Convex projective structures on non-hyperbolic three-manifolds

TL;DR

This paper explores the relationship between convex projective structures and hyperbolic structures on three-manifolds, showing potential deformations and constructions that extend Benoist's classification.

Contribution

It provides initial results suggesting a converse to Benoist's theorem, demonstrating deformations of hyperbolic manifolds into convex projective structures with boundary.

Findings

Cusped hyperbolic three-manifolds can be deformed into convex projective structures.

Convex projective structures can be glued along boundaries with matching geometry.

Many doubles of cusped hyperbolic manifolds admit convex projective structures.

Abstract

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.