Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S ⁶

Abstract

In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S ⁶, which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S ⁶. This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in S⁶. This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S ⁶.

Keywords

Totally isotropic Willmore two-spheres / Normalized potential / Iwasawa decompositions

References

[1] Bernard Y, Rivière T. Singularity removability at branch points for Willmore surfaces. Pacific J. Math., 2013, 265(2): 257-311

[2] Bohle C. Constrained Willmore tori in the 4-sphere. J. Differ. Geom., 2010, 86(1): 71-132

[3] Brander D, Rossman W, Schmitt N. Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods. Adv. Math., 2010, 223(3): 949-986

[4] Bryant R. Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Diff. Geom., 1982, 17: 455-473

[5] Bryant R. A duality theorem for Willmore surfaces. J. Diff. Geom., 1984, 20: 23-53

[6] Bryant R. Surfaces in conformal geometry. Proceedings of Symposia in Pure Mathematics, 1988, 48: 227-240

[7] Burstall F, Ferus D, Leschke K Conformal Geometry of Surfaces in S ⁴ and Quaternions, 2002, Berlin: Springer-Verleg 1772

[8] Burstall F, Guest M. Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann., 1997, 309: 541-572

[9] Burstall F, Pedit F, Pinkall U. Schwarzian Derivatives and Flows of Surfaces, 2002, Providence, RI: Amer. Math. Soc. 39-61

[10] Burstall F, Rawnsley J H. Twistor Theory for Riemannian Symmetric Spaces: With Applications to Harmonic Maps of Riemann Surfaces, 1990, Berlin: Springer-Verleg 1424

[11] Calabi E. Minimal immersions of surfaces in Euclidean spheres. J. Diff. Geom., 1967, 1: 111-125

[12] Chen J, Li Y. Bubble tree of branched conformal immersions and applications to the Willmore functional. Amer. J. Math., 2014, 136(4): 1107-1154

[13] Clancey, K. and Gohberg, I., Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser, 1981.

[14] Correia N, Pacheco R. Harmonic maps of finite uniton number into G ₂. Math. Z., 2012, 271(1–2): 13-32

[15] Dorfmeister J, Pedit F, Wu H. Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom., 1998, 6: 633-668

[16] Dorfmeister, J. and Wang, P., Willmore surfaces in spheres via loop groups I: Generic cases and some examples, arXiv:1301.2756, 2013.

[17] Dorfmeister J F, Wang P. Willmore surfaces in spheres: The DPW approach via the conformal Gauss map. Abh. Math. Semin. Univ. Hambg, 2019, 89(1): 77-103

[18] Dorfmeister, J. and Wang, P., Harmonic maps of finite uniton type into non-compact inner symmetric spaces, arXiv:1305.2514, 2013.

[19] Guest, M., An update on Harmonic maps of finite uniton number, via the Zero Curvature Equation, Integrable Systems, Topology, and Physics, A Conference on Integrable Systems in Differential Geometry Contemp. Math., 309, M. Guest et al., eds., Amer. Math. Soc., Providence, R. I., 2002, 85–113.

[20] Ejiri N. Willmore surfaces with a duality in S ⁿ(1). Proc. London Math. Soc., 1988, 57(2): 383-416

[21] Ferreira M J, Simões B A, Wood J C. All harmonic 2-spheres in the unitary group, completely explicitly. Math. Z., 2010, 266(4): 953-978

[22] Hélein F. Willmore immersions and loop groups. J. Differ. Geom., 1998, 50: 331-385

[23] Kellersch, P., Eine Verallgemeinerung der Iwasawa Zerlegung in Loop Gruppen, volume 4 of DGDS, Differential Geometry-Dynamical Systems, Monographs, Geometry Balkan Press, Bucharest, 2004. Dissertation, Technische Universität München, 1999, http://vectron.mathem.pub.ro/dgds/mono/dgdsmono.htm.

[24] Kuwert E, Schätzle R. Branch points of Willmore surfaces. Duke Math. J., 2007, 138: 179-201

[25] Lamm T, Nguyen H T. Branched willmore spheres. J. Reine Angew. Math., 2015, 2015(701): 169-194

[26] Ma, X., Willmore Surfaces in S ⁿ: Transforms and Vanishing Theorems, Dissertation, TU Berlin, 2005.

[27] Ma X. Adjoint transforms of Willmore surfaces in S ⁿ. Manuscripta Math., 2006, 120: 163-179

[28] Montiel S. Willmore two spheres in the four-sphere. Trans. Amer. Math. Soc., 2000, 352(10): 4469-4486

[29] Musso E. Willmore surfaces in the four-sphere. Ann. Global Anal. Geom., 1990, 8(1): 21-41

[30] Pressley A, Segal G. Loop Groups, Oxford Mathematical Monographs, 1986, New York: Oxford Science Publications, The charendon Press, Oxford University Press

[31] Segal G. Loop groups and harmonic maps. LMS Lecture Notes Ser., 1989, 139: 153-164

[32] Svensson M, Wood J C. Filtrations, factorizations and explicit formulae for harmonic maps. Communications in Math. Phys., 2012, 310(1): 99-134

[33] Uhlenbeck K. Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom., 1989, 30: 1-50

[34] Wang, P., Willmore surfaces in spheres via loop groups II: A coarse classification of Willmore two-spheres by potentials, arXiv:1412.6737, 2014.

[35] Wood J C. Explicit construction and parametrization of harmonic two-spheres in the unitary group. Proc. Lond. Math. Soc., 1989, 58(3): 608-624

[36] Wu H Y. A simple way for determining the normalized potentials for harmonic maps. Ann. Global Anal. Geom., 1999, 17: 189-199