Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T02:04:25.463Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE WILLMORE FUNCTIONAL OF 2-TORI IN SOME PRODUCT RIEMANNIAN MANIFOLDS

Published online by Cambridge University Press:  30 March 2012

PENG WANG
PENG WANG*
Affiliation:
Department of Mathematics, Tongji University, Siping Road 1239, Shanghai, 200092, People's Republic of China. e-mail: netwangpeng@tongji.edu.cn, netwangpeng@hotmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the minimum of Willmore functional of torus in a Riemannian manifold N, especially for the case that N is a product manifold. We show that when N = S2 × S1, the minimum of W(T2) is 0, and when N = R2 × S1, there exists no torus having least Willmore functional. When N = H2(−c) × S1, and x = γ × S1, the minimum of W(x) is .

Keywords

MSC(2000): 53A3053B30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 517 - 528
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Arroyo, J., Barros, M. and Garay, O. J., Willmore-Chen tubes on homogeneous spaces in warped product spaces, Pacific. J. Math. 188 (2) (1999), 201–207.CrossRefGoogle Scholar
2.Barros, M., Free elasticae and Willmore tori in warped product spaces, Glasgow Math. J. 40 (1988), 263–270.Google Scholar
3.Barros, M., Willmore tori in non-standard three spheres, Math. Proc. Camb. Phil. Soc. 121 (1997), 321–324.CrossRefGoogle Scholar
4.Bauer, M. and Kuwert, E., Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 10 (2003), 553–576.CrossRefGoogle Scholar
5.Berdinsky, D. A. and Taimanov, I., Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional, Siberian Math. J. 48 (3) (2007), 395–407.CrossRefGoogle Scholar
6.Bonahon, F., Geometric structures on 3-manifolds, Handbook of geometric topology, 93-164, North-Holland, Amsterdam, 2002.Google Scholar
7.Chen, B. Y., Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. 10 (1974), 380–385.Google Scholar
8.Chen, B. Y., Total mean curvature and submanifolds of finite type (World Scientific, Singapore, 1984).CrossRefGoogle Scholar
9.Chern, S. S., Minimal submanifolds in a riemannian manifold (mimeograph- ed) (University of Kansas, Lawrence, 1968).Google Scholar
10.Hu, Z. J. and Li, H. L., Willmore submanifolds in riemannian manifolds, in Proceedings of the Workshop, Contem. Geom and Related Topics (World Scientific, 2002) 251–275.Google Scholar
11.Kusner, R., Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (2) (1989), 317–345.CrossRefGoogle Scholar
12.Langer, J. and Singer, D., The total squared curvature of closed curves, J. Diff. Geom. 20 (1984), 1–22.Google Scholar
13.Li, P. and Yau, S. T., A new conformal invariant and its applications to the Willmore conjecture and first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269–291.CrossRefGoogle Scholar
14.Mondino, A., Some results about the existence of critical points for the Willmore functional, Math. Z. 266 (3) (2009), 583–622.CrossRefGoogle Scholar
15.Mondino, A.. The Conformal Willmore Functional: a Perturbative Approach, 2010, arXiv: 1010.4151v1.Google Scholar
16.Montiel, S. and Urbano, F., A Willmore functional for compact surfaces in the complex projective plane, J. Reine Angew. Math. 546 (2002), 139–154.Google Scholar
17.O'Neill, B., Semi-riemannian geometry, Vol. 3 (Academic Press, New York London, 1983).Google Scholar
18.Palais, R. S., Critical point theory and the minimax principle, In Global Analysis. Proc. Sympos. Pure Math. 15 (1970), 185–212.CrossRefGoogle Scholar
19.Pinkall, U., Hopf tori in S 3, Invent. Math. 81 (2) (1985), 379–386.CrossRefGoogle Scholar
20.Simon, L., Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (2) (1993), 281–326.CrossRefGoogle Scholar
21.Souamand, R. and Toubiana, E., Totally umbilic surfaces in homogeneous 3-manifolds, Comm. Math. Helv. 84 (2009), 673–704.Google Scholar
22.Wang, C. P., Moebious geometry of submanifolds in Sn, manuscripta math. 96 (4) (1998), 517–534.CrossRefGoogle Scholar