On the spectral theory and dynamics of asymptotically hyperbolic manifolds  [ Sur la théorie spectrale et la dynamique des variétés asymptotiquement hyperboliques ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2461-2492.

Cet article est une présentation rapide de la théorie spectrale et de la dynamique des variétés asymptotiquement hyperboliques à volume infini. Nous commençons par leur géométrie et quelques exemples, nous poursuivons en rappelant leur théorie spectrale, puis continuons sur des développements récents de leur dynamique. Nous concluons par une discussion des résultats qui démontrent un rapport entre leurs mécaniques quantiques et classiques et enfin, nous offrons quelques idées et conjectures.

We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.

DOI : https://doi.org/10.5802/aif.2615
Classification : 37D40,  58J50,  53C22
Mots clés : variété asymptotiquement hyperbolique, conformement compact, courbures negatives, spectre des longeurs géodesiques, flot géodesique, dynamique, formule de trace dynamique, entropie topologique, mécanique quantique et classique
@article{AIF_2010__60_7_2461_0,
     author = {Rowlett, Julie},
     title = {On the spectral theory and dynamics of asymptotically hyperbolic manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2461--2492},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     doi = {10.5802/aif.2615},
     mrnumber = {2849270},
     zbl = {1252.37025},
     language = {en},
     url = {www.numdam.org/item/AIF_2010__60_7_2461_0/}
}
Rowlett, Julie. On the spectral theory and dynamics of asymptotically hyperbolic manifolds. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2461-2492. doi : 10.5802/aif.2615. http://www.numdam.org/item/AIF_2010__60_7_2461_0/

[1] Anderson, Michael T. Geometric aspects of the AdS/CFT correspondence, AdS/CFT correspondence: Einstein metrics and their conformal boundaries (IRMA Lect. Math. Theor. Phys.) Volume 8, Eur. Math. Soc., Zürich, 2005, pp. 1-31 | Article | MR 2160865 | Zbl 1071.81553

[2] Anderson, Michael T. Topics in conformally compact Einstein metrics, Perspectives in Riemannian geometry (CRM Proc. Lecture Notes) Volume 40, Amer. Math. Soc., Providence, RI, 2006, pp. 1-26 | MR 2237104 | Zbl 1110.53031

[3] Anosov, D. V. Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., Volume 90 (1967), pp. 209 | MR 224110 | Zbl 0163.43604

[4] Arthur, James The trace formula and Hecke operators, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 11-27 | MR 993309 | Zbl 0671.10026

[5] Bahuaud, Eric An intrinsic characterization of asymptotically hyperbolic metrics (2007) (Ph. D. Thesis) | MR 2710594

[6] Bahuaud, Eric Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics, Pacific J. Math., Volume 239 (2009) no. 2, pp. 231-249 | Article | MR 2457230 | Zbl 1163.53025

[7] Barreira, Luis; Pesin, Yakov B. Lyapunov exponents and smooth ergodic theory, University Lecture Series, Volume 23, American Mathematical Society, Providence, RI, 2002 | MR 1862379 | Zbl 1195.37002

[8] Bérard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | Article | MR 455055 | Zbl 0341.35052

[9] Bishop, R. L.; O’Neill, B. Manifolds of negative curvature, Trans. Amer. Math. Soc., Volume 145 (1969), pp. 1-49 | Article | MR 251664 | Zbl 0191.52002

[10] Bolton, J. Conditions under which a geodesic flow is Anosov, Math. Ann., Volume 240 (1979) no. 2, pp. 103-113 | Article | MR 524660 | Zbl 0382.58017

[11] Born, M.; Heisenberg, W.; Jordan, P. Zur Quantenmechanik II, Zeitschrift für Physik, Volume 35 (1925), pp. 557-616 | Article

[12] Borthwick, David Scattering theory for conformally compact metrics with variable curvature at infinity, J. Funct. Anal., Volume 184 (2001) no. 2, pp. 313-376 | Article | MR 1851001 | Zbl 1006.58019

[13] Borthwick, David Upper and lower bounds on resonances for manifolds hyperbolic near infinity, Comm. Partial Differential Equations, Volume 33 (2008) no. 7-9, pp. 1507-1539 | Article | MR 2450168 | Zbl 1168.58012

[14] Borthwick, David; Judge, Chris; Perry, Peter A. Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv., Volume 80 (2005) no. 3, pp. 483-515 | Article | MR 2165200 | Zbl 1079.58023

[15] Borthwick, David; Perry, Peter Scattering poles for asymptotically hyperbolic manifolds, Trans. Amer. Math. Soc., Volume 354 (2002) no. 3, p. 1215-1231 (electronic) | Article | MR 1867379 | Zbl 1009.58021

[16] Borthwick, David; Perry, Peter Inverse scattering results for manifolds hyperbolic near infinity, 2009 (arXiv:0906.0542v2)

[17] Bowen, Rufus Periodic orbits for hyperbolic flows, Amer. J. Math., Volume 94 (1972), pp. 1-30 | Article | MR 298700 | Zbl 0254.58005

[18] Bowen, Rufus Maximizing entropy for a hyperbolic flow, Math. Systems Theory, Volume 7 (1974) no. 4, pp. 300-303 | MR 385928 | Zbl 0303.58014

[19] Canary, Richard D.; Minsky, Yair N.; Taylor, Edward C. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, J. Geom. Anal., Volume 9 (1999) no. 1, pp. 17-40 | MR 1760718 | Zbl 0957.57012

[20] do Carmo, Manfredo Perdigão Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992 (Translated from the second Portuguese edition by Francis Flaherty) | MR 1138207 | Zbl 0752.53001

[21] Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA, Volume 103 (2006) no. 8, pp. 2535-2540 | Article | MR 2203156 | Zbl 1160.53356

[22] Chang, Sun-Yung A.; Qing, Jie; Yang, Paul Some progress in conformal geometry, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 3 (2007), pp. Paper 122, 17 | Article | MR 2366900 | Zbl 1133.53031

[23] Chen, Su Shing; Manning, Anthony The convergence of zeta functions for certain geodesic flows depends on their pressure, Math. Z., Volume 176 (1981) no. 3, pp. 379-382 | Article | MR 610218 | Zbl 0437.58016

[24] Dirac, P. The quantum theory of the electron, Proc. R. Soc. London Series A, (1928) no. 778, pp. 610-624 (Containing Papers of a Mathematical and Physical Character 117) | Article

[25] Duistermaat, J. J.; Guillemin, V. W. The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | Article | MR 405514 | Zbl 0307.35071

[26] Eberlein, Patrick Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2), Volume 95 (1972), pp. 492-510 | Article | MR 310926 | Zbl 0217.47304

[27] Eberlein, Patrick Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc., Volume 178 (1973), pp. 57-82 | Article | MR 314084 | Zbl 0264.53027

[28] Eberlein, Patrick When is a geodesic flow of Anosov type? I,II, J. Differential Geometry, Volume 8 (1973), p. 437-463; ibid. 8 (1973), 565–577 | MR 380891 | Zbl 0295.58009

[29] Eberlein, Patrick; Hamenstädt, Ursula; Schroeder, Viktor Manifolds of nonpositive curvature, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) (Proc. Sympos. Pure Math.) Volume 54, Amer. Math. Soc., Providence, RI, 1993, pp. 179-227 | MR 1216622

[30] Eberlein, Patrick; O’Neill, B. Visibility manifolds, Pacific J. Math., Volume 46 (1973), pp. 45-109 | MR 336648 | Zbl 0264.53026

[31] Einstein, P.; Podolsky, B.; Rosen, N. Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev., Volume 47 (1935), pp. 777-780 | Article | Zbl 0012.04201

[32] Fefferman, Charles; Graham, C. Robin Conformal invariants, Astérisque (1985) no. Numero Hors Serie, pp. 95-116 (The mathematical heritage of Élie Cartan (Lyon, 1984)) | MR 837196 | Zbl 0602.53007

[33] Fefferman, Charles; Graham, C. Robin Q-curvature and Poincaré metrics, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 139-151 | MR 1909634 | Zbl 1016.53031

[34] Franco, Ernesto Flows with unique equilibrium states, Amer. J. Math., Volume 99 (1977) no. 3, pp. 486-514 | Article | MR 442193 | Zbl 0368.54014

[35] Freire, A.; Mañé, R. On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., Volume 69 (1982) no. 3, pp. 375-392 | Article | MR 679763 | Zbl 0476.58019

[36] Gangolli, Ramesh; Warner, Garth On Selberg’s trace formula, J. Math. Soc. Japan, Volume 27 (1975), pp. 328-343 | Article | MR 399354 | Zbl 0325.22014

[37] Gangolli, Ramesh; Warner, Garth Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J., Volume 78 (1980), pp. 1-44 http://projecteuclid.org/getRecord?id=euclid.nmj/1118786087 | MR 571435

[38] Graham, C. Robin Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999) (2000) no. 63, pp. 31-42 | MR 1758076 | Zbl 0984.53020

[39] Graham, C. Robin; Jenne, Ralph; Mason, Lionel J.; Sparling, George A. J. Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2), Volume 46 (1992) no. 3, pp. 557-565 | Article | MR 1190438 | Zbl 0726.53010

[40] Graham, C. Robin; Zworski, Maciej Scattering matrix in conformal geometry, Invent. Math., Volume 152 (2003) no. 1, pp. 89-118 | Article | MR 1965361 | Zbl 1030.58022

[41] Guillarmou, Colin Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J., Volume 129 (2005) no. 1, pp. 1-37 | Article | MR 2153454 | Zbl 1099.58011

[42] Guillarmou, Colin Generalized Krein formula, determinants, and Selberg zeta function in even dimension, Amer. J. Math., Volume 131 (2009) no. 5, pp. 1359-1417 | Article | MR 2555844 | Zbl 1207.58023

[43] Guillarmou, Colin; Naud, Frédéric Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds, Comm. Anal. Geom., Volume 14 (2006) no. 5, pp. 945-967 http://projecteuclid.org/getRecord?id=euclid.cag/1175790874 | MR 2287151 | Zbl 1127.58028

[44] Guillemin, Victor Wave-trace invariants and a theorem of Zelditch, Internat. Math. Res. Notices (1993) no. 12, pp. 303-308 | Article | MR 1253645 | Zbl 0798.58073

[45] Guillemin, Victor Wave-trace invariants, Duke Math. J., Volume 83 (1996) no. 2, pp. 287-352 | Article | MR 1390650 | Zbl 0858.58051

[46] Guillopé, Laurent Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J., Volume 53 (1986) no. 3, pp. 827-848 | Article | MR 860674 | Zbl 0611.53042

[47] Guillopé, Laurent; Zworski, Maciej Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Anal., Volume 11 (1995) no. 1, pp. 1-22 | MR 1344252 | Zbl 0859.58028

[48] Guillopé, Laurent; Zworski, Maciej Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., Volume 129 (1995) no. 2, pp. 364-389 | Article | MR 1327183 | Zbl 0841.58063

[49] Guillopé, Laurent; Zworski, Maciej The wave trace for Riemann surfaces, Geom. Funct. Anal., Volume 9 (1999) no. 6, pp. 1156-1168 | Article | MR 1736931 | Zbl 0947.58022

[50] Gutzwiller, M. C. Periodic orbits and classical quantization conditions, J. Math. Phys., Volume 12 (1971) no. 3, pp. 343-358 | Article

[51] Hejhal, Dennis A. The Selberg trace formula for congruence subgroups, Bull. Amer. Math. Soc., Volume 81 (1975), pp. 752-755 | Article | MR 371818 | Zbl 0304.10018

[52] Hejhal, Dennis A. The Selberg trace formula for PSL ( 2 , R ) . Vol. 1 and 2, Lecture Notes in Mathematics, Volume 548 and 1001, Springer-Verlag, Berlin, 1976 and 1983 | Zbl 0543.10020

[53] Hörmander, Lars The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003 (Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]) | MR 1996773 | Zbl 1028.35001

[54] Jakobson, Dmitry; Polterovich, Iosif; Toth, John A. A lower bound for the remainder in Weyl’s law on negatively curved surfaces, Int. Math. Res. Not. IMRN (2008) no. 2, pp. Art. ID rnm142, 38 | MR 2418855 | Zbl 1161.58010

[55] Joshi, Mark S.; Sá Barreto, Antônio Inverse scattering on asymptotically hyperbolic manifolds, Acta Math., Volume 184 (2000) no. 1, pp. 41-86 | Article | MR 1756569 | Zbl 1142.58309

[56] Joshi, Mark S.; Sá Barreto, Antônio The wave group on asymptotically hyperbolic manifolds, J. Funct. Anal., Volume 184 (2001) no. 2, pp. 291-312 | Article | MR 1851000 | Zbl 0997.58010

[57] Karnaukh, A. Spectral count on compact negatively curved surfaces (1996) (Ph. D. Thesis) | MR 2695000

[58] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, Volume 54, Cambridge University Press, Cambridge, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza) | MR 1326374 | Zbl 0878.58019

[59] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), Volume 99 (1974), pp. 1-13 | Article | MR 377980 | Zbl 0272.53025

[60] Lalley, S. P. The “prime number theorem” for the periodic orbits of a Bernoulli flow, Amer. Math. Monthly, Volume 95 (1988) no. 5, pp. 385-398 | Article | MR 937528 | Zbl 0645.28013

[61] Lax, Peter D.; Phillips, Ralph S. Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Comm. Pure Appl. Math., Volume 37 (1984) no. 3, pp. 303-328 | Article | MR 739923 | Zbl 0544.10024

[62] Lee, John M. The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom., Volume 3 (1995) no. 1-2, pp. 253-271 | MR 1362652 | Zbl 0934.58029

[63] Lee, John M. Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc., Volume 183 (2006) no. 864, pp. vi+83 | MR 2252687 | Zbl 1112.53002

[64] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | Article | MR 554385 | Zbl 0426.58016

[65] Manning, Anthony private correspondence, 2008

[66] Mazzeo, Rafe The Hodge cohomology of a conformally compact metric, J. Differential Geom., Volume 28 (1988) no. 2, pp. 309-339 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442281 | MR 961517 | Zbl 0656.53042

[67] Mazzeo, Rafe; Pacard, Frank Maskit combinations of Poincaré-Einstein metrics, Adv. Math., Volume 204 (2006) no. 2, pp. 379-412 | Article | MR 2249618 | Zbl 1097.53029

[68] Mazzeo, Rafe R.; Melrose, Richard B. 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

[69] Melrose, Richard B. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, Volume 4, A K Peters Ltd., Wellesley, MA, 1993 | MR 1348401 | Zbl 0796.58050

[70] Müller, Werner Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math., Volume 109 (1992) no. 2, pp. 265-305 | Article | MR 1172692 | Zbl 0772.58063

[71] Naud, Frédéric Classical and quantum lifetimes on some non-compact Riemann surfaces, J. Phys. A, Volume 38 (2005) no. 49, pp. 10721-10729 | Article | MR 2197679 | Zbl 1082.81026

[72] Parry, William; Pollicott, Mark An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2), Volume 118 (1983) no. 3, pp. 573-591 | Article | MR 727704 | Zbl 0537.58038

[73] Patterson, S. J. Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) (London Math. Soc. Lecture Note Ser.) Volume 111, Cambridge Univ. Press, Cambridge, 1987, pp. 281-323 | MR 903855 | Zbl 0611.30036

[74] Patterson, S. J.; Perry, Peter A. Divisor of the Selberg zeta function for Kleinian groups in even dimensions, Duke Math. J., Volume 326 (2001), pp. 321-390 (with an appendix by C. Epstein) | MR 1813434 | Zbl 1012.11083

[75] Perry, Peter A. The Laplace operator on a hyperbolic manifold. I. Spectral and scattering theory, J. Funct. Anal., Volume 75 (1987) no. 1, pp. 161-187 | Article | MR 911204 | Zbl 0631.58030

[76] Perry, Peter A. Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume, Geom. Funct. Anal., Volume 11 (2001) no. 1, pp. 132-141 | Article | MR 1829645 | Zbl 0986.11059

[77] Perry, Peter A. A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds, Int. Math. Res. Not. (2003) no. 34, pp. 1837-1851 | Article | MR 1988782 | Zbl 1035.58020

[78] Phillips, R. S.; Sarnak, P. The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math., Volume 155 (1985) no. 3-4, pp. 173-241 | Article | MR 806414 | Zbl 0611.30037

[79] Phillips, Ralph The spectrum of the Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) (Internat. Schriftenreihe Numer. Math.) Volume 65, Birkhäuser, Basel, 1984, pp. 521-525 | MR 820548 | Zbl 0559.10017

[80] Phillips, Ralph; Rudnick, Zeév The circle problem in the hyperbolic plane, J. Funct. Anal., Volume 121 (1994) no. 1, pp. 78-116 | Article | MR 1270589 | Zbl 0812.11035

[81] Rosenberg, Steven The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, Volume 31, Cambridge University Press, Cambridge, 1997 (An introduction to analysis on manifolds) | Article | MR 1462892 | Zbl 0868.58074

[82] Rowlett, Julie Dynamics of asymptotically hyperbolic manifolds, Pacific J. Math., Volume 242 (2009) no. 2, pp. 377-397 | Article | MR 2546718 | Zbl 1198.37036

[83] Rubinstein, Michael; Sarnak, Peter Chebyshev’s bias, Experiment. Math., Volume 3 (1994) no. 3, pp. 173-197 http://projecteuclid.org/getRecord?id=euclid.em/1048515870 | MR 1329368 | Zbl 0823.11050

[84] Sá Barreto, Antônio Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds., Duke Math. J., Volume 129 (2005) no. 3, pp. 407-480 | Article | MR 2169870 | Zbl 1154.58310

[85] Schrödinger, E. An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., Volume 28 (1926) no. 6, pp. 1049-1070 | Article

[86] Selberg, A. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), Volume 20 (1956), pp. 47-87 | MR 88511 | Zbl 0072.08201

[87] Shi, Yuguang; Tian, Gang Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys., Volume 259 (2005) no. 3, pp. 545-559 | Article | MR 2174416 | Zbl 1092.53033

[88] Smale, S. Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 | Article | MR 228014 | Zbl 0202.55202

[89] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 171-202 | Article | Numdam | MR 556586 | Zbl 0439.30034

[90] Walters, Peter A variational principle for the pressure of continuous transformations, Amer. J. Math., Volume 97 (1975) no. 4, pp. 937-971 | Article | MR 390180 | Zbl 0318.28007

[91] Yue, Chengbo The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., Volume 348 (1996) no. 12, pp. 4965-5005 | Article | MR 1348871 | Zbl 0864.58047

[92] Zelditch, Steven On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Volume 160 (1994) no. 1, pp. 81-92 http://projecteuclid.org/getRecord?id=euclid.cmp/1104269516 | Article | MR 1262192 | Zbl 0788.58043

[93] Zelditch, Steven Wave invariants at elliptic closed geodesics, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 145-213 | Article | MR 1437476 | Zbl 0876.58010

[94] Zelditch, Steven Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal., Volume 8 (1998) no. 1, pp. 179-217 | Article | MR 1601862 | Zbl 0908.58022