Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds

Communications in Analysis and Geometry

Volumn 14, Issue 5, 2006, Pages 945-967

  Guillarmou, Colin ^a   Naud, Frédéric ^b  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33846651297     PISSN: 10198385     EISSN: None     Source Type: Journal    
DOI: 10.4310/CAG.2006.v14.n5.a5     Document Type: Article

References (41)

1. W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. 138 (1993), 213-242.
2. D. Borthwick and P. Perry, Scattering poles for asymptotically hyperbolic manifolds, Trans. A.M.S. 354 (2002), 1215-1231.
3. D. Borthwick, C. Judge and P. Perry, Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Preprint.
4. R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25.
5. T. Branson, The Functional Determinant, Lecture Notes Series, 4, Seoul National University, 1993.
6. U. Bunke and M. Olbrich, Fuchsian groups of the second kind and representations carried by the limit set, Invent. Math. 127 (1997), 127-154.
7. U. Bunke and M. Olbrich, Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group, Ann. Math. 149 (1999), 627-689.
8. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Birkhauser Boston, 1992.
9. R. Gover, Laplacian operators and Q curvature on conformally Einstein manifolds, ArXiv:math.DG/0506037.
10. R. Gover and K. Hirachi, Conformally invariant powers of the Laplacian - A complete non-existence theorem, J. Amer. Math. Soc. 17 (2004), no. 2, 389-405.
11. C. R. Graham, R. Jenne, L. J. Manson and G. A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. 46 (1992), no. 2, 557-565.
12. C. R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo, (Suppl.), 63 Ser. II (2000), 31-42.
13. C. R. Graham, Private communication.
14. C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118.
15. C. Guillarmou, Resonances and scattering poles on asymptotically hyperbolic manifolds, Math. Res. Lett. 12 (2005), 103-119.
16. L. Guillopé and M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal. 129 (1995), no. 2, 364-389.
17. L. Guillopé and M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymp. Anal. 11 (1995), 1-22.
18. L. Guillopé and M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. Math. 145 (1997), 597-660.
19. L. Guillopé and M. Zworski, The wave trace for Riemann surfaces, G.A.F.A. 9 (1999), 1156-1168.
20. L. Guillopé, K. Lin and M. Zworski, The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245 (2004), no. 1, 149-176.
21. S. Helgason, Groups and Geometric Analysis, Academic Press, 1984.
22. D. Hejhal, The Selberg Trace Formula for PSL(2, ℝ), Lecture Notes in Mathematics, 548, Springer-Verlag, Berlin, 1976.
23. H. Huber, Zur analytischen theorie hyperbolisher raumformen und bewegungsgruppen 2, Math. Ann. 142 (1961), 385-398.
24. M. Joshi and A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41-86.
25. M. Joshi and A. Sá Barreto, The wave group on asymptotically hyperbolic manifolds, J. Funct. Anal. 184 (2001), no. 2, 291-312.
26. B. Maskit, A characterization of Schottky groups, J. Anal. Math. 19 (1967), 227-230.
27. R. Mazzeo and R. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310.
28. R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Diff. Geom. 28 (1988), 309-339.
29. R. Mazzeo, Elliptic theory of differential edge operators. I, Commun. Partial Diff. Equations 16 (1991), 1615-1664.
30. F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 116-153.
31. F. Naud, Precise asymptotics of the length spectrum for finite geometry Riemann surfaces, Int. Math. Res. Notices (2005), no. 5, 299-310.
32. S. Patterson and P. Perry, The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein, Duke Math. J. 106 (2001), 321-391.
33. P. Perry, The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix, J. Reine Angew. Math. 398 (1989), 67-91.
34. P. Perry, Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume, GAFA, Geom. Funct. Anal. 11 (2001), 132-141.
35. P. Perry, A Poisson formula and lower bounds for resonances on hyperbolic manifolds, Int. Math. Res. Notices 34 (2003), 1837-1851.
36. B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241-247.
37. P. Sarnak, Prime geodesic theorems, Phd Thesis, Stanford University, 1980.
38. D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171-202.
39. D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups., Acta Math. 153 (1984), nos. 3, 4, 259-277.
40. L. Stoyanov, Ruelle zeta functions and spectra of transfer operators for some axiom A flows, Preprint (2005).
41. M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Invent. Math. 136 (1999), no. 2, 353-409.