Complex points of minimal surfaces in almost Kähler manifolds

If (N, ω, J, g) is an almost Kähler manifold and M is a branched minimal immersion which is not a J-holomorphic curve, we show that the complex tangents are isolated and that each has a negative index, which extends the results in the Kähler case by S. S. Chern and J. Wolfson [2] and S. Webster [7] to almost Kähler manifolds. As an application, we get lower estimates for the genus of embedded minimal surfaces in almost Kähler manifolds. The proofs of these results are based on the well-known Cartan's moving frame methods as in [2, 7]. In our case, we must compute the torsion of the almost complex structures and find a useful representation of torsion. Finally, we prove that the minimal surfaces in complex projective plane with any almost complex structure is a J-holomorphic curve if it is homologous to the complex line.