We study almost complex surfaces in the nearly Kähler S 3 × S 3 . We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente [10]. 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 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.
A Lagrangian surface in a Lorentzian Kähler surface is called marginally trapped if its mean curvature vector is lightlike at each point. In this paper we classify marginally trapped Lagrangian surfaces in Lorentzian complex space forms. Our main results state that there exist three families of marginally trapped Lagrangian surfaces in C12, nine families in CP12, and nine families in CH12. Conversely, all marginally trapped Lagrangian surfaces in Lorentzian complex space forms are obtained from these 21 families.
In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.
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