On Willmore surfaces in $S^n$ of flat normal bundle
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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 $S^6$.References
- Manuel Barros, Oscar J. Garay, and David A. Singer, 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, DOI 10.2748/tmj/1178224810
- Robert L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), no. 1, 23–53. MR 772125
- F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Conformal geometry of surfaces in $\textit {S}^4$ and quaternions, Lecture Notes in Mathematics, vol. 1772, Springer-Verlag, Berlin, 2002. MR 1887131, DOI 10.1007/b82935
- Francis Burstall, Franz Pedit, and Ulrich Pinkall, Schwarzian derivatives and flows of surfaces, Differential geometry and integrable systems (Tokyo, 2000) Contemp. Math., vol. 308, Amer. Math. Soc., Providence, RI, 2002, pp. 39–61. MR 1955628, DOI 10.1090/conm/308/05311
- J. L. Cabrerizo and M. Fernández, 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, DOI 10.1216/rmjm/1021477244
- Dorfmeister, J., Wang, P. Willmore surfaces in $S^{n+2}$ via loop group methods I: Generic cases and some examples, arXiv:1301.2756.
- Norio Ejiri, Willmore surfaces with a duality in $S^N(1)$, Proc. London Math. Soc. (3) 57 (1988), no. 2, 383–416. MR 950596, DOI 10.1112/plms/s3-57.2.383
- Dirk Ferus and Franz Pedit, $S^1$-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, DOI 10.1007/BF02570873
- Frédéric Hélein, Willmore immersions and loop groups, J. Differential Geom. 50 (1998), no. 2, 331–385. MR 1684984
- Udo Hertrich-Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. MR 2004958, DOI 10.1017/CBO9780511546693
- K. Leschke, F. Pedit, and U. Pinkall, Willmore tori in the 4-sphere with nontrivial normal bundle, Math. Ann. 332 (2005), no. 2, 381–394. MR 2178067, DOI 10.1007/s00208-005-0630-x
- Haizhong Li and Luc Vrancken, New examples of Willmore surfaces in $S^n$, Ann. Global Anal. Geom. 23 (2003), no. 3, 205–225. MR 1966845, DOI 10.1023/A:1022825513863
- Xiang Ma, Adjoint transform of Willmore surfaces in $\Bbb S^n$, Manuscripta Math. 120 (2006), no. 2, 163–179. MR 2234246, DOI 10.1007/s00229-006-0635-0
- Ma, X. Willmore surfaces in $S^{n}$: Transforms and vanishing theorems, dissertation, Technischen Universität Berlin, 2005.
- Xiang Ma and Changping Wang, Willmore surfaces of constant Möbius curvature, Ann. Global Anal. Geom. 32 (2007), no. 3, 297–310. MR 2336179, DOI 10.1007/s10455-007-9065-9
- Xiang Ma and Peng Wang, Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, Sci. China Ser. A 51 (2008), no. 9, 1561–1576. MR 2426054, DOI 10.1007/s11425-008-0052-9
- Sebastián Montiel, Willmore two-spheres in the four-sphere, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4469–4486. MR 1695032, DOI 10.1090/S0002-9947-00-02571-X
- Emilio Musso, Willmore surfaces in the four-sphere, Ann. Global Anal. Geom. 8 (1990), no. 1, 21–41. MR 1075237, DOI 10.1007/BF00055016
- U. Pinkall, Hopf tori in $S^3$, Invent. Math. 81 (1985), no. 2, 379–386. MR 799274, DOI 10.1007/BF01389060
- Marco Rigoli, The conformal Gauss map of submanifolds of the Möbius space, Ann. Global Anal. Geom. 5 (1987), no. 2, 97–116. MR 944775, DOI 10.1007/BF00127853
- Changping Wang, Moebius geometry of submanifolds in $S^n$, Manuscripta Math. 96 (1998), no. 4, 517–534. MR 1639852, DOI 10.1007/s002290050080
- Peng Wang, Generalized polar transforms of spacelike isothermic surfaces, J. Geom. Phys. 62 (2012), no. 2, 403–411. MR 2864488, DOI 10.1016/j.geomphys.2011.11.003
- Qiao Ling Xia and Yi Bing Shen, Weierstrass type representation of Willmore surfaces in $S^n$, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 1029–1046. MR 2130369, DOI 10.1007/s10114-004-0389-0
- Kichoon Yang, Homogeneous minimal surfaces in Euclidean spheres with flat normal connections, Proc. Amer. Math. Soc. 94 (1985), no. 1, 119–122. MR 781068, DOI 10.1090/S0002-9939-1985-0781068-1
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3068977