Let $\Sigma$ be a Riemann surface with symplectic form $\omega$ and complex structure $J$, and denote by $g$ the induced metric. My question is
Is there a nice expression of the Gaussian curvature of $g$ in terms of $\omega$ and $J$?
I have though about this problem for a while, but I haven't been able to find anything. Any solution, hint or reference would be greatly appreciated.
For Riemann surfaces, the formulas are very simple. If $z=x+iy$ is a local holomorphic coordinate, the Kähler form can be written $\omega = \tfrac i 2\, h^2\, dz\wedge d{\bar z}$ for some smooth positive function $h$. The Riemannian metric is then $g= h^2\, dz\,d\bar z$ (where juxtaposition denotes the symmetric product), and the Gaussian curvature is $K = - \Delta(\log h)/h^2$, where $\Delta$ is the usual Laplace operator in $(x,y)$-coordinates.
[One source: Griffiths & Harris, Principles of Algebraic Geometry, Chapter 0.]