Volume maximization and the extended hyperbolic space

TL;DR

This paper explores a volume maximization approach for constructing hyperbolic structures on 3-manifolds, revealing that critical points correspond to extended hyperbolic space structures, including degenerate Euclidean cases.

Contribution

It establishes a connection between volume maximization and extended hyperbolic structures, broadening the understanding of geometric decompositions in 3-manifolds.

Findings

Critical points relate to extended hyperbolic space structures.

Extended structures include decompositions along totally geodesic surfaces.

Degenerate cases correspond to Euclidean simplices.

Abstract

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space -- the natural extension of hyperbolic space by the de Sitter space -- except for the degenerate case where all simplices are Euclidean in a generalized sense. Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as connected sum, along embedded spheres (or projective planes) which are totally geodesic, space-like surfaces in the de Sitter part of the extended hyperbolic structure.