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Exceptional minimal surfaces in spheres

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Abstract

We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy (Rend Semin Mat Univ Padova, 79:185–202, 1998), we obtain a description of exceptional surfaces in terms of a set of absolute value type functions, the a-invariants, that determine the geometry of the higher order curvature ellipses and satisfy certain Ricci-type conditions. We show that the a-invariants determine these surfaces up to a multiparameter family of isometric minimal deformations, where the number of the parameters is precisely the number of non-vanishing Hopf differentials. We give applications to superconformal surfaces and pseudoholomorphic curves in the nearly Kähler sphere \({\mathbb{S}^{6}}\). Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show a relation to pseudoholomorphic curves in \({\mathbb{S}^{6}}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, 45110, Ioannina, Greece

    Theodoros Vlachos

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  1. Theodoros Vlachos
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Correspondence to Theodoros Vlachos.

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The author was supported by the Alexander von Humboldt Foundation.

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Vlachos, T. Exceptional minimal surfaces in spheres. manuscripta math. 150, 73–98 (2016). https://doi.org/10.1007/s00229-015-0792-0

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