Minimal surfaces in spheres and a Ricci-like condition
- PDF / 403,143 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 73 Downloads / 180 Views
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Amalia-Sofia Tsouri · Theodoros Vlachos
Minimal surfaces in spheres and a Ricci-like condition Received: 11 January 2020 / Accepted: 30 October 2020 Abstract. We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic S5 in the nearly Kähler sphere S6 . Being locally isometric to a pseudoholomorphic curve in S5 turns out to be equivalent to the Ricci-like condition 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 S5 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 S5 . 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 S5 .
1. Introduction A fundamental problem in the theory of isometric immersions is the study of rigidity and deformability of a given isometric immersion. A particular aspect of this problem is to classify those minimal surfaces that are isometric to a given one. More precisely, the following question has been addressed by Lawson in [25]: Given a minimal surface f : M → Qnc in a n-dimensional space form of curvature c, what is the moduli space of all noncongruent minimal surfaces , any m, which are isometric to f . f˜ : M → Qn+m c Partial answers to this problem were provided by several authors. For instance, see [4,21,24,25,29,30,33–35,37]. 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. A.-S. Tsouri (B) · T. Vlachos: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece. e-mail: [email protected] T. Vlachos e-mail: [email protected] Mathematics Subject Classification: Primary 53A10 · Secondary 53C42
https://doi.org/10.1007/s00229-020-01254-7
A.-S. Tsouri, T. Vlachos
A classical result due to Ricci-Curbastro [32] asserts that the Gaussian curvature K ≤ 0 of any minimal surface in R3 satisfies the so-called Ricci condition log(−K ) = 4K , away from totally geodesic points, where is the Laplacian operator of the surface with respect to the induced metric ds 2 . This condition is equivalent to the flatness of the metric d sˆ 2 = (−K )1/2 ds 2 . Conversely (see [23]), a metric on a simply connected 2-dimensional
Data Loading...
