Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

We show that totally geodesic subvarieties of the moduli space ${\mathcal{M}}_{g,n}$ of genus $g$ curves with $n$ marked points, endowed with the Weil–Petersson metric, are locally rigid. This implies that covering constructions – examples of totally geodesic subvarieties of ${\mathcal{M}}_{g,n}$ endowed with the Teichmüller metric – are locally rigid. We deduce the local rigidity statement from a more general rigidity result for a class of orbifold maps to ${\mathcal{M}}_{g,n}$.