Title: Quantitative marked length spectrum rigidity
Speaker: Karen Butt (University of Michigan)
Abstract: The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. Conjecturally, the marked length spectrum determines the metric up to isometry (Burns-Katok). This is known to be true in some special cases, namely in dimension 2 (Otal, Croke), in dimension at least 3 if one of the metrics is locally symmetric (Hamenstadt, Besson-Courtois-Gallot), and in any dimension if the metrics are assumed to be sufficiently close in a suitable C^k topology (Guillarmou-Knieper-Lefeuvre). Even in these cases, there is more to be understood about to what extent the marked length spectrum determines the metric. Namely, if two manifolds have marked length spectra which are not equal but are close, is there some sense in which the metrics are close to being isometric? In this talk, we will provide some (quantitative) answers to this question, refining the known rigidity results for surfaces and for locally symmetric spaces of dimension at least 3.