The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 924–950 | Cite as

Heat Invariants of the Steklov Problem

  • Iosif Polterovich
  • David A. SherEmail author


We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with the Steklov eigenvalues. As an application, it is proved that a three-dimensional ball is uniquely defined by its Steklov spectrum among all Euclidean domains with smooth connected boundary.


Steklov problem Heat trace Dirichlet-to-Neumann operator Riemannian manifold Spectral rigidity 

Mathematics Subject Classification

58J50 58J35 58J40 



The authors would like to thank P. Hislop and P. Perry for useful discussions, and the anonymous referee for helpful remarks. Research of I.P. was supported in part by NSERC, FQRNT, and the Canada Research Chairs Program. Research of D.S. was supported in part by the CRM-ISM postdoctoral fellowship.


  1. 1.
    Agranovich, M.S.: Some asymptotic formulas for elliptic pseudodifferential operators. Funkc. Anal. Prilozh. 21, 53–56 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alexandrov, A.D.: Uniqueness theorem for surfaces in the large I. Vestn. Leningr. Univ. 11, 5–17 (1956) Google Scholar
  3. 3.
    Alias, L., de Lira, J., Malacarne, J.M.: Constant higher-order mean curvature hypersurfaces in Riemannian spaces. J. Inst. Math. Jussieu 5(4), 527–562 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Besse, A.: Manifolds All of Whose Geodesics are Closed. Ergeb, Math., vol. 93. Springer, New York (1978) CrossRefzbMATHGoogle Scholar
  5. 5.
    Binoy, Santhanam, G.: Sharp upper bound and a comparison theorem for the first nonzero Steklov eigenvalue. arXiv:1208.1690
  6. 6.
    Brock, F.: An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81, 69–71 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992) CrossRefzbMATHGoogle Scholar
  8. 8.
    Colbois, B., El Soufi, A., Girouard, A.: Isoperimetric control of the Steklov spectrum. J. Funct. Anal. 261(5), 1384–1399 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Duistermaat, H., Guillemin, V.: Spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Edward, J.: An inverse spectral result for the Neumann operator on planar domains. J. Funct. Anal. 111, 312–322 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Edward, J., Wu, S.: Determinant of the Neumann operator on smooth Jordan curves. Proc. Am. Math. Soc. 111(2), 357–363 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fraser, A., Schoen, R.: The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math. 226, 4011–4030 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fraser, A., Schoen, R.: Eigenvalue bounds and minimal surfaces in the ball. arXiv:1209.3789
  14. 14.
    Gilkey, P.: Asymptotic Formulae in Spectral Geometry. CRC Press, Boca Raton (2004) zbMATHGoogle Scholar
  15. 15.
    Gilkey, P., Grubb, G.: Logarithmic terms in asymptotic expansions of heat operator traces. Commun. Partial Differ. Equ. 23(5–6), 777–792 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Girouard, A., Polterovich, I.: Shape optimization for low Neumann and Steklov eigenvalues. Math. Methods Appl. Sci. 33(4), 501–516 (2010) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Girouard, A., Polterovich, I.: Upper bounds for Steklov eigenvalues on surfaces. ERA-MS 19, 77–85 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2008) Google Scholar
  19. 19.
    Grubb, G., Seeley, R.: Weakly parametric pseudodifferential operators and Atiyah–Patodi–Singer boundary problems. Invent. Math. 121, 481–529 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Guillemin, V.: The Radon transform on Zoll surfaces. Adv. Math. 22, 85–119 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hassell, A., Zworski, M.: Resonant rigidity of S 2. J. Funct. Anal. 169, 604–609 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
  23. 23.
    Hörmander, L.: The Analysis of Partial Differential Operators, vol. IV. Grundlehren, vol. 275. Springer, New York (1984) Google Scholar
  24. 24.
    Jammes, P.: Prescription du spectre de Steklov dans une classe conforme. arXiv:1209.4571
  25. 25.
    Karpukhin, M., Kokarev, G., Polterovich, I.: Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. arXiv:1209.4869
  26. 26.
    Lee, J., Uhlmann, G.: Determining isotropic real-analytic conductivities by boundary measurements. Commun. Pure Appl. Math. 42, 1097–1112 (1989) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Lee, Y.: Burghelea–Friedlander–Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion. Trans. Am. Math. Soc. 355, 4093–4110 (2003) CrossRefzbMATHGoogle Scholar
  28. 28.
    Polterovich, I.: Combinatorics of the heat trace on spheres. Can. J. Math. 54, 1086–1099 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Seeley, R.: Complex Powers of an Elliptic Operator. In: Singular Integrals, Providence, RI. Proc. Symp. Pure Math., pp. 288–307 (1967) CrossRefGoogle Scholar
  30. 30.
    Tanno, S.: Eigenvalues of the Laplacian of Riemannian manifolds. Tohoku Math. J. (2) 25(3), 391–403 (1973) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Taylor, M.: Partial Differential Equations II. Qualitative Studies of Linear Equations. Applied Mathematical Sciences, vol. 116. Springer, New York (1996) CrossRefGoogle Scholar
  32. 32.
    Viaclovsky, J.: Topics in Riemannian geometry Lecture notes. Available online at
  33. 33.
    Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3, 343–356 (1954) MathSciNetGoogle Scholar
  34. 34.
    Zelditch, S.: Maximally degenerate Laplacians. Ann. Inst. Fourier 46(2), 547–587 (1996) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Département de Mathématiques et de StatistiqueUniversité de MontréalMontrealCanada
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Centre de Recherches MathematiquesUniversité de MontréalMontrealCanada

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