Convex real projective structures and Hilbert metrics

TL;DR

This paper reviews convex real projective structures, their deformation theory, and associated Hilbert metrics, highlighting their geometric properties, parameterizations, and analogies with hyperbolic geometry and Teichmüller theory.

Contribution

It synthesizes recent developments on convex real projective structures, including deformation parameters, character varieties, and geodesic flows, emphasizing new insights and analogies with classical geometries.

Findings

Parameterization of deformation space using Hilbert lengths

Identification of projective structures with holomorphic differentials

Comparison of Hilbert and hyperbolic geometries via geodesic currents

Abstract

We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichm\"uller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichm\"uller space of the surface.