On almost complex surfaces in the nearly Kähler S³× S³

Original · EN

We study almost complex surfaces in the nearly Kähler S³× S³. 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 R³. 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.