Advertisement

Local Marked Boundary Rigidity Under Hyperbolic Trapping Assumptions

  • Thibault LefeuvreEmail author
Article

Abstract

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points, and a hyperbolic trapped set are locally marked boundary rigid.

Keywords

Inverse problem Geodesic flow X-ray transform Boundary rigidity Marked boundary distance function 

Mathematics Subject Classification

37D20 37D40 35R30 

Notes

Acknowledgements

We warmly thank Colin Guillarmou for fruitful discussions during the redaction of this paper. We are also grateful to the anonymous referee for helpful comments. This research is partially supported by the ERC IPFLOW project.

References

  1. 1.
    Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5), 731–799 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burago, D., Ivanov, S.: Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann. Math. (2) 171(2), 1183–1211 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Burns, K., Katok, A.: Manifolds with nonpositive curvature. Ergod. Theory Dyn. Syst. 5(2), 307–317 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Croke, C.B., Dairbekov, N.S., Sharafutdinov, V.A.: Local boundary rigidity of a compact Riemannian manifold with curvature bounded above. Trans. Am. Math. Soc. 352(9), 3937–3956 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Croke, C.B.: Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1), 150–169 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Croke, C.B.: Rigidity theorems in Riemannian geometry. In: Geometric Methods in Inverse Problems and PDE Control. The IMA Volumes in Mathematics and its Applications, vol. 137, pp. 47–72. Springer, New York (2004)Google Scholar
  7. 7.
    Dyatlov, S., Guillarmou, C.: Pollicott-Ruelle resonances for open systems. Ann. Henri Poincaré 17(11), 3089–3146 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dyatlov, S., Zworski, M.: Mathematical Theory of Resonances. math.mit.edu/ dyatlov/res/, **Google Scholar
  9. 9.
    Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. (4) 49(3), 543–577 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308(2), 325–364 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gérard, P., Golse, F.: Averaging regularity results for PDEs under transversality assumptions. Commun. Pure Appl. Math. 45(1), 1–26 (1992)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guillarmou, C., Lefeuvre, T.: The marked length spectrum of Anosov manifolds. ArXiv e-prints (2018)Google Scholar
  13. 13.
    Guillarmou, C., Mazzucchelli, M.: Marked boundary rigidity for surfaces. Ergod. Theory Dyn. Syst. 38(4), 1459–1478 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Guillarmou, C.: Invariant distributions and X-ray transform for Anosov flows. J. Differ. Geom. 105(2), 177–208 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guillarmou, C.: Lens rigidity for manifolds with hyperbolic trapped sets. J. Am. Math. Soc. 30(2), 561–599 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lefeuvre, T.: On the s-injectivity of the X-ray transform on manifolds with hyperbolic trapped set. ArXiv e-prints. arXiv:1807.03680 (2018)
  18. 18.
    Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65(1), 71–83, (1981/1982)Google Scholar
  19. 19.
    Otal, J.-P.: Le spectre marqué des longueurs des surfaces à courbure négative. Ann. Math. (2) 131(1), 151–162 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. (2) 161(2), 1093–1110 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Paternain, G.P., Zhou, H.: Invariant distributions and the geodesic ray transform. Anal. PDE 9(8), 1903–1930 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sharafutdinov, V.A.: Integral geometry of tensor fields. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994)Google Scholar
  23. 23.
    Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123(3), 445–467 (2004)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Stefanov, P., Uhlmann, G.: Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds. J. Differ. Geom. 82(2), 383–409 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stefanov, P., Uhlmann, G., Vasy, A.: Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge. ArXiv e-prints. arXiv:1702.03638 (2017)
  26. 26.
    Taylor, M.E.: Partial differential equations I. Basic Theory, 2 edn. Applied Mathematical Sciences, vol. 115. Springer, New York (2011)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

Personalised recommendations