Almost complex surfaces in the nearly K\"ahler $S^3\times S^3$

TL;DR

This paper systematically studies almost complex surfaces in the nearly Kähler $S^3 imes S^3$, establishing a global holomorphic differential, a correspondence with solutions of the $H$-system, and classifying totally geodesic surfaces.

Contribution

It introduces a global holomorphic differential on such surfaces, links them to the $H$-system solutions, and classifies totally geodesic and topologically spherical surfaces.

Findings

Existence of a global holomorphic differential induced by an almost product structure.

A correspondence between almost complex surfaces and solutions of the $H$-system.

Classification of totally geodesic almost complex surfaces and proof that almost complex 2-spheres are totally geodesic.

Abstract

In this paper almost complex surfaces of the nearly K\"ahler $S^3\times S^3$ are studied in a systematic way. 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 K\"ahler $S^3\times S^3$. We also find a correspondence between almost complex surfaces in the nearly K\"ahler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente, 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 K\"ahler $S^3\times S^3$ with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we will 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.