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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
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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  • [AnRo] Yurii E. Anikonov and Vladimir G. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl. 5 (1997), no. 6, 487-490 (1998). MR 1623603,
  • [Be] Mikhail I. Belishev, The Calderon problem for two-dimensional manifolds by the BC-method, SIAM J. Math. Anal. 35 (2003), no. 1, 172-182 (electronic). MR 2001471,
  • [BoRu] Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181-202. MR 0380889
  • [BuIv] Dmitri Burago and Sergei Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Ann. of Math. (2) 171 (2010), no. 2, 1183-1211. MR 2630062,
  • [BuLi] Oliver Butterley and Carlangelo Liverani, Smooth Anosov flows: correlation spectra and stability, J. Mod. Dyn. 1 (2007), no. 2, 301-322. MR 2285731,
  • [Cr1] Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150-169. MR 1036134,
  • [Cr2] Christopher B. Croke, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), no. 2, 445-464. MR 1094465
  • [Cr3] Christopher Croke, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 826-836. MR 3199795,
  • [CrHe] Christopher B. Croke and Pilar Herreros, Lens rigidity with trapped geodesics in two dimensions, Asian J. Math. 20 (2016), no. 1, 47-57. MR 3460758,
  • [CrKl] Christopher B. Croke and Bruce Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, J. Differential Geom. 39 (1994), no. 3, 659-680. MR 1274134
  • [DaUh] Nurlan Dairbekov and Gunther Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Probl. Imaging 4 (2010), no. 3, 397-409. MR 2671103,
  • [DKSU] David Dos Santos Ferreira, Carlos E. Kenig, Mikko Salo, and Gunther Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), no. 1, 119-171. MR 2534094,
  • [DKLS] David Dos Santos Ferreira, Yaroslav Kurylev, Matti Lassas, and Mikko Salo, The Calderon problem in transversally anisotropic geometries, to appear in J. Eur. Math. Soc.
  • [DyGu1] Semyon Dyatlov and Colin Guillarmou, Microlocal limits of plane waves and Eisenstein functions, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 2, 371-448 (English, with English and French summaries). MR 3215926
  • [DyGu2] Semyon Dyatlov, Colin Guillarmou, Pollicott-Ruelle resonances for open systems. Ann. Henri Poincaré, 1-18, DOI 10.1007/s00023-016-0491-8.
  • [DyZw] Semyon Dyatlov, Maciej Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. l'ENS 49 (2016), 543-577.
  • [FaSj] Frédéric Faure and Johannes Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308 (2011), no. 2, 325-364 (English, with English and French summaries). MR 2851145,
  • [FrJo] Friedrich G. Friedlander and Mark S. Joshi, Introduction to the Theory of Distributions, Cambridge Univ. Press. (1999), pp. 188.
  • [GeGo] Patrick Gérard and François Golse, Averaging regularity results for PDEs under transversality assumptions, Comm. Pure Appl. Math. 45 (1992), no. 1, 1-26. MR 1135922,
  • [GrSj] Alain Grigis and Johannes Sjöstrand, Microlocal Analysis for Differential Operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107
  • [Gu] Colin Guillarmou, Invariant distributions and X-ray transform for Anosov flows, J. Differential Geom., to appear. arXiv:1408.4732.
  • [GuKa1] Victor Guillemin and David Kazhdan, Some inverse spectral results for negatively curved $ 2$-manifolds, Topology 19 (1980), no. 3, 301-312. MR 579579,
  • [GuKa2] Victor Guillemin and David Kazhdan, Some inverse spectral results for negatively curved $ n$-manifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 153-180. MR 573432
  • [HaKa] Anatole Katok and Boris Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • [HPPS] Morris Hirsch, Jacob Palis, Charles Pugh, and Michael Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/1970), 121-134. MR 0262627
  • [Hö] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • [KMPT] Tosio Kato, Marius Mitrea, Gustavo Ponce, and Michael Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett. 7 (2000), no. 5-6, 643-650. MR 1809290,
  • [Kl] Wilhelm Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1-13. MR 0377980
  • [Kl2] Wilhelm P. A. Klingenberg, Riemannian Geometry, 2nd ed., de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin, 1995. MR 1330918
  • [LSU] Matti Lassas, Vladimir Sharafutdinov, and Gunther Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann. 325 (2003), no. 4, 767-793. MR 1974568,
  • [Mi] René Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), no. 1, 71-83 (French). MR 636880,
  • [MuRo] Ravil G. Mukhometov and Vladimir G. Romanov, On the problem of finding an isotropic Riemannian metric in an $ n$-dimensional space, Dokl. Akad. Nauk SSSR 243 (1978), no. 1, 41-44 (Russian). MR 511273
  • [Mu] Ravil G. Mukhometov, On a problem of reconstructing Riemannian metrics, Sibirsk. Mat. Zh. 22 (1981), no. 3, 119-135, 237 (Russian). MR 621466
  • [NoSt] Lyle Noakes and Luchezar Stoyanov, Rigidity of scattering lengths and travelling times for disjoint unions of strictly convex bodies, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3879-3893. MR 3359579,
  • [Pa] Gabriel P. Paternain, Geodesic Flows, Progress in Mathematics, vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1712465
  • [PSU1] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Tensor tomography on surfaces, Invent. Math. 193 (2013), no. 1, 229-247. MR 3069117,
  • [PSU2] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Math. Ann. 363 (2015), no. 1-2, 305-362. MR 3394381,
  • [PeUh] Leonid Pestov and Gunther Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2) 161 (2005), no. 2, 1093-1110. MR 2153407,
  • [PeSh] Leonid N. Pestov and Vladimir A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sibirsk. Mat. Zh. 29 (1988), no. 3, 114-130, 221 (Russian); English transl., Sib. Math. J. 29 (1988), no. 3, 427-441 (1989). MR 953028,
  • [Ro] Clark Robinson, Structural stability on manifolds with boundary, J. Differential Equations 37 (1980), no. 1, 1-11. MR 583334,
  • [SaUh] Mikko Salo and Gunther Uhlmann, The attenuated ray transform on simple surfaces, J. Differential Geom. 88 (2011), no. 1, 161-187. MR 2819758
  • [Sh] Vladimir A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR 1374572
  • [Sm] Stephen Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014
  • [StUh1] Plamen Stefanov and Gunther Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (2005), no. 4, 975-1003. MR 2163868,
  • [StUh2] Plamen Stefanov and Gunther Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom. 82 (2009), no. 2, 383-409. MR 2520797
  • [SUV] Plamen Stefanov, Gunther Uhlmann, and Andras Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc. 29 (2016), no. 2, 299-332. MR 3454376,
  • [Ot] Jean-Pierre Otal, Sur les longueurs des géodésiques d'une métrique à courbure négative dans le disque, Comment. Math. Helv. 65 (1990), no. 2, 334-347 (French). MR 1057248,
  • [Sa] Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364
  • [Su] Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3-4, 259-277. MR 766265,
  • [Ta] Michael E. Taylor, Partial Differential Equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148
  • [Ta2] Michael E. Taylor, Partial Differential Equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
  • [UhVa] Gunther Uhlmann and András Vasy, The inverse problem for the local geodesic ray transform, Invent. Math. 205 (2016), no. 1, 83-120. MR 3514959,
  • [Wa] Peter Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
  • [Yo] Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525-543. MR 975689,
  • [Zw] Maciej Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218

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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

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

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