Hyperbolic manifolds with polyhedral boundary

TL;DR

This paper studies hyperbolic 3-manifolds with polyhedral boundary, exploring the relationship between boundary geometry and the hyperbolic metric, and extends classical results to non-smooth, polyhedral cases.

Contribution

It provides a comprehensive analysis of hyperbolic manifolds with polyhedral boundary, including unique correspondences between boundary dihedral angles and the hyperbolic metric, extending smooth boundary results.

Findings

Characterization of dihedral angles for polyhedral boundaries

Extension of the third fundamental form results to polyhedral cases

Connections to Teichmüller space and circle packing theorems

Abstract

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$ has curvature $K>-1$, and each such metric on $\dr M$ is obtained for a unique choice of $g$. A dual statement is that, for each $g$ as above, the third fundamental form of $\dr M$ has curvature $K<1$, and its closed geodesics which are contractible in $M$ have length $L>2\pi$. Conversely, any such metric on $\dr M$ is obtained for a unique choice of $g$. We are interested here in the similar situation where $\partial M$ is not smooth, but rather looks locally like an ideal polyhedron in $H^3$. We can give a fairly complete answer to the question on the third fundamental form -- which in this case concerns the dihedral angles -- and some partial results about the induced metric. This has some by-products, like an affine piecewise flat structure on the Teichmueller space of a surface with some marked points, or an extension of the Koebe circle packing theorem to many 3-manifolds with boundary.