Dual metrics on the boundary of strictly polyhedral hyperbolic 3-manifolds

TL;DR

The paper proves a sharp realization theorem for dual metrics on the boundary of compact hyperbolic 3-manifolds with polyhedral boundary. Using a continuity method in the spirit of Alexandrov, it introduces a dual boundary map $\mathcal{I}_V$ from isotopy classes of strictly polyhedral metrics to concave large spherical cone-metrics on $\partial M$, and establishes that $\mathcal{I}_V$ is a $C^1$-diffeomorphism by proving infinitesimal rigidity and properness. The main result extends Rivin–Hodgson–Schlenker-type duality from smooth/bounded settings to general convex cocompact hyperbolic 3-manifolds with boundary, under largeness assumptions on the boundary metric. The proof leverages a doubling construction, $L^2$-cohomology techniques, and a careful topological analysis of the moduli spaces of polyhedral metrics and cone-metrics, culminating in existence, uniqueness up to isotopy, and the boundary dual realization of the polyhedral structure. This has broad implications for rigidity and realization problems in discrete hyperbolic geometry and for understanding convex cores and their dual boundary data in convex cocompact manifolds.

Abstract

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles are greater than $2π$ and the lengths of all closed geodesics that are contractible in $M$ are greater than $2π$ there exists a unique strictly polyhedral hyperbolic metric on $M$ such that $d$ is the induced dual metric on $\partial M$.

Theorems & Definitions (70)

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