Quantitative marked length spectrum rigidity

Butt, Karen

doi:10.2140/gt.2025.29.3995

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Quantitative marked length spectrum rigidity

Karen Butt

Geometry & Topology 29 (2025) 3995–4054

DOI: 10.2140/gt.2025.29.3995

Abstract

We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least $3$ , with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$ . Using the methods of Hamenstädt (1999), we show that the volumes of $M$ and $N$ are approximately equal. We then show that the smooth map $F : M \to N$ constructed by Besson, Courtois and Gallot (1996) is a diffeomorphism with derivative bounds close to 1 and which depend on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson, Courtois and Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for closed negatively curved surfaces using the methods of Otal (1990) and Pugh (1987).

Keywords

marked length spectrum, rigidity, closed geodesics, locally symmetric spaces, negative curvature, volumes

Mathematical Subject Classification

Primary: 37D40, 53C22, 53C24, 53C35

Secondary: 37C27, 37D20

References

Publication

Received: 15 August 2022

Revised: 30 September 2022

Accepted: 3 November 2022

Published: 26 November 2025

Proposed: David Fisher

Seconded: Benson Farb, Mladen Bestvina

Authors

Karen Butt
Department of Mathematics University of Michigan Ann Arbor, MI United States	Department of Mathematics University of Chicago Chicago, IL United States

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Volume 29, issue 8 (2025)