Rigidity of compact Fuchsian manifolds with convex boundary

TL;DR

This paper proves that compact Fuchsian manifolds with convex boundary are uniquely determined by the induced metric on their boundary surface, establishing a rigidity result in hyperbolic geometry.

Contribution

It establishes a new rigidity theorem for compact Fuchsian manifolds with convex boundary, showing they are uniquely determined by boundary metrics without additional restrictions.

Findings

Uniqueness of the manifold given boundary metric

No additional boundary restrictions beyond convexity

Advances understanding of hyperbolic 3-manifold rigidity

Abstract

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.