欧氏空间R~（n+1）中n阶Willmore超曲面
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摘要
设维黎曼流形Mn到n+1维黎曼流形Nn+1的等距浸入超曲面.设为第i个平均曲率泛函Wn(x)=∫MQndM是共形不变的,称为x的n阶Willmore泛函,满足其极值条件的超曲面称为n阶Willmore超曲面,本文主要考虑了欧氏空间中的高阶Willmore超曲面,给出欧氏空间中n阶Willmore旋转超曲面的微分方程,然后具体给出欧氏空间中一类n阶Willmore超曲面.
Let x:M n→Nn+1 be an n-dimensional hypersurface immersed in an (n+1)-dimensional Riemannian manifold Nn+1, Letσi (O≤i≤n)be the ith mean curvature and
The functionalWn(x)=∫MQndM is a conformal invariant, and called nth Willmore functional of x, An extremal hypersurface of conformal invariant functional Wn (x) is called nth order Willmore hypersurface. In this paper we considered the higher order Willmore hypersurface in Euclidean space, We established the ordinary differential equation of nth order revolution Willmore hypersurface, and considered the solution of the O.D.E.. Then we give a classes of nth order Willmore hypersurface in Euclidean space.
Let x:M n→Nn+1 be an n-dimensional hypersurface immersed in an (n+1)-dimensional Riemannian manifold Nn+1, Letσi (O≤i≤n)be the ith mean curvature and
The functionalWn(x)=∫MQndM is a conformal invariant, and called nth Willmore functional of x, An extremal hypersurface of conformal invariant functional Wn (x) is called nth order Willmore hypersurface. In this paper we considered the higher order Willmore hypersurface in Euclidean space, We established the ordinary differential equation of nth order revolution Willmore hypersurface, and considered the solution of the O.D.E.. Then we give a classes of nth order Willmore hypersurface in Euclidean space.
引文
[1]M. A. Akivis and V. V. Goldberg. Conformal differential geometry and its generalizations[J]. Wiley, New York,1996.
[2]M. A. Akivis and V. V. Goldberg. A conformal differential invariant and the conformal rigedity of hypersurfaces [J]. Proc. Amer. Math. Soc.125 (1997),2415-2424.
[3]R. Bryant. A duality theorem for Willmore surface [J]. J. Differential Geom.20(1987),23-53.
[4]陈省身,陈维桓.微分几何讲义[M].北京：北京大学出版社.1983.
[5]陈维桓.微分流形初步[M].北京：高等教育出版社,1998.
[6]陈维桓,李兴校.黎曼几何引论[M].北京：北京大学出版社.2002
[7]Chen S. S., Do Carmo M. and Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length [J], in Functional Analysis and Related Fields(F. E. Browder, ed.) Springer-Verlag, New York (1970),59-75.
[8]B. Y. Chen. An invariant of conformal mappings [J]. Proc. Amer. Math. Soc.40, (1973)563-564.
[9]Zhen Guo. Willmore submanifolds in the unit sphere [J], Collect. Math.55,3(2004),279-287.
[10]Zhen Guo. Generalized willmore functions and related variation problems [J]. Diff. Geom. Appl.25(2007)543-551
[11]Zhen Guo. Higher order Willmore hypersurfaces in Euclidean Space [J]. Acta. Math. Sinica. English series.24,5(2008).
[12]Zhen Guo, H. Li and C. P. Wang. The second variation formula for Willmore submanifolds in Sn [J]. Results Math.40(2001),205-225.
[13]H. Li. Willmore hypersurfaces in a sphere [J].Asian J. of Math.5(2001),365-377.
[14]H. Li. Willmore submanifolds in a sphere [J]. Math. Res. Letters 9(2002),771-790.
[15]P. Li and S. T. Yau. A new conformal invariant and its application to Willmore conjecture and the fisrt eigenyalue of compact surface [J]. Invent. Math.69(1982),269-291.
[16]丘成桐,孙理察.微分几何讲义[M].北京：高等教育出版社,2004.
[17]R. Reilly. Extrinsic vigidity theorems for compact submanifolds of the sphere [J],J.Differential Geometry 4(1970)487-497.
[18]R. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space Forms [J], J. Differential Geometry 8(1973)465-477.
[19]R. Reilly. On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space [J], Comment.Math. Helvetici,53(1977)525-533.
[20]Willmore T. J. Note on embeded surfaces. Ann. stiit. Univ." AI. I. Cuza"Iasi Ia. Mat [J] 11,1965, 494-496.
[21]张家玲.子流形的广义Willmore泛函及其变分问题[D].昆明：云南师范大学.2006.
[2]M. A. Akivis and V. V. Goldberg. A conformal differential invariant and the conformal rigedity of hypersurfaces [J]. Proc. Amer. Math. Soc.125 (1997),2415-2424.
[3]R. Bryant. A duality theorem for Willmore surface [J]. J. Differential Geom.20(1987),23-53.
[4]陈省身,陈维桓.微分几何讲义[M].北京：北京大学出版社.1983.
[5]陈维桓.微分流形初步[M].北京：高等教育出版社,1998.
[6]陈维桓,李兴校.黎曼几何引论[M].北京：北京大学出版社.2002
[7]Chen S. S., Do Carmo M. and Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length [J], in Functional Analysis and Related Fields(F. E. Browder, ed.) Springer-Verlag, New York (1970),59-75.
[8]B. Y. Chen. An invariant of conformal mappings [J]. Proc. Amer. Math. Soc.40, (1973)563-564.
[9]Zhen Guo. Willmore submanifolds in the unit sphere [J], Collect. Math.55,3(2004),279-287.
[10]Zhen Guo. Generalized willmore functions and related variation problems [J]. Diff. Geom. Appl.25(2007)543-551
[11]Zhen Guo. Higher order Willmore hypersurfaces in Euclidean Space [J]. Acta. Math. Sinica. English series.24,5(2008).
[12]Zhen Guo, H. Li and C. P. Wang. The second variation formula for Willmore submanifolds in Sn [J]. Results Math.40(2001),205-225.
[13]H. Li. Willmore hypersurfaces in a sphere [J].Asian J. of Math.5(2001),365-377.
[14]H. Li. Willmore submanifolds in a sphere [J]. Math. Res. Letters 9(2002),771-790.
[15]P. Li and S. T. Yau. A new conformal invariant and its application to Willmore conjecture and the fisrt eigenyalue of compact surface [J]. Invent. Math.69(1982),269-291.
[16]丘成桐,孙理察.微分几何讲义[M].北京：高等教育出版社,2004.
[17]R. Reilly. Extrinsic vigidity theorems for compact submanifolds of the sphere [J],J.Differential Geometry 4(1970)487-497.
[18]R. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space Forms [J], J. Differential Geometry 8(1973)465-477.
[19]R. Reilly. On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space [J], Comment.Math. Helvetici,53(1977)525-533.
[20]Willmore T. J. Note on embeded surfaces. Ann. stiit. Univ." AI. I. Cuza"Iasi Ia. Mat [J] 11,1965, 494-496.
[21]张家玲.子流形的广义Willmore泛函及其变分问题[D].昆明：云南师范大学.2006.