The boundary of a totally geodesic subvariety of moduli space

TL;DR

This paper investigates the structure of totally geodesic subvarieties within moduli spaces of Riemann surfaces, demonstrating that their boundary components are also totally geodesic and decompose into prime, locally isometric pieces.

Contribution

It establishes that the boundary of a totally geodesic subvariety in moduli space inherits a totally geodesic structure and decomposes into prime, locally isometric components.

Findings

Boundary loci are totally geodesic within boundary strata.

Boundary decomposes into prime, locally isometric pieces.

Projection to factors preserves local isometry.

Abstract

We consider subvarieties $N$ of $\mathcal{M}_{g,n}$, the moduli space of genus $g$ Riemann surfaces with $n$ marked points, that are totally geodesic with respect to the Teichm\"uller metric. The Deligne-Mumford boundary of $\mathcal{M}_{g,n}$ decomposes into strata, each of which is essentially a product of lower complexity moduli spaces -- in such spaces there is a natural notion of totally geodesic. We show that the boundary locus of $N$ in any such stratum is itself totally geodesic. Furthermore, we prove that each such boundary locus decomposes into prime pieces, and for each such piece the projection to each factor is locally isometric in an appropriate sense.