Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Eigenfunctions of the Laplacian
on rotationally symmetric manifolds

Author: Michel Marias
Journal: Trans. Amer. Math. Soc. 350 (1998), 4367-4375
MSC (1991): Primary 58G25, 60J45
MathSciNet review: 1616007
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold $M=(\mathbf{R}^n,ds^2),$ $ds^2=dr^2+f(r)^2d\theta ^2,$ are constructed explicitly under the assumption that an integral of $f(r)$ converges. This integral is the same one which gives the existence of nonconstant harmonic functions on $M.$

References [Enhancements On Off] (What's this?)

  • 1. J. Cheeger, S.T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math., 34, 465-480, (1981). MR 82i:58065
  • 2. J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of a complete Riemannian manifold, J. Diff. Geom., 17, 15-53, (1982). MR 84b:58109
  • 3. H.I. Choi, Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc., 281, 691-716, (1984). MR 85b:53040
  • 4. P. Eberlein, B. O'Neill, Visibility manifolds, Pac. J. Math., 46, 45-109, (1973). MR 49:1421
  • 5. J. Elliott, Eigenfunction expansions associated with singular differential operators, Trans. Amer. Math. Soc.,78, 406-425, (1955). MR 16:927a
  • 6. A. Erdelyi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, 1954. MR 15:868a
  • 7. W. Feller, The parabolic differential equation and the associated semi-groups of transformations, Ann. of Math., 55, 468-519, (1952). MR 13:948a
  • 8. M. Freidlin, Functional Integration and Partial Differential Equations, Annals of Math. Studies, Princeton University Press, Princeton, 1985. MR 87g:60066
  • 9. R.E. Greene, H. Wu, Function Theory on Manifolds Which Possess a Pole, L.N.M. 699, Springer, Berlin, 1979. MR 81a:53002
  • 10. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, and Kodansha, Tokyo, 1981. MR 84b:60080
  • 11. N. Mandouvalos, M. Marias, Eigenfunctions of the Laplacian and boundary behaviour on manifolds of hyperbolic type, Math. Proc. Camb. Phil. Soc., 122, 551-564, (1997). CMP 97:17
  • 12. P. March, Brownian motion and harmonic functions on rotationally symmetric manifolds, Annals of Prob., 14, 793-801, (1986). MR 87m:60181
  • 13. M. Marias, Generalised Poisson Kernels and Applications, Rev. Roumaine Math. Pures Appl., 35, 417-429, (1990). CMP 91:04
  • 14. J. Milnor, On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly, 84, 43-46, (1977). MR 55:1257
  • 15. L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Inter. Math. Research Notices 1992, No. 2, 27-38. MR 93d:58158
  • 16. S.T. Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures et Appl., 57, 191- 201, (1978). MR 81b:58041

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G25, 60J45

Retrieve articles in all journals with MSC (1991): 58G25, 60J45

Additional Information

Michel Marias
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54.006, Greece

Keywords: Rotationally symmetric manifolds, radial curvature, eigenfunctions, harmonic functions, heat kernels, diffusion processes, subordination measure.
Received by editor(s): July 16, 1995
Received by editor(s) in revised form: January 18, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society