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Journal Articles tohoku math. journal Year : 2015

Almost complex surfaces in the nearly Kähler S3 _ S3

Abstract

In this paper we initiate the study of almost complex surfaces in the nearly Kähler S3×S3. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler S3×S3. We also find a local correspondence between almost complex surfaces in the nearly Kähler S3×S3 and solutions of the general H-system equation introduced by Wente ([13]), thus obtaining a geometric interpretation of solutions of the general H-system equation. From this we deduce a correspondence between constant mean curvature surfaces in R3 and almost complex surfaces in the nearly Kähler S3×S3 with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we prove that almost complex topological 2-spheres in S3×S3 are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.

Dates and versions

hal-03233846 , version 1 (25-05-2021)

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John Bolton, Franki Dillen, Bart Dioos, Luc Vrancken. Almost complex surfaces in the nearly Kähler S3 _ S3. tohoku math. journal, 2015, 67, pp.1-17. ⟨10.2748/tmj/1429549576⟩. ⟨hal-03233846⟩
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