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On Willmore surfaces in $ S^n$ of flat normal bundle


Author: Peng Wang
Journal: Proc. Amer. Math. Soc. 141 (2013), 3245-3255
MSC (2010): Primary 53A30, 53A07, 53B30
DOI: https://doi.org/10.1090/S0002-9939-2013-11683-7
Published electronically: May 16, 2013
MathSciNet review: 3068977
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Abstract: We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $ S^n$ must be located in some $ S^3\subset S^n$, from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in $ S^n$ with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in $ S^n$ is conformal to a minimal surface with embedded planer ends in $ \mathbb{R}^3$. We also point out that for a class of Willmore tori, they have a flat normal bundle if and only if they are located in some $ S^3$. In the end, we show that a Willmore surface with flat normal bundle must locate in some $ S^6$.


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  • 1. Barros, M., Garay, J., Singer, D. A. Elasticae with constant slant in the complex projective plane and new examples of Willmore tori in five spheres, Tohoku Math. J. (2) 51 (1999), no. 2, 177-192. MR 1690015 (2000d:53083)
  • 2. Bryant, R. A duality theorem for Willmore surfaces, J. Diff. Geom. 20 (1984), 23-53. MR 772125 (86j:58029)
  • 3. Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U. Conformal geometry of surfaces in $ S^{4}$ and quaternions, LNM 1772. Springer, Berlin, 2002. MR 1887131 (2004a:53058)
  • 4. Burstall, F., Pedit, F., Pinkall, U. Schwarzian derivatives and flows of surfaces, Contemporary Mathematics 308, 39-61, Providence, RI: Amer. Math. Soc., 2002. MR 1955628 (2004f:53010)
  • 5. Cabrerizo, J. L., Fernández, M. Willmore tori in a wide family of conformal structures on odd dimensional spheres, Rocky Mountain J. Math. 30 (2000), no. 3, 815-822. MR 1797815 (2001k:53101)
  • 6. Dorfmeister, J., Wang, P. Willmore surfaces in $ S^{n+2}$ via loop group methods I: Generic cases and some examples, arXiv:1301.2756.
  • 7. Ejiri, N. Willmore surfaces with a duality in $ S^{n}(1)$, Proc. London Math. Soc. (3), 57(2) (1988), 383-416. MR 950596 (89h:53117)
  • 8. Ferus, D., Pedit, F. $ S^1$-equivariant minimal tori in $ S^4$ and $ S^1$-equivariant Willmore tori in $ S^3$, Math. Z. 204 (1990), no. 2, 269-282. MR 1055991 (91g:53009)
  • 9. Hélein, F. Willmore immersions and loop groups, J. Differ. Geom. 50 (1998), 331-385. MR 1684984 (2000c:53076)
  • 10. Hertrich-Jeromin, U. Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series 300, Cambridge University Press, Cambridge, 2003. MR 2004958 (2004g:53001)
  • 11. Leschke, K., Pedit, F., Pinkall, U. Willmore tori in the $ 4$-sphere with nontrivial normal bundle, Math. Ann. 332, 381-394 (2005). MR 2178067 (2008c:53058)
  • 12. Li, H., Vrancken, L. New examples of Willmore surfaces in $ S^n$, Ann. Global Anal. Geom. 23 (2003), no. 3, 205-225. MR 1966845 (2004b:53105)
  • 13. Ma, X. Adjoint transforms of Willmore surfaces in $ S^{n}$, Manuscripta Math. 120 (2006), 163-179. MR 2234246 (2007k:53083)
  • 14. Ma, X. Willmore surfaces in $ S^{n}$: Transforms and vanishing theorems, dissertation, Technischen Universität Berlin, 2005.
  • 15. Ma, X., Wang, C. P. Willmore surfaces of constant Möbius curvature, Ann. Global Anal. Geom. 32 (2007), no. 3, 297-310. MR 2336179 (2008h:53113)
  • 16. Ma, X., Wang, P. Spacelike Willmore surfaces in $ 4$-dimensional Lorentzian space forms, Sci. in China: Ser. A, Math. 51(9) (2008), 1561-1576. MR 2426054 (2009h:53122)
  • 17. Montiel, S. Willmore two spheres in the four-sphere, Trans. Amer. Math. Soc. 352(10) (2000), 4469-4486. MR 1695032 (2001b:53080)
  • 18. Musso, E. Willmore surfaces in the four-sphere, Ann. Global Anal. Geom. 8(1) (1990), 21-41. MR 1075237 (92g:53059)
  • 19. Pinkall, U. Hopf tori in $ S^3$, Invent. Math. 81(2) (1985), 379-386. MR 799274 (86k:53075)
  • 20. Rigoli, M. The conformal Gauss map of submanifolds of the Möbius space, Ann. Global Anal. Geom. 5(2) (1987), 97-116. MR 944775 (89e:53083)
  • 21. Wang, C. P. Moebious geometry of submanifolds in $ S^{n}$, Manuscripta Math. 96 (1998), no. 4, 517-534. MR 1639852 (2000a:53019)
  • 22. Wang, P. Generalized polar transforms of spacelike isothermic surfaces, J. Geom. Phys. 62 (2012), no. 2, 403-411. MR 2864488
  • 23. Xia, Q. L., Shen, Y. B. Weierstrass type representation of Willmore surfaces in $ S^n$, Acta Math. Sinica 20(6) (2004), 1029-1046. MR 2130369 (2005m:53111)
  • 24. Yang, K. Homogeneous minimal surfaces in Euclidean spheres with flat normal connections, Proc. AMS 94 (1985), no. 1, 119-122. MR 781068 (86e:53046)

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Additional Information

Peng Wang
Affiliation: Department of Mathematics, Tongji University, Siping Road 1239, Shanghai, 200092, People’s Republic of China
Email: netwangpeng@tongji.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11683-7
Keywords: Willmore surfaces, S-Willmore surfaces, Willmore sphere, Clifford torus, flat normal bundle
Received by editor(s): November 22, 2011
Published electronically: May 16, 2013
Additional Notes: This work was supported by the Program for Young Excellent Talents in Tongji University, the Tianyuan Foundation of China, grant 10926112, and Project 10901006 of NSFC
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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