On almost complex surfaces in the nearly Kähler $S^3\times S^3$

Abstract

We study almost complex surfaces in the nearly Kähler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic differential on every almost complex surface, and show that when this differential vanishes, then the corresponding solution of the H-surface equation gives a constant mean curvature surface in ${\mathbb{R}}^3$. We use this, together with a theorem of Hopf, to classify all almost complex 2-spheres. In fact there is essentially only one, and it is totally geodesic. More details, as well as the proofs of the various theorems are given in [1]. Finally, we state two theorems, one of which states that locally there are just two almost complex surfaces with parallel second fundamental form.

Appeared in the Proceedings of Pure and Applied Differential Geometry PADGE 2012 - in honor of Franki Dillen