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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This is an elliptic differential operator with zero kernel and cokernel satisfying the Lopatinskii’s transmission condition (see [22, Theorem 3.3.2]). It can thus be used in order to define the scale of Sobolev spaces.
Indeed, \(\tau \mapsto g_\tau \) depends smoothly on \(\tau \), so \(\xi _\tau := \left( \exp _p^{g_\tau }\right) ^{-1}(q)\) depends smoothly on \(\tau \). Thus \((t,\tau ) \mapsto \varphi _t^{g_\tau }(p,\xi _\tau )\) is smooth in both variables and by the implicit function theorem, the length \(l_+^{g_\tau }(p,\xi _\tau )\) is smooth in \(\tau \). Thus, the reparametrized geodesic \(\gamma _\tau \) depends smoothly on \(\tau \)
References
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)
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)
Burns, K., Katok, A.: Manifolds with nonpositive curvature. Ergod. Theory Dyn. Syst. 5(2), 307–317 (1985)
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)
Croke, C.B.: Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1), 150–169 (1990)
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)
Dyatlov, S., Guillarmou, C.: Pollicott-Ruelle resonances for open systems. Ann. Henri Poincaré 17(11), 3089–3146 (2016)
Dyatlov, S., Zworski, M.: Mathematical Theory of Resonances. math.mit.edu/ dyatlov/res/, **
Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. (4) 49(3), 543–577 (2016)
Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308(2), 325–364 (2011)
Gérard, P., Golse, F.: Averaging regularity results for PDEs under transversality assumptions. Commun. Pure Appl. Math. 45(1), 1–26 (1992)
Guillarmou, C., Lefeuvre, T.: The marked length spectrum of Anosov manifolds. ArXiv e-prints (2018)
Guillarmou, C., Mazzucchelli, M.: Marked boundary rigidity for surfaces. Ergod. Theory Dyn. Syst. 38(4), 1459–1478 (2018)
Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)
Guillarmou, C.: Invariant distributions and X-ray transform for Anosov flows. J. Differ. Geom. 105(2), 177–208 (2017)
Guillarmou, C.: Lens rigidity for manifolds with hyperbolic trapped sets. J. Am. Math. Soc. 30(2), 561–599 (2017)
Lefeuvre, T.: On the s-injectivity of the X-ray transform on manifolds with hyperbolic trapped set. ArXiv e-prints. arXiv:1807.03680 (2018)
Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65(1), 71–83, (1981/1982)
Otal, J.-P.: Le spectre marqué des longueurs des surfaces à courbure négative. Ann. Math. (2) 131(1), 151–162 (1990)
Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. (2) 161(2), 1093–1110 (2005)
Paternain, G.P., Zhou, H.: Invariant distributions and the geodesic ray transform. Anal. PDE 9(8), 1903–1930 (2016)
Sharafutdinov, V.A.: Integral geometry of tensor fields. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994)
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)
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)
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)
Taylor, M.E.: Partial differential equations I. Basic Theory, 2 edn. Applied Mathematical Sciences, vol. 115. Springer, New York (2011)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Throughout this paper, we shall work in the smooth category, that is all the manifolds and coordinate charts are considered to be smooth.
Rights and permissions
About this article
Cite this article
Lefeuvre, T. Local Marked Boundary Rigidity Under Hyperbolic Trapping Assumptions. J Geom Anal 30, 448–465 (2020). https://doi.org/10.1007/s12220-019-00149-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-019-00149-8