Quantitative marked length spectrum rigidity

TL;DR

This paper demonstrates that if a negatively curved manifold's marked length spectrum closely matches that of a locally symmetric space, then the two are nearly isometric, refining previous rigidity results.

Contribution

It extends marked length spectrum rigidity to a quantitative setting, showing near-spectrum equality implies near-isometry, with explicit bounds and diffeomorphism construction.

Findings

Volumes of close spectra are approximately equal

The Besson-Courtois-Gallot map is nearly an isometry

Results apply to both higher dimensions and surfaces

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 Hamenst\"adt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenst\"adt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh.