Approximate rigidity of the marked length spectrum
Karen Butt · Mathematics Research Reports · 2024
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.