Asymptotic behaviour of harmonic 1-forms on Riemannian surfaces of increasing genus

$2$-dimensional compact oriented Riemannian manifolds $M_\varepsilon$ consisting of one or several copies of some base surface $\Gamma$ with a large number of thin tubes, endowed with a metric depending on a small parameter $\varepsilon$ are considered. The asymptotic behaviour of harmonic 1-forms on $M_\varepsilon$ is studied when the number of tubes increases and their thickness vanishes, as $\varepsilon\to 0$. We obtain the homogenized equations on the base surface $\Gamma$ describing the leading term of the asymptotics.