## Tohoku Mathematical Journal 2015 March SECOND SERIES VOL. 67, NO. 1

 Tohoku Math. J. 67 (2015), 1-17

Title ALMOST COMPLEX SURFACES IN THE NEARLY KÄHLER $S^3\times S^3$

Author John Bolton, Franki Dillen, Bart Dioos and Luc Vrancken

Abstract. In this paper we initiate the study of almost complex surfaces in the nearly Kähler $S^3\times S^3$. 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 Kähler $S^3\times S^3$. We also find a local correspondence between almost complex surfaces in the nearly Kähler $S^3\times S^3$ 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 Kähler $S^3\times S^3$ 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.
Key words and phrases. Almost complex surface, constant mean curvature surface, $H$-surface equation, holomorphic differential, minimal surface, nearly Kähler manifold.