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Minimal surfaces in spheres and a Ricci-like condition

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Abstract

We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic \({\mathbb {S}}^{5}\) in the nearly Kähler sphere \({\mathbb {S}}^6\). Being locally isometric to a pseudoholomorphic curve in \({\mathbb {S}}^5\) turns out to be equivalent to the Ricci-like condition \(\Delta \log (1-K)=6K,\) where K is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in \({\mathbb {S}}^5\) do satisfy this Ricci-like condition. Surfaces in both classes are exceptional surfaces. These are minimal surfaces whose all Hopf differentials are holomorphic, or equivalently the curvature ellipses have constant eccentricity up to the last but one. Under appropriate global assumptions, we prove that minimal surfaces in spheres that satisfy this Ricci-like condition are indeed exceptional. Thus, the classification of these surfaces is reduced to the classification of exceptional surfaces that are locally isometric to a pseudoholomorphic curve in \({\mathbb {S}}^5.\) In fact, we prove, among other results, that such exceptional surfaces in odd dimensional spheres are flat or direct sums of surfaces in the associated family of a pseudoholomorphic curve in \({\mathbb {S}}^5\).

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

    Amalia-Sofia Tsouri & Theodoros Vlachos

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  1. Amalia-Sofia Tsouri
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  2. Theodoros Vlachos
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Correspondence to Amalia-Sofia Tsouri.

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The first named author would like to acknowledge financial support by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) Grant No: 133.

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Tsouri, AS., Vlachos, T. Minimal surfaces in spheres and a Ricci-like condition. manuscripta math. 166, 561–588 (2021). https://doi.org/10.1007/s00229-020-01254-7

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