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July 2019 The marked length spectrum of Anosov manifolds
Colin Guillarmou, Thibault Lefeuvre
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Ann. of Math. (2) 190(1): 321-344 (July 2019). DOI: 10.4007/annals.2019.190.1.6

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 new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of 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.

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Colin Guillarmou. Thibault Lefeuvre. "The marked length spectrum of Anosov manifolds." Ann. of Math. (2) 190 (1) 321 - 344, July 2019. https://doi.org/10.4007/annals.2019.190.1.6

Information

Published: July 2019
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2019.190.1.6

Subjects:
Primary: 37C27 , 37D40 , 53C22 , 53C24

Keywords: Anosov geodesic flow , closed geodesics , marked length spectrum , rigidity

Rights: Copyright © 2019 Department of Mathematics, Princeton University

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Vol.190 • No. 1 • July 2019
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