Conformal currents and the entropy of negatively curved three-manifolds

TL;DR

This paper explores the relationship between geodesic and conformal currents on hyperbolic three-manifolds, establishing bounds involving entropy and providing a new proof of Mostow Rigidity.

Contribution

It introduces a novel approach linking currents to entropy bounds and offers a new proof of Mostow Rigidity in three dimensions.

Findings

Established sharp bounds involving Liouville entropy, minimal surface entropy, and area ratio.

Connected conformal currents with geometric entropy measures.

Provided a new proof of the Mostow Rigidity Theorem.

Abstract

In this paper, we describe the intersection between geodesic and conformal currents on closed hyperbolic three-manifolds. We use this to prove some sharp bounds which involve the Liouville entropy of a negatively curved metric, the minimal surface entropy, and the area ratio. Using these ideas we also give a new proof of the Mostow Rigidity Theorem in the three-dimensional case.