On some variational problems for 2-dimensional Hermitian metrics

Let (M, J, g) be a compact complex 2-dimensional Hermitian manifold with the Kähler form Ω, and the torsion 1-form ω defined by dΩ = ω ∧ Ω. In this note we obtain the Euler-Lagrange equations for the variational functionals defined by {norm of matrix}ω{norm of matrix}² and {norm of matrix}dω{norm of matrix}², where g runs in the space of all the Hermitian metrics on M. In the first case, the extremals are precisely the Kähler metrics [Gd]. In the second case, we also write down a formula for the second variation.