On the spectral theory and dynamics of asymptotically hyperbolic manifolds
[Sur la théorie spectrale et la dynamique des variétés asymptotiquement hyperboliques]
Rowlett, Julie
Annales de l'Institut Fourier, Tome 60 (2010), p. 2461-2492 / Harvested from Numdam

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.

Publié le : 2010-01-01
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
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     title = {On the spectral theory and dynamics of asymptotically hyperbolic manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {60},
     year = {2010},
     pages = {2461-2492},
     doi = {10.5802/aif.2615},
     zbl = {1252.37025},
     mrnumber = {2849270},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2010__60_7_2461_0}
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Rowlett, Julie. On the spectral theory and dynamics of asymptotically hyperbolic manifolds. Annales de l'Institut Fourier, Tome 60 (2010) pp. 2461-2492. doi : 10.5802/aif.2615. http://gdmltest.u-ga.fr/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, Eur. Math. Soc., Zürich (IRMA Lect. Math. Theor. Phys.) Tome 8 (2005), pp. 1-31 | Article | MR 2160865 | Zbl 1071.81553

[2] Anderson, Michael T. Topics in conformally compact Einstein metrics, Perspectives in Riemannian geometry, Amer. Math. Soc., Providence, RI (CRM Proc. Lecture Notes) Tome 40 (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., Tome 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, University of Washington (2007) (Ph. D. Thesis) | MR 2710594

[6] Bahuaud, Eric Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics, Pacific J. Math., Tome 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, American Mathematical Society, Providence, RI, University Lecture Series, Tome 23 (2002) | MR 1862379 | Zbl 1195.37002

[8] Bérard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Tome 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., Tome 145 (1969), pp. 1-49 | Article | MR 251664 | Zbl 0191.52002

[10] Bolton, J. Conditions under which a geodesic flow is Anosov, Math. Ann., Tome 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, Tome 35 (1925), pp. 557-616 | Article

[12] Borthwick, David Scattering theory for conformally compact metrics with variable curvature at infinity, J. Funct. Anal., Tome 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, Tome 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., Tome 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., Tome 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., Tome 94 (1972), pp. 1-30 | Article | MR 298700 | Zbl 0254.58005

[18] Bowen, Rufus Maximizing entropy for a hyperbolic flow, Math. Systems Theory, Tome 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., Tome 9 (1999) no. 1, pp. 17-40 | MR 1760718 | Zbl 0957.57012

[20] Do Carmo, Manfredo Perdigão Riemannian geometry, Birkhäuser Boston Inc., Boston, MA, Mathematics: Theory & Applications (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, Tome 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., Tome 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., Tome 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., Tome 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), Tome 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., Tome 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, Tome 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), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 54 (1993), pp. 179-227 | MR 1216622

[30] Eberlein, Patrick; O’Neill, B. Visibility manifolds, Pacific J. Math., Tome 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., Tome 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., Tome 9 (2002) no. 2-3, pp. 139-151 | MR 1909634 | Zbl 1016.53031

[34] Franco, Ernesto Flows with unique equilibrium states, Amer. J. Math., Tome 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., Tome 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, Tome 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., Tome 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), Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 12 (1971) no. 3, pp. 343-358 | Article

[51] Hejhal, Dennis A. The Selberg trace formula for congruence subgroups, Bull. Amer. Math. Soc., Tome 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, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 548 and 1001 (1976 and 1983) | Zbl 0543.10020

[53] Hörmander, Lars The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, Classics in Mathematics (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., Tome 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., Tome 184 (2001) no. 2, pp. 291-312 | Article | MR 1851000 | Zbl 0997.58010

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

[58] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 54 (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), Tome 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, Tome 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., Tome 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., Tome 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., Tome 183 (2006) no. 864, pp. vi+83 | MR 2252687 | Zbl 1112.53002

[64] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Tome 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., Tome 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., Tome 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., Tome 75 (1987) no. 2, pp. 260-310 | Article | MR 916753 | Zbl 0636.58034

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

[70] Müller, Werner Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math., Tome 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, Tome 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), Tome 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), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 111 (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., Tome 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., Tome 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., Tome 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., Tome 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), Birkhäuser, Basel (Internat. Schriftenreihe Numer. Math.) Tome 65 (1984), pp. 521-525 | MR 820548 | Zbl 0559.10017

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

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

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

[83] Rubinstein, Michael; Sarnak, Peter Chebyshev’s bias, Experiment. Math., Tome 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., Tome 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., Tome 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.), Tome 20 (1956), pp. 47-87 | MR 88511 | Zbl 0072.08201

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

[88] Smale, S. Differentiable dynamical systems, Bull. Amer. Math. Soc., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 8 (1998) no. 1, pp. 179-217 | Article | MR 1601862 | Zbl 0908.58022