X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds  [ Transformée en rayons X et rigidité du bord pour les variétés asymptotiquement hyperboliques ]
Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 2857-2919.

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.

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.

Publié le : 2020-06-26
DOI : https://doi.org/10.5802/aif.3339
Classification : 35R30,  37D40,  53C22
Mots clés: transformée en rayons X, rigidité du bord, variété asymptotiquement hyperbolique
@article{AIF_2019__69_7_2857_0,
     author = {Graham, C. Robin and Guillarmou, Colin and Stefanov, Plamen and Uhlmann, Gunther},
     title = {X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2857--2919},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     doi = {10.5802/aif.3339},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2019__69_7_2857_0/}
}
Graham, C. Robin; Guillarmou, Colin; Stefanov, Plamen; Uhlmann, Gunther. X-Ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 2857-2919. doi : 10.5802/aif.3339. https://aif.centre-mersenne.org/item/AIF_2019__69_7_2857_0/

[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 | Article | MR 2653898 | Zbl 1193.53131

[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 | Article | MR 1623603 | Zbl 0908.35136

[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 | Article | MR 1104811 | Zbl 0742.44002

[4] Besse, Arthur L. Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 10, Springer, 1987, xii+510 pages | Article | MR 867684 | Zbl 0613.53001

[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 | Article | MR 3473907 | Zbl 1351.58018

[6] Coddington, Earl A.; Levinson, Norman Theory of ordinary differential equations, McGraw-Hill, 1955, xii+429 pages | MR 0069338 | Zbl 0064.33002

[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 | Article | MR 2169902 | Zbl 1080.53033

[8] Czech, Bartłomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James Integral geometry and holography, J. High Energy Phys. (2015) no. 10, 175, 41 pages | Article | MR 3435464 | Zbl 1388.83217

[9] Dyatlov, Semyon; Guillarmou, Colin Pollicott–Ruelle resonances for open systems, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3089-3146 | Article | MR 3556517 | Zbl 1367.37038

[10] Eberlein, Patrick Geodesic flow in certain manifolds without conjugate points, Trans. Am. Math. Soc., Volume 167 (1972), pp. 151-170 | Article | MR 295387 | Zbl 0209.53304

[11] Eberlein, Patrick When is a geodesic flow of Anosov type? I, J. Differ. Geom., Volume 8 (1973), pp. 437-463 | Article | MR 0380891 | Zbl 0285.58008

[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 837196 | Zbl 0602.53007

[14] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry, Universitext, Springer, 1987, xii+248 pages | Article | MR 909697 | Zbl 0636.53001

[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 1758076 | Zbl 0984.53020

[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 | Article | MR 1112625 | Zbl 0765.53034

[17] Gromoll, Detlef; Klingenberg, Wilhelm; Meyer, Wolfgang Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, Volume 55, Springer, 1975, vi+287 pages | MR 0365399 | Zbl 0293.53001

[18] Guillarmou, Colin Lens rigidity for manifolds with hyperbolic trapped sets, J. Am. Math. Soc., Volume 30 (2017) no. 2, pp. 561-599 | Article | MR 3600043 | Zbl 1377.53098

[19] Guillarmou, Colin; Mazzucchelli, Marco Marked boundary rigidity for surfaces, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 4, pp. 1459-1478 | Article | MR 3789172 | Zbl 1390.37054

[20] Heil, Konstantin; Moroianu, Andrei; Semmelmann, Uwe Killing and conformal Killing tensors, J. Geom. Phys., Volume 106 (2016), pp. 383-400 | Article | MR 3508929 | Zbl 1342.53066

[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 | Article | MR 1108651 | Zbl 0794.44001

[22] Helgason, Sigurdur Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, Volume 39, American Mathematical Society, 1994, xiv+611 pages | Article | MR 1280714 | Zbl 0809.53057

[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 | Article | MR 3770848 | Zbl 1387.53100

[24] Ivanov, Sergei Volume comparison via boundary distances, Proceedings of the International Congress of Mathematicians. Volume II (2010), pp. 769-784 | MR 2827818 | Zbl 1230.53042

[25] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type, Ann. Math., Volume 99 (1974), pp. 1-13 | Article | MR 377980 | Zbl 0272.53025

[26] Klingenberg, Wilhelm Riemannian geometry, De Gruyter Studies in Mathematics, Volume 1, Walter de Gruyter, 1995, x+409 pages | Article | MR 1330918 | Zbl 0911.53022

[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 | Article | MR 3825848 | Zbl 1394.37056

[28] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol I, Interscience Publishers, 1963, xi+329 pages | MR 0152974 | Zbl 0119.37502

[29] Lassas, Matti; Sharafutdinov, Vladimir; Uhlmann, Gunther Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., Volume 325 (2003) no. 4, pp. 767-793 | Article | MR 1974568 | Zbl 1331.53066

[30] Lehtonen, Jere The geodesic ray transform on two-dimensional Cartan–Hadamard manifolds (2016) (https://arxiv.org/abs/1612.04800) (Ph. D. Thesis) | Zbl 1369.53005

[31] Lehtonen, Jere; Railo, Jesse; Salo, Mikko Tensor tomography on Cartan-Hadamard manifolds, Inverse Probl., Volume 34 (2018) no. 4, 044004, 27 pages | Article | MR 3781677 | Zbl 1394.53078

[32] Mazzeo, Rafe Hodge Cohomology of Negatively Curved Manifolds (1986) (Ph. D. Thesis) | MR 2941112

[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 | Article | MR 916753 | Zbl 0636.58034

[34] Melrose, Richard The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics, Volume 4, A K Peters, 1993, xiv+377 pages | MR 1348401 | Zbl 0796.58050

[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 | Article | MR 3169792 | Zbl 1323.58020

[36] Michel, René Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., Volume 65 (1981) no. 1, pp. 71-83 | Article | MR 636880 | Zbl 0471.53030

[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 | Article | MR 3339183 | Zbl 1319.53086

[38] Muhometov, R. G. On a problem of reconstructing Riemannian metrics, Sib. Mat. Zh., Volume 22 (1981) no. 3, pp. 119-135 | MR 621466

[39] Paternain, Gabriel P. Geodesic flows, Progress in Mathematics, Volume 180, Birkhäuser, 1999, xiv+149 pages | Article | MR 1712465 | Zbl 0930.53001

[40] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther Tensor tomography on surfaces, Invent. Math., Volume 193 (2013) no. 1, pp. 229-247 | Article | MR 3069117 | Zbl 1275.53067

[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 | Article | MR 3394381 | Zbl 1328.53099

[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 | Article | MR 953028 | Zbl 0675.53048

[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 | Article | MR 2153407 | Zbl 1076.53044

[44] Porrati, Massimo; Rabadan, Raul Boundary rigidity and holography, J. High Energy Phys. (2004) no. 1, 034, 24 pages | Article | MR 2045873 | Zbl 1243.53120

[45] Sa Barreto, Antonio; Wang, Yiran The scattering relation on asymptotically hyperbolic manifolds (2014) (https://arxiv.org/abs/1410.6842) | Zbl 1409.35163

[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 | Article | MR 3532393 | Zbl 1349.35260

[47] Sá Barreto, Antônio; Wang, Yiran The scattering operator on asymptotically hyperbolic manifolds, J. Spectr. Theory, Volume 9 (2019) no. 1, pp. 269-313 | Article | MR 3900787 | Zbl 1409.35163

[48] Sharafutdinov, Vladimir Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, 1994, 271 pages | Article | MR 1374572

[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 | Article | MR 2302878 | Zbl 1142.53029

[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 | Article | MR 2163868 | Zbl 1079.53061

[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 | Article | MR 2758914 | Zbl 1138.53039

[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 | Article | MR 2520797 | Zbl 1247.53049

[53] Stefanov, Plamen; Uhlmann, Gunther The geodesic X-ray transform with fold caustics, Anal. PDE, Volume 5 (2012) no. 2, pp. 219-260 | Article | MR 2970707 | Zbl 1271.53070

[54] Stefanov, Plamen; Uhlmann, Gunther; Vasy, András Boundary rigidity with partial data, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 299-332 | Article | MR 3454376 | Zbl 1335.53055

[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 | Article | MR 3892472 | Zbl 07008552

[57] Uhlmann, Gunther; Vasy, András The inverse problem for the local geodesic ray transform, Invent. Math., Volume 205 (2016) no. 1, pp. 83-120 | Article | MR 3514959 | Zbl 1350.53098

[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 | Article | MR 2576693 | Zbl 1202.53041