Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T03:14:34.707Z Has data issue: false hasContentIssue false
  • English
  • Français

HEAT KERNEL BOUNDS, POINCARÉ SERIES, AND L2 SPECTRUM FOR LOCALLY SYMMETRIC SPACES

Part of: Parabolic equations and systems Partial differential equations on manifolds; differential operators Global differential geometry

Published online by Cambridge University Press:  01 August 2008

ANDREAS WEBER
ANDREAS WEBER*
Affiliation:
Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstr. 2, 76128 Karlsruhe, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We derive upper Gaussian bounds for the heat kernel on complete, noncompact locally symmetric spaces M=Γ∖X with nonpositive curvature. Our bounds contain the Poincaré series of the discrete group Γ and therefore we also provide upper bounds for this series.

Keywords

heat kernelslocally symmetric spacesPoincaré series

MSC classification

Secondary: 58J35: Heat and other parabolic equation methods 35K05: Heat equation 53C35: Symmetric spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 1 , August 2008 , pp. 73 - 86
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Anker, J.-P. and Ji, L., ‘Heat kernel and Green function estimates on noncompact symmetric spaces’, Geom. Funct. Anal. 9(6) (1999), 1035–1091.Google Scholar
[2]Anker, J.-P. and Ostellari, P., ‘The heat kernel on noncompact symmetric spaces’, in: Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl. Ser. 2, 210 (American Mathematical Society, Providence, RI, 2003), pp. 27–46.Google Scholar
[3]Chavel, I. and Karp, L., ‘Large time behavior of the heat kernel: the parabolic λ-potential alternative’, Comment. Math. Helv. 66(4) (1991), 541–556.CrossRefGoogle Scholar
[4]Corlette, K., ‘Hausdorff dimensions of limit sets. I’, Invent. Math. 102(3) (1990), 521–541.Google Scholar
[5]Davies, E. B. and Mandouvalos, N., ‘Heat kernel bounds on manifolds with cusps’, J. Funct. Anal. 75(2) (1987), 311–322.CrossRefGoogle Scholar
[6]Davies, E. B., ‘Heat kernel bounds on hyperbolic space and Kleinian groups’, Proc. London Math. Soc. (3) 57(1) (1988), 182–208.CrossRefGoogle Scholar
[7]Elstrodt, J., ‘Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I’, Math. Ann. 203 (1973), 295–300.CrossRefGoogle Scholar
[8]Elstrodt, J., ‘Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II’, Math. Z. 132 (1973), 99–134.Google Scholar
[9]Elstrodt, J., ‘Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. III’, Math. Ann. 208 (1974), 99–132.Google Scholar
[10]Grigor’yan, A., ‘Estimates of heat kernels on Riemannian manifolds’, in: Spectral Theory and Geometry (Edinburgh, 1998), London Mathematical Society Lecture Note Series, 273 (Cambridge University Press, Cambridge, 1999), pp. 140–225.Google Scholar
[11]Knieper, G., ‘On the asymptotic geometry of nonpositively curved manifolds’, Geom. Funct. Anal. 7(4) (1997), 755–782.CrossRefGoogle Scholar
[12]Leuzinger, E., ‘Critical exponents of discrete groups and L 2-spectrum’, Proc. Amer. Math. Soc. 132(3) (2004), 919–927.Google Scholar
[13]Li, P., ‘Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature’, Ann. of Math. (2) 124(1) (1986), 1–21.CrossRefGoogle Scholar
[14]Nicholls, P. J., The Ergodic Theory of Discrete Groups, London Mathematical Society Lecture Note Series, 143 (Cambridge University Press, Cambridge, 1989).Google Scholar
[15]Ostellari, P., ‘Estimations globales du noyau de la chaleur’, PhD Thesis, Université Henri Poincaré, Nancy 1, 2003.Google Scholar
[16]Patterson, S. J., ‘The limit set of a Fuchsian group’, Acta Math. 136(3–4) (1976), 241–273.CrossRefGoogle Scholar
[17]Simon, B., ‘Large time behavior of the heat kernel: on a theorem of Chavel and Karp’, Proc. Amer. Math. Soc. 118(2) (1993), 513–514.CrossRefGoogle Scholar
[18]Sullivan, D., ‘The density at infinity of a discrete group of hyperbolic motions’, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202.CrossRefGoogle Scholar
[19]Sullivan, D., ‘Related aspects of positivity in Riemannian geometry’, J. Differential Geom. 25(3) (1987), 327–351.CrossRefGoogle Scholar