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Sequences of Willmore surfaces

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Abstract

In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the Twistor projection of a holomorphic curve into \({\mathbb{C}}{\mathbb{P}}^3\) or the inversion of a minimal surface with planar ends in \({\mathbb{R}}^4\). These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors.

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References

  1. Bobenko A.I. (1991). All constant mean curvature tori in R 3, S 3, H 3 in terms of theta-functions. Math. Ann. 290: 209–245

    Article  MATH  MathSciNet  Google Scholar 

  2. Bolton J., Jensen G., Rigoli M. and Woodward L. (1988). On conformal minimal immersions of S 2 into CP n. Math. Ann. 279: 599–620

    Article  MATH  MathSciNet  Google Scholar 

  3. Bryant R.L. (1984). A duality theorem for Willmore surfaces. J. Diff. Geom. 20: 23–53

    MATH  Google Scholar 

  4. Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal Geometry of Surfaces in S 4 and Quaternions. Lecture Notes in Mathematics, Springer, Berlin (2002)

  5. Din A.M. and Zakrzewski W.J. (1980). General classical solutions in the CP n-1 model. Nuclear Phys. B 174(2–3): 397–406

    Article  MathSciNet  Google Scholar 

  6. Eells J. and Wood J.C. (1983). Harmonic maps from surfaces into projective spaces. Adv. Math. 49: 217–263

    Article  MATH  MathSciNet  Google Scholar 

  7. Ejiri N. (1988). Willmore surfaces with a duality in S n(1). Proc. Lond. Math. Soc., III Ser. 57(2): 383–416

    Article  MATH  MathSciNet  Google Scholar 

  8. Glaser, V., Stora, R.: Regular solutions of the CP n models and further generalizations. CERN preprint (1980)

  9. Hitchin N. (1990). Harmonic maps from a 2-torus to the 3-sphere. J. Diff. Geom. 31(3): 627–710

    MATH  MathSciNet  Google Scholar 

  10. Leschke, K.: Transformations on Willmore surfaces. Habilitationsschrift (2006)

  11. Leschke K. and Pedit F. (2005). Bäcklund transforms of conformal maps into the 4–sphere. Banach Center Publications 69: 103–118

    Article  MathSciNet  Google Scholar 

  12. Leschke K. and Pedit F. (2007). Envelopes and Osculates of Willmore surfaces. J. Lond. Math. Soc. (2) 75: 199–212

    Article  MATH  MathSciNet  Google Scholar 

  13. Leschke K., Pedit F. and Pinkall U. (2005). Willmore tori with non-trivial normal bundle. Math. Ann. 332(2): 381–394

    Article  MATH  MathSciNet  Google Scholar 

  14. Montiel S. (2000). Willmore two-spheres in the four sphere. Trans. Amer. Math. Soc. 352: 4449–4486

    Article  MathSciNet  Google Scholar 

  15. Pinkall U. and Sterling I. (1989). On the classification of constant mean curvature tori. Ann. Math. 130: 407–451

    Article  MathSciNet  Google Scholar 

  16. Uhlenbeck K. (1989). Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30: 1–50

    MATH  MathSciNet  Google Scholar 

  17. Wolfson J.G. (1988). Harmonic sequences and harmonic maps of surfaces into complex Grassman manifolds. J. Diff. Geom. 27: 161–178

    MATH  MathSciNet  Google Scholar 

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Correspondence to Katrin Leschke.

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K. Leschke thanks the Department of Mathematics and Statistics at the University of Massachusetts, Amherst, and the Center for Geometry, Analysis, Numerics and Graphics for their support and hospitality.

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Leschke, K., Pedit, F. Sequences of Willmore surfaces. Math. Z. 259, 113–122 (2008). https://doi.org/10.1007/s00209-007-0214-4

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  • DOI: https://doi.org/10.1007/s00209-007-0214-4

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