乘积流形上两个存在性结果
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摘要
本文主要对于乘积流形上的预给定曲率的问题做了一些研究。我们主要研究了两个问题。第一个是在乘积单位球面上,我们给出自然的由乘积单位球面到高一维球面的嵌入映射,然后考虑了在乘积单位球面上预给定Gauss-Kronecker曲率后,我们所考虑的嵌入的存在性。我们得到了:当乘积单位球面不是由同维数的球面构成的话,在所谓PHC-域上,预给定光滑的曲率函数存在这类嵌入。第二个是在任意乘积流形上,引入了我们称为体积元保持变换的一种度量变换,我们研究了对于流形上预给定的数量曲率,是否存在体积元保持变换将流形上原有度量变为预给定数量曲率的度量。我们得到了:当流形是一个闭流形乘上较小的一段区间时存在这类变换或是一个闭流形乘上任意有限区间,但是预给定的数量曲率比较小的时候也存在这类变换。
The present report mainly studies prescribed curvature problems on product manifolds. We consider two problems.On the product of unit spheres,we give a kind of natural embeddings from the product unit spheres to the unit sphere in which the product of unit spheres can be viewed as a hypersurface.Now the first problem is:for a given positive function on the product of unit spheres,can we find an embedding of this kind such that its Gauss-Kronecker curvature is the given function.We obtain that on so-called PHC-domains,the existence is hold with the hypothesis that the product of unit spheres is not composed by the same dimensional unit sphere.On arbitrary product manifolds,we introduce a class of metric deformations which are called the volume element preserving deformation.Now the second problem is:for given scalar curvatures on some product manifold,can we find some volume element preserving deformation to satisfying that the scalar curvature of deformed metric coincides with the prescribed curvature.We obtain the existence in two cases.In the first case,the product manifold is a closed manifold timing a sufficiently small interval.In the second case, the product manifold is a closed manifold timing a finite interval,but the prescribed curvature should be sufficiently small.
The present report mainly studies prescribed curvature problems on product manifolds. We consider two problems.On the product of unit spheres,we give a kind of natural embeddings from the product unit spheres to the unit sphere in which the product of unit spheres can be viewed as a hypersurface.Now the first problem is:for a given positive function on the product of unit spheres,can we find an embedding of this kind such that its Gauss-Kronecker curvature is the given function.We obtain that on so-called PHC-domains,the existence is hold with the hypothesis that the product of unit spheres is not composed by the same dimensional unit sphere.On arbitrary product manifolds,we introduce a class of metric deformations which are called the volume element preserving deformation.Now the second problem is:for given scalar curvatures on some product manifold,can we find some volume element preserving deformation to satisfying that the scalar curvature of deformed metric coincides with the prescribed curvature.We obtain the existence in two cases.In the first case,the product manifold is a closed manifold timing a sufficiently small interval.In the second case, the product manifold is a closed manifold timing a finite interval,but the prescribed curvature should be sufficiently small.
引文
[CCL]S.S.Chern,W.H.Chen,K.S.Lam,Lectures on Differential Geometry,Series on University Mathematics Vol 1,World Scientific,1998.
[CL]陈维桓,李兴校编著,黎曼几何引论(上)(下),北京大学出版社,2002。
[CNS1]L.A.Caffarelli,L.Nirenberg,J.Spruck,The Dirichlet problem for nonlinear second order elliptic equation I.Monge-Ampere equation,Communications on Pure and Applied Mathematics ⅩⅩⅩⅥ(1984),pp.369-402.
[CNS2]L.A.Caffarelli,L.Nirenberg,J.Spruck,The Dirichlet problem for nonlinear second order elliptic equations,Ⅲ:Functions of the eigenvalues of the Hessian,Acta Mathematics 155,(1985) 6,pp.261-301.
[CNS3]L.A.Caffarelli,L.Nirenberg,J.Spruck,Nonlinear second order elliptic equations,Ⅳ:Starshaped compact Weingarden hypersurface,Current Topics in Partial Differentail Equation,Y.Ohya,K.Kasahara and N.Shikmakura,eds.,Kinokunize Co.,Tokyo,1986,pp.1-26.
[CNS4]L.A.Caffarelli,L.Nirenberg,J.Spruck,Nonlinear second order elliptic equations.Ⅴ.The Dirichlet problem for Weingarten hypersurfaces,Communications on Pure and Applied Mathematics 41(1988),41-70.
[CY]S.Y.Cheng,S.T.Yau On the regularity of the solution of the n-dimensional Minkowski problem,Communications on Pure and Applied Mathematics ⅩⅩⅨ,495-516,1976.
[EK1]A.El Kacimi Alaoui,Operateurs transversalement elliptiques sur un feuilletage riemannien et applications,Compositio Mathematica 73,57-106,1990.
[EK2]A.El Kacimi Alaoui,Stabilite des V-varietes kahleriennes,Lecture Notes in Mathematics 1345,111-123,Springer,1988.
[Fu] W. Fulton, Introduction to toric varieties, Princeton University Press, 1993.
[G] H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Mathematis- che Annalen, 146 (1962), pp. 331-368.
[Gu] B. Guan, The Dirichlet problem for Monge-Ampere equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Thanscations of the American Mathematical Society 350,4955-4971, 1998.
[GHS] J. Girbau, A. Haefliger, D. Sundararaman, On deformations of transversely holomorphic foliations, Journal fur die Reine und Angewandte Mathematik, 345 (1983), pp. 122-147.
[GS] B, Guan, J. Spruke, Boundary-value problems on S~n for surfaces of constant Gauss curvature , Annals of Mathematics 138(1993), pp. 601-624.
[GT] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[H] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equation, Mathematiques & Applications 26, Springer-Verlag, 1997.
[Hu] X. Huang, Generalized Clifford torus in S~(n+1) and prescribed mean curvature function, Chiness Annual of Mathematics, 12B, 73-79, 1991.
[HKM] T. J. R. Hughes, T. Kato, and J. E. Marsden, Well-Posed Quasi-Linear Second -Order Hyperbolic Systems with Application to Nonlinear Elastodynamics and General Relativity, Archive for Rational Mechanics and Analysis, 1977, 63: 273-294.
[Jo] F. John, Decayed Singularity Formation in Solution of Nonlinear Wave Equations in Higher Dimension, Communications on Pure and Applied Mathematics, 1976, 29: 649-681.
[Jos] J. Jost, Riemannian geometry and geometric analysis, Third edition, Universitext, Springer, 2002.
[Kl] S. Klainerman, Global Existence for Nonlinear Wave Equations, Communications on Pure and Applied Mathematics, 1980, Vol XXXIII: 43-401.
[Ma] X. Masa, Duality and minimality in Riemannian foliations, Commentarii Mathematici Helvetici, 67, 17-27,1992.
[Mo] P. Molino, Riemannian foliations, Progress in Mathematics 73, Birkhuser, Boston, 1988.
[N1] L. Nirenberg, The Wey1 and Minkowski problems in differential geometry in the large, Communications on Pure and Applied Mathematics, 6, 337-394 (1953).
[N2] L. Nirenberg, On Elliptic Partial Differential Equation, Annali della Scuola Normale Superiore di Pisa, 1959 , 13(3): 115-162.
[O] V. I. Oliker, Hypersurfaces in R~(n+1) with prescribed Gaussian curvature and related equation of Monge-Ampere type, Communications on Partial Differentail Equation, 9(1984), 807-838.
[Pf] M. J. Pflaum, Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, 1768. Springer-Verlag, Berlin, 2001.
[Rum] H. Rummler, Quelques notions simples en geometrie riemannienne et leurs applications aux feuilletages compacts, Commentarii Mathematici Helvetici, 54(1979), 224-239.
[S] M. Sanchez, Geodesic connectedness of semi-Riemannian manifolds, Nonlinear Analysis, 47(2001), 3085-3102.
[Sp]M,Spivak,A comprehensive introduction to differential geometry,Vol:Ⅰ-Ⅴ,Second Edition,Public or Perish inc.Berkeley,1979.
[ST]H.F.Smith and D.Tataru,Sharp Local Well-Posedness Results for the Nonlinear Wave Equation,Annals of Mathematics,2005,163:291-366.
[SY]R.Schoen,S.T.Yau,Lectures on Differential Geometry,Conference Proceedings and Lecture Notes in Geometry and Topology,Volume 1,International Press,1994.
[Ton]P.Tondeur,Geometry of foliations,Monographs in Mathematics,90.Birkh(a|¨)user Verlag,Basel,1997.
[Tru]N.S.Trndinger,On the Dirichlet problem for Hessian equations,Acta Mathematica,175(1995),151-164.
[X]辛杰,非线性弹性动力学方程组的外问题,复旦大学博士毕业论文,2004.
[Y]S.T.Yau,On the Ricci curvature of a compact K(a|¨)hler manifold and the complex Monge-Ampere equation I,Communications on Pure and Applied Mathematics ⅩⅩⅪ,339-411,1978.
[Z]F.Zheng,Complex differential geometry,Studies in Advanced Mathematics,Vol.18,International Press,2000.
[Zh]S.Zheng,Nonlinear Evolution Equations,Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 133,Chapman & Hall/CRC,2004.
[CL]陈维桓,李兴校编著,黎曼几何引论(上)(下),北京大学出版社,2002。
[CNS1]L.A.Caffarelli,L.Nirenberg,J.Spruck,The Dirichlet problem for nonlinear second order elliptic equation I.Monge-Ampere equation,Communications on Pure and Applied Mathematics ⅩⅩⅩⅥ(1984),pp.369-402.
[CNS2]L.A.Caffarelli,L.Nirenberg,J.Spruck,The Dirichlet problem for nonlinear second order elliptic equations,Ⅲ:Functions of the eigenvalues of the Hessian,Acta Mathematics 155,(1985) 6,pp.261-301.
[CNS3]L.A.Caffarelli,L.Nirenberg,J.Spruck,Nonlinear second order elliptic equations,Ⅳ:Starshaped compact Weingarden hypersurface,Current Topics in Partial Differentail Equation,Y.Ohya,K.Kasahara and N.Shikmakura,eds.,Kinokunize Co.,Tokyo,1986,pp.1-26.
[CNS4]L.A.Caffarelli,L.Nirenberg,J.Spruck,Nonlinear second order elliptic equations.Ⅴ.The Dirichlet problem for Weingarten hypersurfaces,Communications on Pure and Applied Mathematics 41(1988),41-70.
[CY]S.Y.Cheng,S.T.Yau On the regularity of the solution of the n-dimensional Minkowski problem,Communications on Pure and Applied Mathematics ⅩⅩⅨ,495-516,1976.
[EK1]A.El Kacimi Alaoui,Operateurs transversalement elliptiques sur un feuilletage riemannien et applications,Compositio Mathematica 73,57-106,1990.
[EK2]A.El Kacimi Alaoui,Stabilite des V-varietes kahleriennes,Lecture Notes in Mathematics 1345,111-123,Springer,1988.
[Fu] W. Fulton, Introduction to toric varieties, Princeton University Press, 1993.
[G] H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Mathematis- che Annalen, 146 (1962), pp. 331-368.
[Gu] B. Guan, The Dirichlet problem for Monge-Ampere equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Thanscations of the American Mathematical Society 350,4955-4971, 1998.
[GHS] J. Girbau, A. Haefliger, D. Sundararaman, On deformations of transversely holomorphic foliations, Journal fur die Reine und Angewandte Mathematik, 345 (1983), pp. 122-147.
[GS] B, Guan, J. Spruke, Boundary-value problems on S~n for surfaces of constant Gauss curvature , Annals of Mathematics 138(1993), pp. 601-624.
[GT] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[H] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equation, Mathematiques & Applications 26, Springer-Verlag, 1997.
[Hu] X. Huang, Generalized Clifford torus in S~(n+1) and prescribed mean curvature function, Chiness Annual of Mathematics, 12B, 73-79, 1991.
[HKM] T. J. R. Hughes, T. Kato, and J. E. Marsden, Well-Posed Quasi-Linear Second -Order Hyperbolic Systems with Application to Nonlinear Elastodynamics and General Relativity, Archive for Rational Mechanics and Analysis, 1977, 63: 273-294.
[Jo] F. John, Decayed Singularity Formation in Solution of Nonlinear Wave Equations in Higher Dimension, Communications on Pure and Applied Mathematics, 1976, 29: 649-681.
[Jos] J. Jost, Riemannian geometry and geometric analysis, Third edition, Universitext, Springer, 2002.
[Kl] S. Klainerman, Global Existence for Nonlinear Wave Equations, Communications on Pure and Applied Mathematics, 1980, Vol XXXIII: 43-401.
[Ma] X. Masa, Duality and minimality in Riemannian foliations, Commentarii Mathematici Helvetici, 67, 17-27,1992.
[Mo] P. Molino, Riemannian foliations, Progress in Mathematics 73, Birkhuser, Boston, 1988.
[N1] L. Nirenberg, The Wey1 and Minkowski problems in differential geometry in the large, Communications on Pure and Applied Mathematics, 6, 337-394 (1953).
[N2] L. Nirenberg, On Elliptic Partial Differential Equation, Annali della Scuola Normale Superiore di Pisa, 1959 , 13(3): 115-162.
[O] V. I. Oliker, Hypersurfaces in R~(n+1) with prescribed Gaussian curvature and related equation of Monge-Ampere type, Communications on Partial Differentail Equation, 9(1984), 807-838.
[Pf] M. J. Pflaum, Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, 1768. Springer-Verlag, Berlin, 2001.
[Rum] H. Rummler, Quelques notions simples en geometrie riemannienne et leurs applications aux feuilletages compacts, Commentarii Mathematici Helvetici, 54(1979), 224-239.
[S] M. Sanchez, Geodesic connectedness of semi-Riemannian manifolds, Nonlinear Analysis, 47(2001), 3085-3102.
[Sp]M,Spivak,A comprehensive introduction to differential geometry,Vol:Ⅰ-Ⅴ,Second Edition,Public or Perish inc.Berkeley,1979.
[ST]H.F.Smith and D.Tataru,Sharp Local Well-Posedness Results for the Nonlinear Wave Equation,Annals of Mathematics,2005,163:291-366.
[SY]R.Schoen,S.T.Yau,Lectures on Differential Geometry,Conference Proceedings and Lecture Notes in Geometry and Topology,Volume 1,International Press,1994.
[Ton]P.Tondeur,Geometry of foliations,Monographs in Mathematics,90.Birkh(a|¨)user Verlag,Basel,1997.
[Tru]N.S.Trndinger,On the Dirichlet problem for Hessian equations,Acta Mathematica,175(1995),151-164.
[X]辛杰,非线性弹性动力学方程组的外问题,复旦大学博士毕业论文,2004.
[Y]S.T.Yau,On the Ricci curvature of a compact K(a|¨)hler manifold and the complex Monge-Ampere equation I,Communications on Pure and Applied Mathematics ⅩⅩⅪ,339-411,1978.
[Z]F.Zheng,Complex differential geometry,Studies in Advanced Mathematics,Vol.18,International Press,2000.
[Zh]S.Zheng,Nonlinear Evolution Equations,Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 133,Chapman & Hall/CRC,2004.