The boundary of an affine invariant submanifold

TL;DR

This paper investigates the boundary structure of affine invariant submanifolds in the moduli space of translation surfaces, providing formulas for tangent spaces, finiteness results, and generalizations of classical theorems.

Contribution

It introduces a formula for the tangent space to the boundary of affine invariant submanifolds and extends understanding of cylinder structures and Veech dichotomy in this context.

Findings

Derived a formula for the tangent space at the boundary

Proved finiteness results for cylinders in the boundary

Generalized aspects of the Veech dichotomy

Abstract

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove finiteness results concerning cylinders, a partial converse to the Cylinder Deformation Theorem, and a result generalizing part of the Veech dichotomy.