Mathematical Physics, Analysis and Geometry

, Volume 11, Issue 3–4, pp 365–379 | Cite as

Spectrum of the Lichnerowicz Laplacian on Asymptotically Hyperbolic Surfaces



We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian Δ L contains the ray \(\big[\frac{1}{4},+\infty\big[\). If moreover the scalar curvature is constant then − 2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality \(\langle \Delta u, u\rangle_{L^2}\geqslant \frac{1}{4}||u||^2_{L^2}\) holds for all smooth compactly supported function u, then there is no other value in the spectrum.


Asymptotically hyperbolic surfaces Lichnerowicz Laplacian Symmetric 2-tensor Essential spectrum Asymptotic behavior 

Mathematics Subject Classifications (2000)

35P15 58J50 47A53 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson, L.: Elliptic systems on manifolds with asymptotically negative curvature. Indiana Univ. Math. J. 42(4), 1359–1388 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Avez, A.: Le laplacien de Lichnerowicz. Rend. Sem. Mat. Univ. Politec. Torino (35), 123–127 (1976–1977)Google Scholar
  3. 3.
    Avez, A.: Le laplacien de Lichnerowicz sur les tenseurs. C. R. Acad. Sci. Paris Sér. A (284), 1219–1220 (1977)Google Scholar
  4. 4.
    Besse, A.L.: Einstein manifolds. Ergebnisse d. Math. 3. folge, vol. 10. Springer, Berlin (1987)Google Scholar
  5. 5.
    Buzzanca, C.: Le laplacien de Lichnerowicz sur les surfaces à coubure négative constante. C. R. Acad. Sci. Paris Sér. A (285), 391–393 (1977)Google Scholar
  6. 6.
    Buzzanca, C.: Il laplaciano di Lichnerowicz sui tensori. Boll. Un. Mat. Ital. 6(3-B), 531–541 (1984)MathSciNetGoogle Scholar
  7. 7.
    Delay, E.: Essential spectrum of the Lichnerowicz laplacian on two tensor on asymptotically hyperbolic manifolds. J. Geom. Phys. 43, 33–44 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Delay, E.: TT-eigentensors for the Lichnerowicz laplacian on some asymptotically hyperbolic manifolds with warped products metrics. Manuscripta Math. 123(2), 147–165 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Koiso, N.: On the second derivative of the total scalar curvature. Osaka J. Math. 16(2), 413–421 (1979)MATHMathSciNetGoogle Scholar
  11. 11.
    Lee, J.M.: The spectrum of an asymptotically hyperbolic Einstein manifold. Comm. Anal. Geom. 3, 253–271 (1995)MATHMathSciNetGoogle Scholar
  12. 12.
    Mazzeo, R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28, 309–339 (1988)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Laboratoire d’analyse non linéaire et géométrie, Faculté des SciencesAvignonFrance

Personalised recommendations