Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 169–216 | Cite as

Asymptotics of relative heat traces and determinants on open surfaces of finite area

  • Clara L. AldanaEmail author


The goal of this article is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps \((M,g)\) and a metric \(h\) on the surface that is a conformal transformation of the initial metric \(g\). We prove the existence of the relative determinant of the pair \((\Delta _{h},\Delta _{g})\) under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips, and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov’s formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.


Surfaces with asymptotically cusp ends Heat kernels  Asymptotic expansion of heat traces Relative determinants 



This article is registered at the MPG, AEI-2012-200. This paper expands part of my doctoral thesis. I thank my supervisor Werner Müller for his guidance. I am grateful to Rafe Mazzeo, Eugenie Hunsicker, and Sylvie Paycha for helpful discussions and their interest in this work. The author thanks an anonymous referee for the suggestions and comments. Finally, the author thanks the Mathematical Institute at the University of Bonn for hosting her during her graduate studies.


  1. 1.
    Albin, P., Aldana, C.L., Rochon, F.: Ricci flow and the determinant of the Laplacian on non-compact surfaces, ArXiv:0909.0807Google Scholar
  2. 2.
    Aldana, C.L.: Inverse spectral theory and relative determinants of elliptic operators on surfaces with cusps, Ph.D. dissertation, Bonner Math. Schriften, Univ. Bonn, Mathematisches Institut, Bonn (2009)Google Scholar
  3. 3.
    Branson, T.P., Gilkey, P.B., Ørsted, B.: Leading terms in the heat invariants. Proc. Am. Math. Soc. 109(2), 437–450 (1990)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bruneau, V.: Propriétés asymptotiques du spectre continu d’opérateurs de Dirac. Université de Nantes, These de Doctorat (1995)Google Scholar
  5. 5.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Clarendon Press, Oxford (1999)Google Scholar
  6. 6.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, New York (1984)zbMATHGoogle Scholar
  7. 7.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17, 15–53 (1982)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cheng, S.Y., Li, P., Yau, S.T.: On the upper estimate of the heat kernel of a complete Riemannian manifold. Am. J. Math. 103, 1021–1063 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Colin de Verdière, Y.: Une nouvelle démonstration du prolongement méromorphe des séries d’Eisenstein. C. R. Acad. Sci. Paris Ser. I 293, 361–363 (1981)zbMATHGoogle Scholar
  10. 10.
    Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32, 703–716 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lundelius, R.: Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume. Duke Math. J. 71, 211–242 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Müller, W.: Spectral theory for Riemannian manifolds with cusps and a related trace formula. Math. Nachr. 111, 197–288 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Müller, W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109, 265–303 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Müller, W.: Relative zeta functions, relative determinants, and scattering theory. Comm. Math. Phys. 192, 309–347 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Müller, W., Salomonsen, G.: Scattering theory for the Laplacian on manifolds with bounded curvature. J. Funct. Anal. 253, 158–206 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Osgood, B., Phillips, R., Sarnak, P.: Extremal of determinants of Laplacians. J. Funct. Anal. 80, 148–211 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Osgood, B., Phillips, R., Sarnak, P.: Compact Isospectral sets of surfaces. J. Funct. Anal. 80, 212–234 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ray, D.B., Singer, I.M.: \(R\)-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sakai, T.: On eigen-values of Laplacian and curvature of Riemannian manifold. Tôhoku Math. J. (2) 23, 589–603 (1971)zbMATHCrossRefGoogle Scholar
  20. 20.
    Vaillant, B.: Index and spectral theory for manifolds with generalized fibred cusps, Ph.D. dissertation, Bonner Math. Schriften 344, Univ. Bonn, Mathematisches Institut, Bonn (2001)Google Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Max P lanck Institute for Gravitational Physics(Albert Einstein Institute), MPGPotsdamGermany

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