Properly convex bending of hyperbolic manifolds

TL;DR

This paper demonstrates that bending finite volume hyperbolic manifolds along totally geodesic hypersurfaces yields properly convex projective structures with finite volume, expanding the class of known convex manifolds in various dimensions.

Contribution

It establishes a method to produce properly convex projective structures from hyperbolic manifolds via bending, including non-compact examples in all dimensions ≥ 3.

Findings

Bending hyperbolic manifolds produces properly convex structures.

Existence of finite volume, non-compact convex manifolds in all dimensions ≥ 3.

Examples can be strictly convex or non-strictly convex.

Abstract

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.