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.
It is probably something "classical", so any reference is welcome. My initial goal is to understand the behaviour of such vector field, when the conformal class degenerate, so I don't need explicit formula, but estimate in the collar.
Thanks in advance,
The vector field $X$ is dual to a co-closed $1$-form $\omega$ (i.e., $d{\star}\omega = 0$). This means that ${\star}\omega$ represents an element of $H^1(\Sigma,\Bbb R)$. Except for the case of genus $0$, there are $\Bbb R^{2(\text{genus}\Sigma)}$ options.