Hyperbolic cusps with convex polyhedral boundary

TL;DR

This paper proves the unique determination of hyperbolic cusps with convex polyhedral boundary by boundary metrics and shows that any such boundary metric can be realized, using a scalar curvature functional approach.

Contribution

It establishes the rigidity and realizability of convex polyhedral cusps in hyperbolic space via boundary metrics and introduces a novel proof method using scalar curvature functionals.

Findings

Hyperbolic cusps are uniquely determined by boundary metrics.

Any hyperbolic metric with cone singularities on the torus can be realized as a boundary metric.

Convex polyhedral cusps with particles are rigid with respect to boundary metrics and singular curvatures.

Abstract

We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The proof uses the total scalar curvature functional on the space of ``cusps with particles'', which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus. Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.