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

TL;DR

This work proves local rigidity for almost Weil–Petersson subvarieties $N$ of $\\mathcal{M}_{g,n}$ when $3g-3+n>0$, by introducing $\\Gamma_M$-deformations and analyzing the associated orbifold fundamental group. The authors extend rigidity from curves to general orbifolds via an Imayoshi–Shiga–type energy argument, showing any nonconstant deformation factors through the WP lift $M$ and hence is trivial after normalization. They also connect covering constructions to WP geometry, showing totally marked and regular covers yield WP-geodesic embeddings or fixed-point WP submanifolds, thereby producing and validating almost WP subvarieties as geometric realizations of covering data. The results unify the study of Teichmüller and WP subvarieties, clarify the rigidity landscape for coverings, and provide a framework for understanding when subvarieties arise from covering constructions. Together, these findings advance the understanding of the geometry of moduli spaces under WP metrics and its interactions with Teichmüller theory.

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 $\mathcal M_{g,n}$.

Theorems & Definitions (50)

• Theorem 1.1: Local Rigidity
• Proposition 1.2
• Remark 1.3
• Corollary 1.5: Local rigidity of Covering constructions
• Remark 1.6
• Remark 1.7
• Corollary 1.10: Maximality
• Remark 1.11
• Definition 2.1: Almost Weil--Petersson
• Lemma 3.1: cf. Diez-Harvey*Theorem 1