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

(Received June 6, 2013)
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.

Mathematics Subject Classification. Primary 53C40; Secondary 53C42.
Key words and phrases. Almost complex surface, constant mean curvature surface, $H$-surface equation, holomorphic differential, minimal surface, nearly Kähler manifold.

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