×

On symmetric Willmore surfaces in spheres. II: The orientation reversing case. (English) Zbl 1436.53042

Summary: In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in \(S^4\) and derive a necessary condition for such (possibly branched) isotropic surfaces to descend to (possibly branched) maps from \(\mathbb{R} P^2\) to \(S^4\). The Veronese sphere and several other examples of non-branched Willmore immersions from \(\mathbb{R} P^2\) to \(S^4\) are derived as an illustration of the general theory. The Willmore immersions of \(\mathbb{R} P^2\), just mentioned and different from the Veronese sphere, are new to the authors’ best knowledge.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C43 Differential geometric aspects of harmonic maps
53A31 Differential geometry of submanifolds of Möbius space
53C35 Differential geometry of symmetric spaces
49Q10 Optimization of shapes other than minimal surfaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Brander, D., Björling’s problem and DPW for Willmore surfaces
[2] Bryant, R., Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differ. Geom., 17, 455-473 (1982) · Zbl 0498.53046
[3] Bryant, R., A duality theorem for Willmore surfaces, J. Differ. Geom., 20, 23-53 (1984) · Zbl 0555.53002
[4] Burstall, F. E.; Guest, M. A., Harmonic two-spheres in compact symmetric spaces, revisited, Math. Ann., 309, 541-572 (1997) · Zbl 0897.58012
[5] Burstall, F.; Pedit, F.; Pinkall, U., Schwarzian Derivatives and Flows of Surfaces, Contemporary Mathematics, vol. 308, 39-61 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1031.53026
[6] Dorfmeister, J.; Pedit, F.; Wu, H., Weierstrass type representation of harmonic maps into symmetric spaces, Commun. Anal. Geom., 6, 633-668 (1998) · Zbl 0932.58018
[7] Dorfmeister, J.; Wang, P., Willmore surfaces in spheres via loop groups I: generic cases and some examples
[8] Dorfmeister, J.; Wang, P., Willmore surfaces in spheres: the DPW approach via the conformal Gauss map, Abh. Math. Semin. Univ. Hamb., 89, 1, 77-103 (2019) · Zbl 1421.53015
[9] Dorfmeister, J.; Wang, P., On symmetric Willmore surfaces in spheres I: the orientation preserving case, Differ. Geom. Appl., 43, 102-129 (2015) · Zbl 1328.53012
[10] J. Dorfmeister, P. Wang, On equivariant Willmore surfaces in \(S^{n + 2}\), in preparation. · Zbl 1475.53021
[11] Ejiri, N., Equivariant minimal immersions of \(S^2\) into \(S^{2 m}(1)\), Trans. Am. Math. Soc., 297, 1, 105-124 (1986) · Zbl 0636.53066
[12] Ejiri, N., Willmore surfaces with a duality in \(S^n(1)\), Proc. Lond. Math. Soc. (3), 57, 2, 383-416 (1988) · Zbl 0671.53043
[13] Eujalance, E.; Javier Cirre, F.; Gamboa, J. M.; Gromadzki, G., Symmetries of Compact Riemann Surfaces (2010), LMN, Springer · Zbl 1208.30002
[14] Hélein, F., Willmore immersions and loop groups, J. Differ. Geom., 50, 2, 331-385 (1998) · Zbl 0938.53033
[15] Ishihara, T., Harmonic maps of nonorientable surfaces to four-dimensional manifolds, Tohoku Math. J., 45, 1, 1-12 (1993) · Zbl 0770.58006
[16] Ma, X., Willmore surfaces in \(S^n\): transforms and vanishing theorems (2005), Technische Universität Berlin, Dissertation · Zbl 1233.53003
[17] Wang, P., Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials, submitted for publication
[18] Wang, P., Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms, Tohoku Math. J. (2), 69, 1, 141-160 (2017) · Zbl 1368.53010
[19] P. Wang, A Weierstrass type representation of isotropic Willmore surfaces in \(S^4\), in preparation.
[20] Wu, H. Y., A simple way for determining the normalized potentials for harmonic maps, Ann. Glob. Anal. Geom., 17, 189-199 (1999) · Zbl 0954.58017
[21] Yang, K., Meromorphic functions on a compact Riemann surface and associated complete minimal surfaces, Proc. Am. Math. Soc., 105, 3, 706-711 (1989) · Zbl 0669.53007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.