Hyperbolic 3-manifolds with boundary of polyhedral type

TL;DR

This paper investigates hyperbolic 3-manifolds with boundary, establishing existence, uniqueness, and rigidity results for convex hyperbolic cone-metrics and bent metrics, advancing understanding of their geometric structures and partial progress on Thurston's conjectures.

Contribution

It introduces the concept of bent metrics, proves existence and uniqueness results for convex hyperbolic cone-metrics, and demonstrates rigidity properties for a broad class of hyperbolic 3-manifolds.

Findings

Existence of bent metrics for convex hyperbolic cone-metrics on boundary

Uniqueness of controllably polyhedral bent realizations up to isotopy

Rigidity of convex cocompact metrics with polyhedral convex cores

Abstract

Let $M$ be a compact orientable 3-manifold with hyperbolizable interior and non-empty boundary such that all boundary components have genii at least 2. We study an Alexandrov-Weyl-type problem for convex hyperbolic cone-metrics on $\partial M$. We consider a class of hyperbolic metrics on M with convex boundary, which we call bent metrics, and which naturally generalize hyperbolic metrics on $M$ with convex polyhedral boundary. We show that for each convex hyperbolic cone-metric $d$ on $\partial M$, with few simple exceptions, there exists a bent metric on $M$ such that the induced intrinsic metric on $\partial M$ is $d$. Next, we prove that if a bent realization is what we call controllably polyhedral, then it is unique up to isotopy. We exhibit a large subclass of hyperbolic cone-metrics on $\partial M,$ called balanced, which is open and dense among all convex hyperbolic cone-metrics in the sense of Lipschitz topology, and for which we show that their bent realizations are controllably polyhedral. We additionally prove that any convex realization of a convex hyperbolic cone-metric on $\partial M$ is bent. Finally, we deduce that there exists an open subset of the space of convex cocompact metrics on the interior of $M$, including all metrics with polyhedral convex cores, such that the metrics in this subset are (1) globally rigid with respect to the induced intrinsic metrics on the boundaries of their convex cores; (2) infinitesimally rigid with respect to their bending laminations. This gives partial progress towards conjectures of W. Thurston.