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© 2018, Springer International Publishing AG, part of Springer Nature. By employing a nice adapted frame we prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kähler manifold S3× S3. The proof uses a local correspondence between almost complex surfaces in S3× S3 and surfaces in R3 that satisfy the Wente H-surface equation. Furthermore we give a complete classification of flat almost complex surfaces in the homogeneous nearly Kähler S3× S3. ispartof: Results in Mathematics vol:73 issue:1 status: published

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S3 × S3. ispartof: International Journal of Geometric Methods in Modern Physics vol:12 issue:8 pages:1560012-1560017 ispartof: location:SPAIN, Univ Granada, Granada status: published

In this paper we initiate the study of almost complex surfaces in the nearly Kahler $S^3\times S^3$. 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 Kahler $S^3\times S^3$. We also find a local correspondence between almost complex surfaces in the nearly Kahler $S^3\times S^3$ 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 $\mathbb R^3$ and almost complex surfaces in the nearly Kahler $S^3\times S^3$ 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 $S^3\times S^3$ are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.

International audience; In this talk we show how to study almost complex submanifolds of S3 × S3. Compact 6-dimensional non-K¨ahler nearly K¨ahler manifolds do not admit 4-dimensional almost complex submanifolds, so the almost complex submanifolds are surfaces. Previously, a classification of the totally geodesic almost complex surfaces has been obtained as well as the result that an almost complex immersion of a topological 2 sphere is necessary totally geodesic. In this lecture we will show how to obtain a classification, as well as explicit expressions of all almost complex flat surfaces.