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Lens rigidity for manifolds with hyperbolic trapped sets


Author: Colin Guillarmou
Journal: J. Amer. Math. Soc. 30 (2017), 561-599
MSC (2010): Primary 35R30; Secondary 53C24, 53C65
DOI: https://doi.org/10.1090/jams/865
Published electronically: September 6, 2016
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Abstract: 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.


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Additional Information

Colin Guillarmou
Affiliation: DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 rue d’Ulm, 75230 Paris cedex 05, France
Email: cguillar@dma.ens.fr

DOI: https://doi.org/10.1090/jams/865
Keywords: X-ray transform, lens rigidity, scattering rigidity, hyperbolic dynamics
Received by editor(s): January 16, 2015
Received by editor(s) in revised form: July 12, 2016
Published electronically: September 6, 2016
Additional Notes: The research is partially supported by grants ANR-13-BS01-0007-01 and ANR-13-JS01-0006.
Article copyright: © Copyright 2016 American Mathematical Society