The Compactification of the Moduli Space of Convex RP(2) Surfaces, I

TL;DR

This paper explores the compactification of the moduli space of convex real projective structures on surfaces by extending to stable nodal curves with cubic differentials, linking geometric structures with algebraic degenerations.

Contribution

It introduces a partial compactification of the moduli space of convex RP(2) surfaces using stable nodal curves and relates the structures' holonomy to prior Goldman work.

Findings

Constructed convex real projective structures on degenerate surfaces.

Established relations between holonomy and algebraic degenerations.

Analyzed degenerations approaching the boundary of the moduli space.

Abstract

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of Goldman. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of Cheng-Yau and C.P. Wang and a result of Dunkel on the asymptotics of systems of ODEs.