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

Key Points

• This research investigates the local rigidity of totally geodesic subvarieties in the moduli space of curves under the Weil–Petersson metric.
• Analyzed totally geodesic subvarieties of the moduli space $ {\mathcal{M}}_{g,n}$ with the Weil–Petersson metric.
• Demonstrated local rigidity through a broader rigidity result applicable to orbifold maps.
• Explored covering constructions as specific instances of totally geodesic subvarieties within $ {\mathcal{M}}_{g,n}$.
• Establishes that totally geodesic subvarieties of $ {\mathcal{M}}_{g,n}$ are locally rigid under the Weil–Petersson metric.
• Confirms that covering constructions are also locally rigid, extending this property from the Weil–Petersson metric to the Teichmüller metric.