球面中的Willmore环面的谱刻画
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摘要
等谱问题是微分几何中具有悠久历史同时也是十分重要的一个研究方向,该问题是:黎曼流形上的Laplace算子的谱是否决定它的几何?一般而言,这个问题的回答是否定的.因而该问题的研究就被分成两个部分:一是研究和刻画具有否定回答(等谱问题)的黎曼流形的例子和它们的性质;二是研究和刻画具有肯定回答(等谱问题)的黎曼流形的特征.本文中我们就单位球中的Willmore环面来研究它们的等谱性质.我们的结论如下:
设M是球面S~(n+1)(1)(n≥2)中闭Willmore超曲面,如果M满足下面两个条件:
(1)Spec~p(M)=Spec~p(W_(m,n-m)(P=0,1,2),其中W_(m,n-m)是S~(n+1)(1)中的Willmore环面;
(2)M与W_(m,n-m)有同样的常平均曲率.那么M=W_(m,n-m).
Isospectral problem is an important subject in geometry. The problem is: Does the geometry of a Riemannian manifold be decided by the spectrum of the Laplace operator of the manifold? In this paper we study the isospectral problem of Willmore hypersurfaces in (n+1)-dimensional sphere S~(n+1)(1). The conclusion is as follows:
Let M be a closed Willmore hypersurface in the sphere S(n+1)(1) (n≥2) with the same constant mean curvature of the Willmore torus W_(m,n-m)=S~(n-m)((?))×S~m((?)).If Spec~p(M)=Spec~p(W_(m,n-m)) (p= 0,1,2), then M=W_(m,n-m).
设M是球面S~(n+1)(1)(n≥2)中闭Willmore超曲面,如果M满足下面两个条件:
(1)Spec~p(M)=Spec~p(W_(m,n-m)(P=0,1,2),其中W_(m,n-m)是S~(n+1)(1)中的Willmore环面;
(2)M与W_(m,n-m)有同样的常平均曲率.那么M=W_(m,n-m).
Isospectral problem is an important subject in geometry. The problem is: Does the geometry of a Riemannian manifold be decided by the spectrum of the Laplace operator of the manifold? In this paper we study the isospectral problem of Willmore hypersurfaces in (n+1)-dimensional sphere S~(n+1)(1). The conclusion is as follows:
Let M be a closed Willmore hypersurface in the sphere S(n+1)(1) (n≥2) with the same constant mean curvature of the Willmore torus W_(m,n-m)=S~(n-m)((?))×S~m((?)).If Spec~p(M)=Spec~p(W_(m,n-m)) (p= 0,1,2), then M=W_(m,n-m).
引文
[1]Berger P,Gauduchon P et Mazet E,Le spactre d'une variete Remannienne,J.Lecture Notes in Math.194 (1971),Springer-Verlag.
[2]Chern S S,Do Carmo M and Kobayashi S,Minimal submanifolds of a sphere with second fundamental form of constant length,Functional Analysis and Related Fields,ed,by F.Browder,Berlin:Springer(1970),59-75.
[3]Chen Hongchong and Xu Hongwei,Global pinching theorems for Willmore hypersurface in a sphere,浙江大学2007年硕士论文．
[4]Ding Qing,On spectral characterizations of minimal hypersurface in a sphere,Kodai Math.J.17 (1994),320-328.
[5]Edmunds D E and Evans W D,Spectral theory and differential operator,Oxford University Press,1987.
[6]Hasegawa T,Spectral geometry of closed submanifolds in a space form,real or complex,Kodai Math.J.3 (1980),224-252.
[7]Ikeda A,On the spectrum of a Riemannian manifold of positive constant curvature 77,Osaka J.Math.17 (1980),691-762.
[8]Ikeda A,On spherical space forms which are isospectral but not isometric,J.Math.Soc.Japan.35 (1983),437-404.
[9]Ikeda A and Yoshihiko Yamamoto,On the spectra of 3-dimensional lens spaces,Osaka J.Math.16 (1979),447-469.
[10]Isaac Chavel,Eigenvalues in Riemannian geometry,Academic Press.INC.,1984.
[11]Li Haizhong,Willmore hypersurfaces in a sphere,Asian J.Math.5 (2001),365-377.
[12]Milnor J,Eigenvalues of Laplace operator of certain manifolds,Proc.Nat.Acad.Sci.USA.51 no.4 (1964),542.
[13]Tanaka M,compact Riemannian manifolds which are isospectral to three dimensional lens spaces I,in Minimal Submanifolds and Geodesies,Kaigai Publication,Tokyo,1978,273-282.
[14]Patodi U K,Curvature of the fundamental solution of the heat operator,J.Indian Math.Soc.34 (1970),269-285.
[15]Rosenberg S,The Laplacian on a Riemannian manifolds,Cambridge University Press,1997.
[16]Tanno S and Masuda K,A characterization of product of two 3-sphere by the spectrum,Kodai Math.J.8 (1985),420-429.
[17]Tanno S,Eigenvalues of the Laplacian of Riemannian manifolds,Tohoku Math.J.25 (1973),391-403.
[18] Tanno S, A characterization of the canonical spheres by the spectrum, Math.Zeit.175 (1980), 267-274.
[19] Witt E, Eine Identit(?)t zwischen Modulformen zweiten Grades, Abh.Math.Sem.Univ.Hamburg. 14(1956), 323-337.
[20] Yuri Egorov and Vladimir Kondratiev, On spectral theory of elliptic operators, Basel Boston BerlinBirkhauser, 1996.
[21]童裕孙,泛函分析教程,复旦大学出版社,2003.
[22]白正国,沈一兵,Riemann几何,高等教育出版社,2004.
[23]伍鸿熙,沈纯理,虞言林,黎曼几何初步,北京大学出版社,2003.
[24]丘成桐,孙理察,微分几何,科学出版社,1988.
[25]张恭庆,林源渠,泛函分析讲义,北京大学出版社,1986.
[2]Chern S S,Do Carmo M and Kobayashi S,Minimal submanifolds of a sphere with second fundamental form of constant length,Functional Analysis and Related Fields,ed,by F.Browder,Berlin:Springer(1970),59-75.
[3]Chen Hongchong and Xu Hongwei,Global pinching theorems for Willmore hypersurface in a sphere,浙江大学2007年硕士论文．
[4]Ding Qing,On spectral characterizations of minimal hypersurface in a sphere,Kodai Math.J.17 (1994),320-328.
[5]Edmunds D E and Evans W D,Spectral theory and differential operator,Oxford University Press,1987.
[6]Hasegawa T,Spectral geometry of closed submanifolds in a space form,real or complex,Kodai Math.J.3 (1980),224-252.
[7]Ikeda A,On the spectrum of a Riemannian manifold of positive constant curvature 77,Osaka J.Math.17 (1980),691-762.
[8]Ikeda A,On spherical space forms which are isospectral but not isometric,J.Math.Soc.Japan.35 (1983),437-404.
[9]Ikeda A and Yoshihiko Yamamoto,On the spectra of 3-dimensional lens spaces,Osaka J.Math.16 (1979),447-469.
[10]Isaac Chavel,Eigenvalues in Riemannian geometry,Academic Press.INC.,1984.
[11]Li Haizhong,Willmore hypersurfaces in a sphere,Asian J.Math.5 (2001),365-377.
[12]Milnor J,Eigenvalues of Laplace operator of certain manifolds,Proc.Nat.Acad.Sci.USA.51 no.4 (1964),542.
[13]Tanaka M,compact Riemannian manifolds which are isospectral to three dimensional lens spaces I,in Minimal Submanifolds and Geodesies,Kaigai Publication,Tokyo,1978,273-282.
[14]Patodi U K,Curvature of the fundamental solution of the heat operator,J.Indian Math.Soc.34 (1970),269-285.
[15]Rosenberg S,The Laplacian on a Riemannian manifolds,Cambridge University Press,1997.
[16]Tanno S and Masuda K,A characterization of product of two 3-sphere by the spectrum,Kodai Math.J.8 (1985),420-429.
[17]Tanno S,Eigenvalues of the Laplacian of Riemannian manifolds,Tohoku Math.J.25 (1973),391-403.
[18] Tanno S, A characterization of the canonical spheres by the spectrum, Math.Zeit.175 (1980), 267-274.
[19] Witt E, Eine Identit(?)t zwischen Modulformen zweiten Grades, Abh.Math.Sem.Univ.Hamburg. 14(1956), 323-337.
[20] Yuri Egorov and Vladimir Kondratiev, On spectral theory of elliptic operators, Basel Boston BerlinBirkhauser, 1996.
[21]童裕孙,泛函分析教程,复旦大学出版社,2003.
[22]白正国,沈一兵,Riemann几何,高等教育出版社,2004.
[23]伍鸿熙,沈纯理,虞言林,黎曼几何初步,北京大学出版社,2003.
[24]丘成桐,孙理察,微分几何,科学出版社,1988.
[25]张恭庆,林源渠,泛函分析讲义,北京大学出版社,1986.