# Divergence free vector field on compact surface

I get a free divergence field $X$ on a compact surface $(\Sigma, g)$ and I would like to integrate it.

On the sphere $X=\nabla^\bot f$ since the spher is simply connected.($\nabla^\bot =J\circ \nabla$, where $J$ is the compex structure).

On the Torus $X=\nabla^\bot + C$ where $C$ is a constant vector field.

In higher genus, I have no idea of the form of the rest which of course becomes more and more complicated when genus increases.

In fact I know that the divergence free vector field space is isomorphic to the $H^1(\Sigma,\mathbb{R})$ whose dimension is $2* genus$, but nothing is explicit in the proof I know. I think it is link to the space of holomorphic and meromorphic functions, and then to the Teichmuller space, but I am not an expert in that fields.

It is probably something "classical", so any reference is welcome. My initial goal consist in understanding the behaviour of such vector field, when the conformal class degenerate, so I don't need explicit formula, but only asymptotic behaviour in the collar.

This question has been posted on mathstackexchange but without sucess

• By "integration", you mean writing $X=J\circ\nabla f$? In general, you will have to write $X=J\circ\nabla f+Y$, where $Y$ is dual to a harmonic one-form (for a vector field, this probably means no div and no curl). – Sebastian Goette Nov 25 '15 at 19:29