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

Abstract

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 ${\mathcal{M}}_{g,n}$.

18 pages. Rephrased main theorem in terms of maps, corrected typos, improved exposition. Incorporated feedback from referee