Higher Order Willmore Revolution Hypersurfaes in Rn+1
Abstract
Let x : Mn→Nn 1 be an n-dimensional hypersurface immersed in an (n 1)-dimensional Riemannian manifold Nn 1. Let σi (0≤i≤n) be the ith mean curvature and Qn = , where Cin is binomial coefficient. The second author showed that functional Wn(x)=∫MQndM is a conformal invariant and gave the Euler-Lagrange equation. Wn is called the nth Willmore functional of x. A hypersurface is called the nth order Willmore hypersurfaces if it is a solution of the Euler-Lagrange equation. In this paper, we establish the ordinary differential equation of the nth order Willmore revolution hypersurfaces and present standard examples of the nth order Willmore hypersurfaces.References
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. 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.
5. B. Y. Chen: An invariant of conformal mappings [J]. Proc. Amer. Math. Soc. ,40, (1973) 563-564.
6. Zhen. Guo, Willmore submanifolds in the unit sphere [J],Collect. Math. 55, 3 (2004), 279-287.
7. Zhen. Guo, Generalized willmore functions and related variation problems [J]. Diff. Geom. Appl. 25 (2007) 543-
551
8. Zhen. Guo. Higher order Willmore hypersurfaces in Euclidean space [J]. Acta. Math. Sinica. English series. 24,5
(2008).
9. Zhen. Guo, H. Li and C. P. Wang: The second variation formula for Willmore submanifolds in Sn
[J]. Results
Math. 40 (2001), 205-225.
10. H. Li . Willmore hypersurfaces in a sphere [J]. Asian J. of Math. 5 (2001), 365-377.
11. H. Li . Willmore submanifolds in a sphere [J]. Math. Res. Letters 9 (2002), 771-790.
12. R. Reilly, Extrinsic, vigidity theorems for compact submanifolds of the sphere [J], J. Differential Geometry 4
(1970) 487-497.
13. 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.
14. R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms [J], J.
Differential Geometry 8 (1973) 465-477.
15. R. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space [J], Comment.
Math. Helvetici 53(1977)525-533.
16. Willmore. T. J. Note on embeded surfaces. Ann. stiit. Univ.†AI. I. Cuza†Iasi Ia. Mat [J]. 11, 1965, 494-496.
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