The marked length spectrum of Anosov manifolds

TL;DR

This paper proves that for certain negatively curved Riemannian manifolds with Anosov geodesic flow, the marked length spectrum locally determines the metric and provides stability estimates, with results extending to 2D and a finiteness conclusion.

Contribution

It establishes local rigidity and stability results for the marked length spectrum in Anosov manifolds, including a new stability estimate and solutions to conjectures.

Findings

Marked length spectrum locally determines the metric.

Provided a new stability estimate relating spectrum and metric distance.

Finiteness of negatively curved metrics with same spectrum under curvature bounds.

Abstract

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stability estimate quantifying how the marked length spectrum control the distance between the metrics. In dimension $2$ we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of $C^\infty$ is finite.