Suppose $R$ is orientable compact Riemann surface with boundary $\partial R$ a collection $C_1, \dots C_k$ of oriented circles, so that for $i=1,\dots,j$, $C_i$ has boundary orientation, and for $i=j+1,\dots k$, $C_i$ has nonboundary orientation.
Suppose moreover that have parametrized neighborhoods $U_i$ of $C_i$ in $R$ so that $U_i$ is halfcylinder, so that for $i=1,\dots j$, then $U_i$ is $]-\infty,0] \times \mathbf{S}^1$, and for $i=j+1,\dots k$, then $U_i$ is $[0,\infty[\times \mathbf{S}^1$.
Pick complex coordinate $s+it$ on each $U_i$, pick numbers $a_i$ such that
$a_1 + \dots + a_j - a_{j+1} - \dots - a_k = 0.$
This condition implies that it is possible to find 1-form $\omega$ on $R$ so that $d \omega=0$ and so that on $U_i$ the 1-form $\omega$ is constant 1-form $a_i dt$.
Question For given fixed set $(a_1,\dots a_k)$ of numbers satisfying above condition, let $\Omega(a_1,\dots,a_k)$ denote space of all such 1-forms $\omega$. Is it possible to give nice description (structure of manifold) of $\Omega(a_1,\dots a_k)$?