关于共形平坦空间中的双调和超曲面的研究
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摘要
双调和映射是黎曼流形之间的双能量泛函的临界点映射.它是调和映射的推广.自姜国英在文[1]中给出双调和映射的方程以来,双调和映射的研究吸引了世界上越来越多的数学家,成为了一个有趣的研究领域.它的一个重要研究方向是双调和子流形(即子流形的包含映射是双调和映射).在欧氏空间、球面空间、双曲空间中得到了很多关于双调和子流形的构造方法和分类的结果.然而,下列猜想仍未解决:
Chen猜想:欧氏空间中的双调和子流形都是极小的.
广义的Chen猜想:非正截面曲率的黎曼流形中的双调和子流形都是极小的.
在本文,我们研究共形平坦空间中的双调和超曲面,主要工作包括以下几点:
首先,我们利用文[2]的黎曼流形中的双调和超曲面方程推导出了共形平坦空间中的双调和超曲面方程.
其次,我们给出了所得的方程在以下几个方面的应用:
(1)获得了共形平坦空间中的全脐双调和超曲面方程和常平均曲率双调和超曲面方程;
(2)将所得方程应用于球面空间和双曲空间,我们重新获得了R.Caddeo,S.Montaldo,C.Oniciuc用别的方法获得的常曲率空间中的双调和超曲面方程,也获得了:Sm((?))是Sm+1中的非平凡的双调和超曲面;
(3)确定了共形度量h =f~(-2)(z)(dx~2+dy~2+dz~2),使得(R~3, h)中的线性函数的图所决定的平面是非平凡的双调和超曲面;
(4)给出了S~m×R中的双调和超曲面方程,也获得了:S~(m-1)(?)×R是S~m×R中的非平凡的双调和超曲面和一些其它的结果.
Biharmonic maps are maps between Riemannian manifolds which are critical pointsof the bi-energy functional. They are generalizations of harmonic maps. Since thederivation of the biharmonic equations appeared in [1], the study of biharmonic mapshas attracted more and more attention of mathematicians in the world and has becomea fascinating area of research. One of the central topics in this area is the study of bi-harmonic submanifolds (i.e., those submanifolds whose inclusion maps are biharmonic).Many results about constructions and classification of submanifolds in Euclidean spaces,spheres, hyperbolic space forms have been obtained. However, the following conjecturesremain open:
Chen’s conjecture: any biharmonic submanifold in Euclidean space Rm is mini-mal.
The generalized Chen’s conjecture: any biharmonic submanifold in a space ofnon-positive sectional curvature is minimal.
In this thesis, we study biharmonic hypersurfaces in a conformally ?at space. Ourmain work includes the following:
First, by using the equation of biharmonic hypersurfaces in Riemannian manifoldsin [2], we deduced the equation of biharmonic hypersurfaces in a conformally ?at space.Next, we give applications of the equation in the following areas:
(1)we obtained the equation of constant mean curvature biharmonic hypersurfacesand totally umbilical biharmonic hypersurfaces in a conformally ?at space ;
(2)Using the equation in spheres and hyperbolic space forms, we recovered theequation of biharmonic hypersurfaces in constant curvature spaces which R. Caddeo,S. Montaldo, C.Oniciuc obtained by other means. We also recovered the biharmonichyperplane Sm(?) is a proper biharmonic hypersurface in Sm+1 ;
(3)Identified a family of conformally ?at metric h = f~(-2)(z)(dx~2 +dy~2 +dz~2), whichturn the graphs of a linear function in (R~3, h) into a proper biharmonic surface;
(4)We obtained the equation of biharmonic hypersurfaces in S~m×R. We alsoobtained S~(m-1)( ?)×R is a proper biharmonic hypersurface in S~m×R and some otherresults.
Chen猜想:欧氏空间中的双调和子流形都是极小的.
广义的Chen猜想:非正截面曲率的黎曼流形中的双调和子流形都是极小的.
在本文,我们研究共形平坦空间中的双调和超曲面,主要工作包括以下几点:
首先,我们利用文[2]的黎曼流形中的双调和超曲面方程推导出了共形平坦空间中的双调和超曲面方程.
其次,我们给出了所得的方程在以下几个方面的应用:
(1)获得了共形平坦空间中的全脐双调和超曲面方程和常平均曲率双调和超曲面方程;
(2)将所得方程应用于球面空间和双曲空间,我们重新获得了R.Caddeo,S.Montaldo,C.Oniciuc用别的方法获得的常曲率空间中的双调和超曲面方程,也获得了:Sm((?))是Sm+1中的非平凡的双调和超曲面;
(3)确定了共形度量h =f~(-2)(z)(dx~2+dy~2+dz~2),使得(R~3, h)中的线性函数的图所决定的平面是非平凡的双调和超曲面;
(4)给出了S~m×R中的双调和超曲面方程,也获得了:S~(m-1)(?)×R是S~m×R中的非平凡的双调和超曲面和一些其它的结果.
Biharmonic maps are maps between Riemannian manifolds which are critical pointsof the bi-energy functional. They are generalizations of harmonic maps. Since thederivation of the biharmonic equations appeared in [1], the study of biharmonic mapshas attracted more and more attention of mathematicians in the world and has becomea fascinating area of research. One of the central topics in this area is the study of bi-harmonic submanifolds (i.e., those submanifolds whose inclusion maps are biharmonic).Many results about constructions and classification of submanifolds in Euclidean spaces,spheres, hyperbolic space forms have been obtained. However, the following conjecturesremain open:
Chen’s conjecture: any biharmonic submanifold in Euclidean space Rm is mini-mal.
The generalized Chen’s conjecture: any biharmonic submanifold in a space ofnon-positive sectional curvature is minimal.
In this thesis, we study biharmonic hypersurfaces in a conformally ?at space. Ourmain work includes the following:
First, by using the equation of biharmonic hypersurfaces in Riemannian manifoldsin [2], we deduced the equation of biharmonic hypersurfaces in a conformally ?at space.Next, we give applications of the equation in the following areas:
(1)we obtained the equation of constant mean curvature biharmonic hypersurfacesand totally umbilical biharmonic hypersurfaces in a conformally ?at space ;
(2)Using the equation in spheres and hyperbolic space forms, we recovered theequation of biharmonic hypersurfaces in constant curvature spaces which R. Caddeo,S. Montaldo, C.Oniciuc obtained by other means. We also recovered the biharmonichyperplane Sm(?) is a proper biharmonic hypersurface in Sm+1 ;
(3)Identified a family of conformally ?at metric h = f~(-2)(z)(dx~2 +dy~2 +dz~2), whichturn the graphs of a linear function in (R~3, h) into a proper biharmonic surface;
(4)We obtained the equation of biharmonic hypersurfaces in S~m×R. We alsoobtained S~(m-1)( ?)×R is a proper biharmonic hypersurface in S~m×R and some otherresults.
引文
[1] G. Y. Jiang, 2-Harmonic maps and their first and second variational formulas, Chin.Ann. Math. Ser. A 7(1986) 389-402.
[2] Y-Lin Ou, Biharmonic hypersurfaces in Riemannian manifold, to appear in PacificJ.Math., 2010.
[3]陈维桓,李兴校,黎曼几何引论,北京:北京大学出版社,2002.
[4]纪永强,子流形几何,北京:科学出版社,2004.
[5]白正国,黎曼几何初步(修订版),北京:高等教育出版社,2006.
[6] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soo-chow J. Math. 17 (1991), no. 2, 169–188.
[7] L. P. Eisenhart, Riemannian geometry, Eighth printing, Princeton Landmarks in Math-ematics, Princeton University Press, Princeton, NJ, 1997.
[8] F. A. Ficken, The Riemannian and a?ne di?erential geometry of product-spaces, Ann.of Math. (2) 40, (1939). 892–913.
[9] R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J.Math. 130 (2002), 109–123.
[10] G. Y. Jiang, Some non-existence theorems of 2-harmonic isometric immersions intoEuclidean spaces , Chin. Ann. Math. Ser. 8A (1987) 376-383.
[11] B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998), no. 1, 167–185.
[12] I. Dimitri′c, Submanifolds of Em with harmonic mean curvature vector, Bull. Inst. Math.Acad. Sinica 20 (1992), no. 1, 53–65.
[13] T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms.Canad. Math. Bull. 51 (2008), no. 3, 448–459.
[14] A. Balmus, S. Montaldo and C. Oniciuc, Classification results for biharmonic submani-folds in spheres, Israel J. Math. 168 (2008), 201–220.
[15] R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of S3. Internat. J.Math. 12 (2001), no. 8, 867–876.
[16] G. Y. Jiang, 2-harmonic isometric immersions between Riemannian manifolds. ChineseAnn. Math. Ser. A 7 (1986), no. 2, 130–144.
[17] A. Balmuus, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spcae forms, Sym-posium Valenceiennes (2008), 25–32.
[18] Y-Lin Ou, Some constructions of biharmonic maps and Chen’s conjecture on biharmonichypersurfaces, Preprint, 2009, arXiv:0912.1141.
[19] G. Walschap, Metric structures in dioerential geometry. Graduate Texts in Mathematics,224. Springer-Verlag, New York, 2004.
[20] L. Habermann, Riemannian metrics of constant mass and moduli spaces of conformalstructures. Lecture Notes in Mathematics, 1743. Springer-Verlag, Berlin, 2000.
[21] Ye-Lin Ou and Liang Tang, The generalized Chen’s conjecture is false, Preprint, 2010.
[22] Ye-Lin Ou and Ze-Ping Wang, Some classifications on biharmonic surfaces and hyper-surfaces, Preprint, 2010.
[23] C. H. Gu, Conformally ?at spaces and solutions to Yang-Mills equations, Phys. Rev. D(3) 21 (1980), no. 4, 970–971.
[2] Y-Lin Ou, Biharmonic hypersurfaces in Riemannian manifold, to appear in PacificJ.Math., 2010.
[3]陈维桓,李兴校,黎曼几何引论,北京:北京大学出版社,2002.
[4]纪永强,子流形几何,北京:科学出版社,2004.
[5]白正国,黎曼几何初步(修订版),北京:高等教育出版社,2006.
[6] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soo-chow J. Math. 17 (1991), no. 2, 169–188.
[7] L. P. Eisenhart, Riemannian geometry, Eighth printing, Princeton Landmarks in Math-ematics, Princeton University Press, Princeton, NJ, 1997.
[8] F. A. Ficken, The Riemannian and a?ne di?erential geometry of product-spaces, Ann.of Math. (2) 40, (1939). 892–913.
[9] R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J.Math. 130 (2002), 109–123.
[10] G. Y. Jiang, Some non-existence theorems of 2-harmonic isometric immersions intoEuclidean spaces , Chin. Ann. Math. Ser. 8A (1987) 376-383.
[11] B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math. 52 (1998), no. 1, 167–185.
[12] I. Dimitri′c, Submanifolds of Em with harmonic mean curvature vector, Bull. Inst. Math.Acad. Sinica 20 (1992), no. 1, 53–65.
[13] T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms.Canad. Math. Bull. 51 (2008), no. 3, 448–459.
[14] A. Balmus, S. Montaldo and C. Oniciuc, Classification results for biharmonic submani-folds in spheres, Israel J. Math. 168 (2008), 201–220.
[15] R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of S3. Internat. J.Math. 12 (2001), no. 8, 867–876.
[16] G. Y. Jiang, 2-harmonic isometric immersions between Riemannian manifolds. ChineseAnn. Math. Ser. A 7 (1986), no. 2, 130–144.
[17] A. Balmuus, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spcae forms, Sym-posium Valenceiennes (2008), 25–32.
[18] Y-Lin Ou, Some constructions of biharmonic maps and Chen’s conjecture on biharmonichypersurfaces, Preprint, 2009, arXiv:0912.1141.
[19] G. Walschap, Metric structures in dioerential geometry. Graduate Texts in Mathematics,224. Springer-Verlag, New York, 2004.
[20] L. Habermann, Riemannian metrics of constant mass and moduli spaces of conformalstructures. Lecture Notes in Mathematics, 1743. Springer-Verlag, Berlin, 2000.
[21] Ye-Lin Ou and Liang Tang, The generalized Chen’s conjecture is false, Preprint, 2010.
[22] Ye-Lin Ou and Ze-Ping Wang, Some classifications on biharmonic surfaces and hyper-surfaces, Preprint, 2010.
[23] C. H. Gu, Conformally ?at spaces and solutions to Yang-Mills equations, Phys. Rev. D(3) 21 (1980), no. 4, 970–971.