Convergence theorems for relative spectral functions on hyperbolic Riemann surfaces of finite volume
Log in :: RSS/Atom Feeds
Duke Mathematical Journal

Convergence theorems for relative spectral functions on hyperbolic Riemann surfaces of finite volume

Jay Jorgenson and Rolf Lundelius

Source: Duke Math. J. Volume 80, Number 3 (1995), 785-819.

First Page PDF: View first page of article (PDF, 94 KB)

Primary Subjects: 58G25
Secondary Subjects: 11F72, 58G26

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077246294
Mathematical Reviews number (MathSciNet): MR1370116
Zentralblatt MATH identifier: 0973.58016
Digital Object Identifier: doi:10.1215/S0012-7094-95-08027-2

References

[A] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44.
Mathematical Reviews (MathSciNet): MR56:679
Zentralblatt MATH: 0347.32010
Digital Object Identifier: doi:10.2307/1971024
JSTOR: links.jstor.org
[Be] A. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983.
Mathematical Reviews (MathSciNet): MR85d:22026
Zentralblatt MATH: 0528.30001
[B] L. Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton Univ. Press, Princeton, N.J., 1974, 43–55. Ann. of Math. Studies, No. 79.
Mathematical Reviews (MathSciNet): MR50:13497
Zentralblatt MATH: 0294.32016
[BGV] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992.
Mathematical Reviews (MathSciNet): MR94e:58130
Zentralblatt MATH: 0744.58001
[Bu] P. Buser, Riemannsche Flächen mit Eigenwerten in $(0,$ $1/4)$, Comment. Math. Helv. 52 (1977), no. 1, 25–34.
Mathematical Reviews (MathSciNet): MR55:7924
Zentralblatt MATH: 0348.53027
[Ch] I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984.
Mathematical Reviews (MathSciNet): MR86g:58140
Zentralblatt MATH: 0551.53001
[CC] B. Colbois and G. Courtois, Les valeurs propres inférieures à $\frac 14$ des surfaces de Riemann de petit rayon d'injectivité, Comment. Math. Helv. 64 (1989), no. 3, 349–362.
Mathematical Reviews (MathSciNet): MR91d:58249
Zentralblatt MATH: 0684.53040
[DJ] J. Dodziuk and J. Jorgenson, Spectral asymptotics on degenerating hyperbolic $3$-manifolds, preprint.
Mathematical Reviews (MathSciNet): MR1434990
[DM] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109.
Mathematical Reviews (MathSciNet): MR41:6850
Zentralblatt MATH: 0181.48803
[DP] E. D'Hoker and D. H. Phong, On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), no. 4, 537–545.
Mathematical Reviews (MathSciNet): MR87i:58159
Zentralblatt MATH: 0599.30073
Digital Object Identifier: doi:10.1007/BF01211063
[E]1 I. Efrat, Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys. 119 (1988), no. 3, 443–451.
Mathematical Reviews (MathSciNet): MR90c:58184
Zentralblatt MATH: 0661.10038
Digital Object Identifier: doi:10.1007/BF01218082
[E]2 I. Efrat, Erratum: “Determinants of Laplacians on surfaces of finite volume”, Comm. Math. Phys. 138 (1991), no. 3, 607.
Mathematical Reviews (MathSciNet): MR92c:58149
Zentralblatt MATH: 0722.11029
Digital Object Identifier: doi:10.1007/BF02102044
[Fa] J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352, Springer-Verlag, Berlin, 1973.
Mathematical Reviews (MathSciNet): MR49:569
Zentralblatt MATH: 0281.30013
[GHJ] J. Grotowski, J. Huntley, and J. Jorgenson, Small eigenvalues on degenerating hyperbolic Riemann surfaces, preprint.
[Ha] W. Harvey, Chabauty spaces of discrete groups, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton Univ. Press, Princeton, N.J., 1974, 239–246. Ann. of Math. Studies, No. 79.
Mathematical Reviews (MathSciNet): MR51:883
Zentralblatt MATH: 0313.32028
[He1] D. A. Hejhal, Regular $b$-groups, degenerating Riemann surfaces, and spectral theory, Mem. Amer. Math. Soc. 88 (1990), no. 437, iv+138.
Mathematical Reviews (MathSciNet): MR92h:11043
Zentralblatt MATH: 0718.11024
[He2] D. A. Hejhal, The Selberg trace formula for ${\rm PSL}(2,R)$. Vol. I, Springer-Verlag, Berlin, 1976.
Mathematical Reviews (MathSciNet): MR55:12641
Zentralblatt MATH: 0347.10018
[He3] D. A. Hejhal, The Selberg trace formula for ${\rm PSL}(2,\,{\bf R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983.
Mathematical Reviews (MathSciNet): MR86e:11040
Zentralblatt MATH: 0543.10020
[He4] D. A. Hejhal, A continuity method for spectral theory on Fuchsian groups, Modular forms (Durham, 1983) ed. R. A. Rankin, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 107–140.
Mathematical Reviews (MathSciNet): MR87g:11063
Zentralblatt MATH: 0559.10016
[I] H. Iwaniec, Nonholomorphic modular forms and their applications, Modular forms (Durham, 1983) ed. R. A. Rankin, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 157–196.
Mathematical Reviews (MathSciNet): MR87f:11030
Zentralblatt MATH: 0558.10018
[HJL1] J. Huntley, J. Jorgenson, and R. Lundelius, On the asymptotic behavior of counting functions associated to degenerating hyperbolic Riemann surfaces, preprint.
Mathematical Reviews (MathSciNet): MR1471099
Digital Object Identifier: doi:10.1006/jfan.1997.3097
[HJL2] J. Huntley, J. Jorgenson, and R. Lundelius, Continuity of small eigenfunctions on degenerating Riemann surfaces with hyperbolic cusps, to appear in Boletin Socidad Mex. Math.
Mathematical Reviews (MathSciNet): MR1387655
[JLa1] J. Jorgenson and S. Lang, On Cramér's theorem for general Euler products with functional equation, Math. Ann. 297 (1993), no. 3, 383–416.
Mathematical Reviews (MathSciNet): MR94k:11101
Zentralblatt MATH: 0789.11054
Digital Object Identifier: doi:10.1007/BF01459509
[JLa2] J. Jorgenson, S. Lang, and D. Goldfeld, Explicit formulas, Lecture Notes in Mathematics, vol. 1593, Springer-Verlag, Berlin, 1994.
Mathematical Reviews (MathSciNet): MR96f:11110
Zentralblatt MATH: 0828.11043
[JL1] J. Jorgenson and R. Lundelius, Convergence of the heat kernel and the resolvent kernel on degenerating hyperbolic Riemann surfaces of finite volume, to appear in Quaestiones Math.
Mathematical Reviews (MathSciNet): MR1354117
[JL2] J. Jorgenson and R. Lundelius, Continuity of relative hyperbolic spectral theory through metric degeneration, to appear in Duke Math. J.
Mathematical Reviews (MathSciNet): MR1394748
Digital Object Identifier: doi:10.1215/S0012-7094-96-08402-1
Project Euclid: euclid.dmj/1077243628
[JL3] J. Jorgenson and R. Lundelius, A regularized heat trace for hyperbolic Riemann surfaces of finite volume, preprint, 1994.
Mathematical Reviews (MathSciNet): MR1600164
Digital Object Identifier: doi:10.1007/s000140050039
[K] T. Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo, 1973.
Mathematical Reviews (MathSciNet): MR55:2759
Zentralblatt MATH: 0268.10012
[L] R. Lundelius, Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume, Duke Math. J. 71 (1993), no. 1, 211–242.
Mathematical Reviews (MathSciNet): MR94j:58178
Zentralblatt MATH: 0790.58044
Digital Object Identifier: doi:10.1215/S0012-7094-93-07109-8
Project Euclid: euclid.dmj/1077289842
[M] H. P. McKean, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225–246.
Mathematical Reviews (MathSciNet): MR57:12843a
[MS] H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69.
Mathematical Reviews (MathSciNet): MR36:828
Zentralblatt MATH: 0198.44301
[MP] S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242–256.
Mathematical Reviews (MathSciNet): MR11,108b
Zentralblatt MATH: 0041.42701
[Sa] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120.
Mathematical Reviews (MathSciNet): MR89e:58116
Zentralblatt MATH: 0618.10023
Digital Object Identifier: doi:10.1007/BF01209019
[Se] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.
Mathematical Reviews (MathSciNet): MR19,531g
Zentralblatt MATH: 0072.08201
[St] R. Strichartz, Review of Heat Kernels And Spectral Theory by E. B. Davies, Bull. Amer. Math. Soc.(N.S.) 23 (1990), 222–227.
[Wo] M. Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), no. 2, 487–539.
Mathematical Reviews (MathSciNet): MR92b:58055
Zentralblatt MATH: 0747.58026
[W] S. A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 2, 283–315.
Mathematical Reviews (MathSciNet): MR89c:58136
Zentralblatt MATH: 0629.58029
Digital Object Identifier: doi:10.1007/BF01217814

2009 © Duke University Press