc.,1973,38:605-608.
[4]Cheng Q.M.Nonexistence of stable currents[J].Ann.Global Anal.Geom.,1995,13:197-205.
[5]Chern S.S.,do Carmo M.,Kobayashi S.Minimal submanifolds of a sphere with second fundamental form of constant length[J].Fun.Ana.Rel.Fie.,1970,59-75.
[6]Choi H.I.,Wang A.N.A first eigenvahle estimate for minimal hypersurfaces[J].J.Diff.Geom.,1983,18:559-562.
[7]EL Soufi A.,Ilias S.Une inegalité du type "Reilly" pour les sous-variétés de l'espace hyperbolique[J].Comm.Math.Helv.,1992,67:167-181.
[8]Federer H.,Fleming W.H.Normal and integral currents[J].Ann.Math.,1960,72:458-520.
[9]Hu Z.J.Complete hypersurfaces with constant mean curvature and non-negative sec-tional curvature[J].Proc.Amer.Math.Soc.,1995,123:2835-2840.
[10]Lawson H.B.,Simons J.On stable currents and their application to global problems in real and complex geometry[J].Ann.Math.,1973,98:427-450.
[11]Leung P.F.An estimate on the Ricci curvature of a submanifold and some applications [J].Proc.Amer.Math.Soc.,1992,114:1051-1061.
[12]Leung P.F.Minimal submanifolds in a sphere[J].Math.Z.,1983,183:75-83.
[13]Leung P.F.On a relation between the topology and the intrinsic and extrinsic geome-tries of a compact submanifold[J].Proc.Edinburgh Math.Soc.,1985,28:305-311.
[14]Li A.M.,Li J.M.An intrinsic rigidity theorem for miniraal submanifolds in a sphere [J].Arch.Math.,1992,58:582-594.
[15]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,1988,9A:530-540.
[16]Obata M.Certain conditions for a Riemannian manifold to be isometric with a sphere [J].J.Math.Soc.Japan,1962,14:333-340.
[17]Ohnita Y.Stable minimal submanifolds in compact rank one symmetric spaces[J].Tóhoku Math.J.,1986,38:199-217.
[18]Peng C.K.,Terng C.L.Minimal hypersurfaces of spheres with constant scalar curva-ture[J].Ann.Math.Stud.,1983,103:177-198.
[19]Reilly R.C.On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space[J].Comm.Math.Helv.,1976,52:525-533.
[20]Shen Y.B.On intrinsic rigidity for minimal submanifolds in a sphere[J].Sci.Sinica.,1989,32A:769-781.
[21]Shen Y.B.On submanifolds in Riemannian manifolds of constant curvature[C].DD2Symp.,1981,Shanghai-Hefei,China.
[22]Shen Y.B.Submanifolds with nonnegative sectional curvature[J].Chinese Ann.Math.,1984,5B:625-632.
[23]Simons J.Minimal varieties in Riemannian manifolds[J].Ann.Math.,1968,88:62-105.
[24]Sjerve D.Homology spheres which are covered by spheres[J].J.London Math.Soc.,1973,6:333-336.
[25]Takahashi T.Minimal immersions of Riemannian manifolds[J].J.Math.Soc.,1966,18:380-385.
[26]Vlachos W.The Ricci curvature of submanifolds and its applications[J].Quart.J.Math.,2004,55:225-230.
[27]Wang Q.L.,Xia C.Y.Rigidity theorems for closed hypersurfaces in a unit sphere[J].J.Geom.and Phy.,2005,55:227-240.
[28]Xin Y.L.An application of integral currents to the vanishing theorems[J].Sci.Sinica,1984,27A:233-241.
[29]Yano K.,Ishihara S.Submanifolds with parallel mean curvature[J].J.Diff.Geom.,1971,6:95-118.
[30]Yau S.T.Problem section of seminar on differential geometry at Tokyo[J].Ann.Math.Stud.,1982,102:669-706.
[31]丘成桐,孙理察.微分几何讲义[M].北京:高等教育出版社,2004.
[32]Zhang X.S.Non-existence of stable currents in hypersurfaces[J].Southeast Asian Bull.Math.,2000,23:147-154.
[2]Chen B.Y.,Total mean curvature and submanifolds of finite type[M].Singapore:World Scientific,1984.
[3]Chen B.Y.,Okuraura M.Scalar curvature,inequalities and submanifolds[J].Proc.Amer.Math.Soc.,1973,38:605-608.
[4]Cheng Q.M.Nonexistence of stable currents[J].Ann.Global Anal.Geom.,1995,13:197-205.
[5]Chern S.S.,do Carmo M.,Kobayashi S.Minimal submanifolds of a sphere with second fundamental form of constant length[J].Fun.Ana.Rel.Fie.,1970,59-75.
[6]Choi H.I.,Wang A.N.A first eigenvahle estimate for minimal hypersurfaces[J].J.Diff.Geom.,1983,18:559-562.
[7]EL Soufi A.,Ilias S.Une inegalité du type "Reilly" pour les sous-variétés de l'espace hyperbolique[J].Comm.Math.Helv.,1992,67:167-181.
[8]Federer H.,Fleming W.H.Normal and integral currents[J].Ann.Math.,1960,72:458-520.
[9]Hu Z.J.Complete hypersurfaces with constant mean curvature and non-negative sec-tional curvature[J].Proc.Amer.Math.Soc.,1995,123:2835-2840.
[10]Lawson H.B.,Simons J.On stable currents and their application to global problems in real and complex geometry[J].Ann.Math.,1973,98:427-450.
[11]Leung P.F.An estimate on the Ricci curvature of a submanifold and some applications [J].Proc.Amer.Math.Soc.,1992,114:1051-1061.
[12]Leung P.F.Minimal submanifolds in a sphere[J].Math.Z.,1983,183:75-83.
[13]Leung P.F.On a relation between the topology and the intrinsic and extrinsic geome-tries of a compact submanifold[J].Proc.Edinburgh Math.Soc.,1985,28:305-311.
[14]Li A.M.,Li J.M.An intrinsic rigidity theorem for miniraal submanifolds in a sphere [J].Arch.Math.,1992,58:582-594.
[15]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,1988,9A:530-540.
[16]Obata M.Certain conditions for a Riemannian manifold to be isometric with a sphere [J].J.Math.Soc.Japan,1962,14:333-340.
[17]Ohnita Y.Stable minimal submanifolds in compact rank one symmetric spaces[J].Tóhoku Math.J.,1986,38:199-217.
[18]Peng C.K.,Terng C.L.Minimal hypersurfaces of spheres with constant scalar curva-ture[J].Ann.Math.Stud.,1983,103:177-198.
[19]Reilly R.C.On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space[J].Comm.Math.Helv.,1976,52:525-533.
[20]Shen Y.B.On intrinsic rigidity for minimal submanifolds in a sphere[J].Sci.Sinica.,1989,32A:769-781.
[21]Shen Y.B.On submanifolds in Riemannian manifolds of constant curvature[C].DD2Symp.,1981,Shanghai-Hefei,China.
[22]Shen Y.B.Submanifolds with nonnegative sectional curvature[J].Chinese Ann.Math.,1984,5B:625-632.
[23]Simons J.Minimal varieties in Riemannian manifolds[J].Ann.Math.,1968,88:62-105.
[24]Sjerve D.Homology spheres which are covered by spheres[J].J.London Math.Soc.,1973,6:333-336.
[25]Takahashi T.Minimal immersions of Riemannian manifolds[J].J.Math.Soc.,1966,18:380-385.
[26]Vlachos W.The Ricci curvature of submanifolds and its applications[J].Quart.J.Math.,2004,55:225-230.
[27]Wang Q.L.,Xia C.Y.Rigidity theorems for closed hypersurfaces in a unit sphere[J].J.Geom.and Phy.,2005,55:227-240.
[28]Xin Y.L.An application of integral currents to the vanishing theorems[J].Sci.Sinica,1984,27A:233-241.
[29]Yano K.,Ishihara S.Submanifolds with parallel mean curvature[J].J.Diff.Geom.,1971,6:95-118.
[30]Yau S.T.Problem section of seminar on differential geometry at Tokyo[J].Ann.Math.Stud.,1982,102:669-706.
[31]丘成桐,孙理察.微分几何讲义[M].北京:高等教育出版社,2004.
[32]Zhang X.S.Non-existence of stable currents in hypersurfaces[J].Southeast Asian Bull.Math.,2000,23:147-154.