# ANNALES DE L'INSTITUT FOURIER

X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds
Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2857-2919.

We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.

On considère le problème de rigidité du bord pour les variétés asymptotiquement hyperboliques. Nous montrons l’injectivité de la transformée en rayons X dans plusieurs cas et considérons le problème inverse non-linéaire qui consiste en la détermination de la métrique à partir de données au bord sur le flot géodésique.

Published online:
DOI: 10.5802/aif.3339
Classification: 35R30, 37D40, 53C22
Keywords: X-ray transform, boundary rigidity, asymptotically hyperbolic manifold
Graham, C. Robin 1; Guillarmou, Colin 2; Stefanov, Plamen 3; Uhlmann, Gunther 4

1 Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 (USA)
2 Laboratoire de Mathématiques d’Orsay UMR 8628 du CNRS, Université Paris-Sud 91405 Orsay Cedex (France)
3 Department of Mathematics Purdue University West Lafayette, IN 47907 (USA)
4 Department of Mathematics University of Washington Seattle, WA 98195-4350 (USA) and Institute for Advanced Study of the HKUST Hong Kong University of Science and Technology Clear Water Bay, New Territories, Hong Kong
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Graham, C. Robin; Guillarmou, Colin; Stefanov, Plamen; Uhlmann, Gunther. X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds. Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2857-2919. doi : 10.5802/aif.3339. https://aif.centre-mersenne.org/articles/10.5802/aif.3339/

[1] Alexakis, Spyridon; Mazzeo, Rafe Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds, Commun. Math. Phys., Volume 297 (2010) no. 3, pp. 621-651 | DOI | MR | Zbl

[2] Anikonov, Yurii E.; Romanov, Vladimir G. On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., Volume 5 (1997) no. 6, pp. 487-490 | DOI | MR | Zbl

[3] Berenstein, Carlos A.; Casadio Tarabusi, Enrico Inversion formulas for the $k$-dimensional Radon transform in real hyperbolic spaces, Duke Math. J., Volume 62 (1991) no. 3, pp. 613-631 | DOI | MR | Zbl

[4] Besse, Arthur L. Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 10, Springer, 1987, xii+510 pages | DOI | MR | Zbl

[5] Chen, Xi; Hassell, Andrew Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy, Commun. Partial Differ. Equations, Volume 41 (2016) no. 3, pp. 515-578 | DOI | MR | Zbl

[6] Coddington, Earl A.; Levinson, Norman Theory of ordinary differential equations, McGraw-Hill, 1955, xii+429 pages | MR | Zbl

[7] Croke, Christopher B. Rigidity theorems in Riemannian geometry, Geometric methods in inverse problems and PDE control (The IMA Volumes in Mathematics and its Applications), Volume 137, Springer, 2004, pp. 47-72 | DOI | MR | Zbl

[8] Czech, Bartłomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James Integral geometry and holography, J. High Energy Phys. (2015) no. 10, 175, 41 pages | DOI | MR | Zbl

[9] Dyatlov, Semyon; Guillarmou, Colin Pollicott–Ruelle resonances for open systems, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3089-3146 | DOI | MR | Zbl

[10] Eberlein, Patrick Geodesic flow in certain manifolds without conjugate points, Trans. Am. Math. Soc., Volume 167 (1972), pp. 151-170 | DOI | MR | Zbl

[11] Eberlein, Patrick When is a geodesic flow of Anosov type? I, J. Differ. Geom., Volume 8 (1973), pp. 437-463 | DOI | MR | Zbl

[12] Eptaminitakis, N.; Graham, C. Robin (in preparation)

[13] Fefferman, Charles; Graham, C. Robin Conformal invariants, The mathematical heritage of Élie Cartan (Lyon, 1984) (Astérisque), Société Mathématique de France, 1985, pp. 95-116 | Numdam | MR | Zbl

[14] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry, Universitext, Springer, 1987, xii+248 pages | DOI | MR | Zbl

[15] Graham, C. Robin Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999) (Supplemento ai Rendiconti del Circolo Matemàtico di Palermo), Volume 63 (2000), pp. 31-42 | MR | Zbl

[16] Graham, C. Robin; Lee, John M. Einstein metrics with prescribed conformal infinity on the ball, Adv. Math., Volume 87 (1991) no. 2, pp. 186-225 | DOI | MR | Zbl

[17] Gromoll, Detlef; Klingenberg, Wilhelm; Meyer, Wolfgang Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, 55, Springer, 1975, vi+287 pages | MR | Zbl

[18] Guillarmou, Colin Lens rigidity for manifolds with hyperbolic trapped sets, J. Am. Math. Soc., Volume 30 (2017) no. 2, pp. 561-599 | DOI | MR | Zbl

[19] Guillarmou, Colin; Mazzucchelli, Marco Marked boundary rigidity for surfaces, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 4, pp. 1459-1478 | DOI | MR | Zbl

[20] Heil, Konstantin; Moroianu, Andrei; Semmelmann, Uwe Killing and conformal Killing tensors, J. Geom. Phys., Volume 106 (2016), pp. 383-400 | DOI | MR | Zbl

[21] Helgason, Sigurdur The totally-geodesic Radon transform on constant curvature spaces, Integral geometry and tomography (Arcata, CA, 1989) (Contemporary Mathematics), Volume 113, American Mathematical Society, 1990, pp. 141-149 | DOI | MR | Zbl

[22] Helgason, Sigurdur Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, 39, American Mathematical Society, 1994, xiv+611 pages | DOI | MR | Zbl

[23] Holman, Sean; Uhlmann, Gunther On the microlocal analysis of the geodesic X-ray transform with conjugate points, J. Differ. Geom., Volume 108 (2018) no. 3, pp. 459-494 | DOI | MR | Zbl

[24] Ivanov, Sergei Volume comparison via boundary distances, Proceedings of the International Congress of Mathematicians. Volume II (2010), pp. 769-784 | MR | Zbl

[25] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type, Ann. Math., Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl

[26] Klingenberg, Wilhelm Riemannian geometry, De Gruyter Studies in Mathematics, 1, Walter de Gruyter, 1995, x+409 pages | DOI | MR | Zbl

[27] Knieper, Gerhard A note on Anosov flows of non-compact Riemannian manifolds, Proc. Am. Math. Soc., Volume 146 (2018) no. 9, pp. 3955-3959 | DOI | MR | Zbl

[28] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol I, Interscience Publishers, 1963, xi+329 pages | MR | Zbl

[29] Lassas, Matti; Sharafutdinov, Vladimir; Uhlmann, Gunther Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., Volume 325 (2003) no. 4, pp. 767-793 | DOI | MR | Zbl

[30] Lehtonen, Jere The geodesic ray transform on two-dimensional Cartan–Hadamard manifolds, Ph. D. Thesis, University of Jyväskylä (Finland) (2016) (https://arxiv.org/abs/1612.04800) | Zbl

[31] Lehtonen, Jere; Railo, Jesse; Salo, Mikko Tensor tomography on Cartan-Hadamard manifolds, Inverse Probl., Volume 34 (2018) no. 4, 044004, 27 pages | DOI | MR | Zbl

[32] Mazzeo, Rafe Hodge Cohomology of Negatively Curved Manifolds, Ph. D. Thesis, Massachusetts Institute of Technology (USA) (1986) | MR

[33] Mazzeo, Rafe; Melrose, Richard Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI | MR | Zbl

[34] Melrose, Richard The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics, 4, A K Peters, 1993, xiv+377 pages | MR | Zbl

[35] Melrose, Richard; Sá Barreto, Antônio; Vasy, András Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces, Commun. Partial Differ. Equations, Volume 39 (2014) no. 3, pp. 452-511 | DOI | MR | Zbl

[36] Michel, René Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., Volume 65 (1981) no. 1, pp. 71-83 | DOI | MR | Zbl

[37] Monard, François; Stefanov, Plamen; Uhlmann, Gunther The geodesic ray transform on Riemannian surfaces with conjugate points, Commun. Math. Phys., Volume 337 (2015) no. 3, pp. 1491-1513 | DOI | MR | Zbl

[38] Muhometov, R. G. On a problem of reconstructing Riemannian metrics, Sib. Mat. Zh., Volume 22 (1981) no. 3, pp. 119-135 | MR

[39] Paternain, Gabriel P. Geodesic flows, Progress in Mathematics, 180, Birkhäuser, 1999, xiv+149 pages | DOI | MR | Zbl

[40] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther Tensor tomography on surfaces, Invent. Math., Volume 193 (2013) no. 1, pp. 229-247 | DOI | MR | Zbl

[41] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Math. Ann., Volume 363 (2015) no. 1-2, pp. 305-362 | DOI | MR | Zbl

[42] Pestov, Leonid N.; Sharafutdinov, Vladimir Integral geometry of tensor fields on a manifold of negative curvature, Sib. Mat. Zh., Volume 29 (1988) no. 3, pp. 114-130 | DOI | MR | Zbl

[43] Pestov, Leonid N.; Uhlmann, Gunther Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. Math., Volume 161 (2005) no. 2, pp. 1093-1110 | DOI | MR | Zbl

[44] Porrati, Massimo; Rabadan, Raul Boundary rigidity and holography, J. High Energy Phys. (2004) no. 1, 034, 24 pages | DOI | MR | Zbl

[45] Sa Barreto, Antonio; Wang, Yiran The scattering relation on asymptotically hyperbolic manifolds (2014) (https://arxiv.org/abs/1410.6842) | Zbl

[46] Sá Barreto, Antônio; Wang, Yiran The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity, Commun. Partial Differ. Equations, Volume 41 (2016) no. 8, pp. 1230-1302 | DOI | MR | Zbl

[47] Sá Barreto, Antônio; Wang, Yiran The scattering operator on asymptotically hyperbolic manifolds, J. Spectr. Theory, Volume 9 (2019) no. 1, pp. 269-313 | DOI | MR | Zbl

[48] Sharafutdinov, Vladimir Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, 1994, 271 pages | DOI | MR

[49] Sharafutdinov, Vladimir Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds, J. Geom. Anal., Volume 17 (2007) no. 1, pp. 147-187 | DOI | MR | Zbl

[50] Stefanov, Plamen; Uhlmann, Gunther Boundary rigidity and stability for generic simple metrics, J. Am. Math. Soc., Volume 18 (2005) no. 4, pp. 975-1003 | DOI | MR | Zbl

[51] Stefanov, Plamen; Uhlmann, Gunther Boundary and lens rigidity, tensor tomography and analytic microlocal analysis, Algebraic analysis of differential equations from microlocal analysis to exponential asymptotics, Springer, 2008, pp. 275-293 | DOI | MR | Zbl

[52] Stefanov, Plamen; Uhlmann, Gunther Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differ. Geom., Volume 82 (2009) no. 2, pp. 383-409 | DOI | MR | Zbl

[53] Stefanov, Plamen; Uhlmann, Gunther The geodesic X-ray transform with fold caustics, Anal. PDE, Volume 5 (2012) no. 2, pp. 219-260 | DOI | MR | Zbl

[54] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Boundary rigidity with partial data, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 299-332 | DOI | MR | Zbl

[55] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017) (https://arxiv.org/abs/1702.03638)

[56] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Inverting the local geodesic X-ray transform on tensors, J. Anal. Math., Volume 136 (2018) no. 1, pp. 151-208 | DOI | MR | Zbl

[57] Uhlmann, Gunther; Vasy, András The inverse problem for the local geodesic ray transform, Invent. Math., Volume 205 (2016) no. 1, pp. 83-120 | DOI | MR | Zbl

[58] Vargo, James A proof of lens rigidity in the category of analytic metrics, Math. Res. Lett., Volume 16 (2009) no. 6, pp. 1057-1069 | DOI | MR | Zbl

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