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Approximate rigidity of the marked length spectrum
Karen Butt1
1 Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637
Mathematics Research Reports, Volume 4 (2023), pp. 63-82.

We report on recent work investigating the extent to which finitely many closed geodesics approximately determine a negatively curved metric on a closed manifold. It is known in certain cases—and conjectured to be true in general—that the lengths of all closed geodesics (as a function of their free homotopy classes) determine the underlying negatively curved metric up to isometry. This length function is known as the marked length spectrum. Here, we consider certain pairs of Riemannian manifolds whose marked length spectra agree—only approximately—on a finite set of closed geodesics. We report on our recent results which show the two metrics are “almost isometric". More precisely, we show the metrics are bi-Lipschitz equivalent with constant close to 1, and we obtain estimates for these constants depending only on concrete Riemannian data.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.18
Classification: 37D40, 37D20, 37C27, 53C24, 53C22
Keywords: marked length spectrum, closed geodesics, geodesic flow, negative curvature, locally symmetric spaces, rigidity
Karen Butt 1

1 Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Karen Butt. Approximate rigidity of the marked length spectrum. Mathematics Research Reports, Volume 4 (2023), pp. 63-82. doi : 10.5802/mrr.18. https://mrr.centre-mersenne.org/articles/10.5802/mrr.18/

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