Lens rigidity for manifolds with hyperbolic trapped sets

Guillarmou, Colin¹

¹DMA, 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

Bibliographie

[1] Anikonov, Yu. E., Romanov, V. G. On uniqueness of determination of a form of first degree by its integrals along geodesics J. Inverse Ill-Posed Probl. 1997

[2] Belishev, M. I. The Calderon problem for two-dimensional manifolds by the BC-method SIAM J. Math. Anal. 2003 172 182

[3] Bowen, Rufus, Ruelle, David The ergodic theory of Axiom A flows Invent. Math. 1975 181 202

[4] Burago, Dmitri, Ivanov, Sergei Boundary rigidity and filling volume minimality of metrics close to a flat one Ann. of Math. (2) 2010 1183 1211

[5] Butterley, Oliver, Liverani, Carlangelo Smooth Anosov flows: correlation spectra and stability J. Mod. Dyn. 2007 301 322

[6] Croke, Christopher B. Rigidity for surfaces of nonpositive curvature Comment. Math. Helv. 1990 150 169

[7] Croke, Christopher B. Rigidity and the distance between boundary points J. Differential Geom. 1991 445 464

[8] Croke, Christopher Scattering rigidity with trapped geodesics Ergodic Theory Dynam. Systems 2014 826 836

[9] Croke, Christopher B., Herreros, Pilar Lens rigidity with trapped geodesics in two dimensions Asian J. Math. 2016 47 57

[10] Croke, Christopher B., Kleiner, Bruce Conjugacy and rigidity for manifolds with a parallel vector field J. Differential Geom. 1994 659 680

[11] Dairbekov, Nurlan, Uhlmann, Gunther Reconstructing the metric and magnetic field from the scattering relation Inverse Probl. Imaging 2010 397 409

[12] Dos Santos Ferreira, David, Kenig, Carlos E., Salo, Mikko, Uhlmann, Gunther Limiting Carleman weights and anisotropic inverse problems Invent. Math. 2009 119 171

[13] Dyatlov, Semyon, Guillarmou, Colin Microlocal limits of plane waves and Eisenstein functions Ann. Sci. Éc. Norm. Supér. (4) 2014 371 448

[14] Faure, Frédéric, Sjöstrand, Johannes Upper bound on the density of Ruelle resonances for Anosov flows Comm. Math. Phys. 2011 325 364

[15] Gérard, Patrick, Golse, François Averaging regularity results for PDEs under transversality assumptions Comm. Pure Appl. Math. 1992 1 26

[16] Grigis, Alain, Sjöstrand, Johannes Microlocal analysis for differential operators 1994

[17] Guillemin, V., Kazhdan, D. Some inverse spectral results for negatively curved 2-manifolds Topology 1980 301 312

[18] Guillemin, Victor, Kazhdan, David Some inverse spectral results for negatively curved 𝑛-manifolds 1980 153 180

[19] Katok, Anatole, Hasselblatt, Boris Introduction to the modern theory of dynamical systems 1995

[20] Hirsch, M., Palis, J., Pugh, C., Shub, M. Neighborhoods of hyperbolic sets Invent. Math. 1969/70 121 134

[21] Hörmander, Lars The analysis of linear partial differential operators. I 1983

[22] Kato, Tosio, Mitrea, Marius, Ponce, Gustavo, Taylor, Michael Extension and representation of divergence-free vector fields on bounded domains Math. Res. Lett. 2000 643 650

[23] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type Ann. of Math. (2) 1974 1 13

[24] Klingenberg, Wilhelm P. A. Riemannian geometry 1995

[25] Lassas, Matti, Sharafutdinov, Vladimir, Uhlmann, Gunther Semiglobal boundary rigidity for Riemannian metrics Math. Ann. 2003 767 793

[26] Michel, René Sur la rigidité imposée par la longueur des géodésiques Invent. Math. 1981/82 71 83

[27] Muhometov, R. G., Romanov, V. G. On the problem of finding an isotropic Riemannian metric in an 𝑛-dimensional space Dokl. Akad. Nauk SSSR 1978 41 44

[28] Muhometov, R. G. On a problem of reconstructing Riemannian metrics Sibirsk. Mat. Zh. 1981

[29] Noakes, Lyle, Stoyanov, Luchezar Rigidity of scattering lengths and travelling times for disjoint unions of strictly convex bodies Proc. Amer. Math. Soc. 2015 3879 3893

[30] Paternain, Gabriel P. Geodesic flows 1999

[31] Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther Tensor tomography on surfaces Invent. Math. 2013 229 247

[32] Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther Invariant distributions, Beurling transforms and tensor tomography in higher dimensions Math. Ann. 2015 305 362

[33] Pestov, Leonid, Uhlmann, Gunther Two dimensional compact simple Riemannian manifolds are boundary distance rigid Ann. of Math. (2) 2005 1093 1110

[34] Pestov, L. N., Sharafutdinov, V. A. Integral geometry of tensor fields on a manifold of negative curvature Sibirsk. Mat. Zh. 1988

[35] Robinson, Clark Structural stability on manifolds with boundary J. Differential Equations 1980 1 11

[36] Salo, Mikko, Uhlmann, Gunther The attenuated ray transform on simple surfaces J. Differential Geom. 2011 161 187

[37] Sharafutdinov, V. A. Integral geometry of tensor fields 1994 271

[38] Smale, S. Differentiable dynamical systems Bull. Amer. Math. Soc. 1967 747 817

[39] Stefanov, Plamen, Uhlmann, Gunther Boundary rigidity and stability for generic simple metrics J. Amer. Math. Soc. 2005 975 1003

[40] Stefanov, Plamen, Uhlmann, Gunther Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds J. Differential Geom. 2009 383 409

[41] Stefanov, Plamen, Uhlmann, Gunther, Vasy, Andras Boundary rigidity with partial data J. Amer. Math. Soc. 2016 299 332

[42] Otal, Jean-Pierre Sur les longueurs des géodésiques d’une métrique à courbure négative dans le disque Comment. Math. Helv. 1990 334 347

[43] Santaló, Luis A. Integral geometry and geometric probability 1976

[44] Sullivan, Dennis Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups Acta Math. 1984 259 277

[45] Taylor, Michael E. Partial differential equations. I 1996

[46] Taylor, Michael E. Partial differential equations. III 1997

[47] Uhlmann, Gunther, Vasy, András The inverse problem for the local geodesic ray transform Invent. Math. 2016 83 120

[48] Walters, Peter An introduction to ergodic theory 1982

[49] Young, Lai-Sang Large deviations in dynamical systems Trans. Amer. Math. Soc. 1990 525 543

[50] Zworski, Maciej Semiclassical analysis 2012