Conformal currents and the entropy of negatively curved three-manifolds

Marques, Fernando C; Neves, André

doi:10.2140/gt.2026.30.2565

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Conformal currents and the entropy of negatively curved three-manifolds

Fernando C Marques and André Neves

Geometry & Topology 30 (2026) 2565–2598

DOI: 10.2140/gt.2026.30.2565

Abstract

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.

Keywords

minimal surfaces, Liouville entropy, conformal currents

Mathematical Subject Classification

Primary: 53A10, 57M50

References

Publication

Received: 3 December 2024

Revised: 23 March 2026

Accepted: 28 March 2026

Published: 13 August 2026

Proposed: Ian Agol

Seconded: Tobias H Colding, David Fisher

Authors

Fernando C Marques
Department of Mathematics Princeton University Princeton, NJ United States

André Neves
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 30, issue 7 (2026)