Lens rigidity for manifolds with hyperbolic trapped sets

Guillarmou, Colin¹

¹ DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 rue d’Ulm, 75230 Paris cedex 05, France

Journal of the American Mathematical Society, Tome 30 (2017) no. 2, pp. 561-599

Résumé

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consist of the set of lengths of geodesics $\gamma$ with end points on $\partial M$, together with their end points $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for such manifolds with a hyperbolic trapped set and no conjugate points. This class contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of end points and exit vectors of geodesics (i.e., the scattering data) determines the Riemann surface up to conformal diffeomorphism.

Détail

DOI : 10.1090/jams/865

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