The Laplace-Beltrami Operator on a Rotationally Symmetric Surface

  • Nikolai Tarkhanov
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)


The aim of this work is to highlight a number of analytic problems which make the analysis on manifolds with true cuspidal points much more difficult than that on manifolds with conic points while such singularities are topologically equivalent. To this end we discuss the Laplace-Beltrami operator on a compact rotationally symmetric surface with a complete metric. Even though the symmetry assumptions made here lead to a simplified situation in which standard separation of variables works, it is hoped that the study of this example can nevertheless bring to light some features which may subsist in the more general framework of the calculus on compact manifolds with cusps due to V. Rabinovich et al. (1997).

Mathematics Subject Classification (2000)

Primary 58-XX Secondary 47-XX. 


Manifolds with singularities pseudodifferential operators ellipticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Baider, A., Noncompact Riemannian manifolds with discrete spectra. J. Differential Geom. 14 (1979), no. 1, 41–58.zbMATHMathSciNetGoogle Scholar
  2. [2]
    Collet, J.F., and Volpert, V.A., Computation of the index of linear elliptic operators in unbounded cylinders. J. Funct. Anal. 164 (1999), 34–59.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Dodziuk, J., L 2-harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc. 77 (1979), no. 3, 395–401.zbMATHMathSciNetGoogle Scholar
  4. [4]
    Daletskii, Yu.L., and Krein, M.G., Stability of Solutions to Differential Equations in a Banach Space. Nauka, Moscow, 1970.Google Scholar
  5. [5]
    Dezin, A.A., General Questions of the Theory of Boundary Problems. Nauka, Moscow, 1980.zbMATHGoogle Scholar
  6. [6]
    Fedosov, B., Schulze, B.-W., and Tarkhanov, N., Analytic index formulas for elliptic corner operators. Annales de l’Institut Fourier (Grenoble) 52 (2002), no. 3, 899–982.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Gokhberg, I.Ts., and Sigal, E.I., An operator generalisation of the logarithmic residue theorem and the theorem of Rouché. Mat. Sb. 84 (126) (1971), no. 4, 607–629.MathSciNetGoogle Scholar
  8. [8]
    Keldysh, M.V., On the characteristic values and characteristic functions of certain classes of non-selfadjoint equations. Dokl. Akad. Nauk SSSR 77 (1951), 11–14.zbMATHGoogle Scholar
  9. [9]
    Kiselev, O., and Shestakov, I., Asymptotics of solutions to the Laplace-Beltrami equation on manifolds with cusps. SIAM J. of Math. Anal. (in preparation).Google Scholar
  10. [10]
    Kleine, R., Discreteness conditions for the Laplacian on complete noncompact Riemannian manifolds. Math. Z. 198 (1988), no. 1, 127–141.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Kozlov, V., and Maz’ya, V., Differential Equations with Operator Coefficients. Springer-Verlag, Berlin et al., 1999.zbMATHGoogle Scholar
  12. [12]
    Kuz’minov, V.I., and Shvedov, I.A., An addition theorem for manifolds with discrete spectrum of the Laplace operator. Siberian Math. J. 47 (2006), no. 3, 557–574.MathSciNetGoogle Scholar
  13. [13]
    Maz’ya, V.G., and Plamenevskii, B.A., On asymptotics of solutions of the Dirichlet problem near an isolated singularity of the boundary. Vestnik Leningrad. Univ., Mat. 13 (1977), 60–65.Google Scholar
  14. [14]
    Olwer, F.W.T., Asymptotics and Special Functions. Academic Press, N.Y., 1974.Google Scholar
  15. [15]
    Plamenevskii, B.A., Algebras of Pseudodifferential Operators. Kluwer Academic Publishers, Dordrecht NL, 1989.zbMATHGoogle Scholar
  16. [16]
    Postnikov, M.M., Riemannian Geometry. Encyclopaedia of Mathematical Sciences, Vol. 91, Springer, Berlin et al., 2001.Google Scholar
  17. [17]
    Rabinovich, V., Schulze, B.-W., and Tarkhanov, N., A calculus of boundary value problems in domains with non-Lipschitz singular points. Math. Nachr. 215 (2000), 115–160.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Schulze, B.-W., and Tarkhanov, N., Euler solutions of pseudodifferential equations. Integral Equations and Operator Theory 33 (1999), 98–123.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Schulze, B.-W., and Tarkhanov, N., Singular functions and relative index for elliptic corner operators. Ann. Univ. Ferrara, Sez. VII, Sc. Mat., Suppl. Vol. XLV (1999), 293–310.MathSciNetGoogle Scholar
  20. [20]
    Vasilevski, N.L., On a general local principle for C *-algebras. Izv. VUZ. North-Caucasian Region, Natural Sciences, Special Issue “Pseudodifferential operators and some problems of mathematical physics,” 2005, 34–42.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Nikolai Tarkhanov
    • 1
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Personalised recommendations