The marked length spectrum of Anosov manifolds – Colin Guillarmou

Colin Guillarmou
LMO, Université Paris-Saclay

Date(s) : 11/01/2019   iCal
11 h 00 min - 13 h 30 min

Exposé commun du séminaire de Probabilités et du séminaire Teich.

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 is finite.



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