Angle structures and hyperbolic $3$-manifolds with totally geodesic boundary

TL;DR

This paper establishes a correspondence between angle structures on ideal triangulations and hyperbolic metrics with totally geodesic boundary in 3-manifolds, providing a new way to understand their geometric structures.

Contribution

It proves that the existence of angle structures is equivalent to the existence of hyperbolic metrics with totally geodesic boundary in certain 3-manifolds.

Findings

Existence of angle structures implies hyperbolic metrics with totally geodesic boundary.

Every hyperbolic 3-manifold with totally geodesic boundary admits an ideal triangulation with angle structures.

The paper bridges combinatorial angle structures and geometric hyperbolic structures.

Abstract

This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal triangulation that admits angle structures.