Revista Matemática Iberoamericana

Polynomial growth harmonic functions on complete Riemannian manifolds

Yong Hah Lee

Full-text: Open access

Abstract

In this paper, we give a sharp estimate on the dimension of the space of polynomial growth harmonic functions with fixed degree on a complete Riemannian manifold, under various assumptions.

Article information

Source
Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 315-332.

Dates
First available in Project Euclid: 17 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1087482017

Mathematical Reviews number (MathSciNet)
MR2073122

Zentralblatt MATH identifier
1059.53036

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 31C05: Harmonic, subharmonic, superharmonic functions

Keywords
polynomial growth harmonic function volume doubling condition Poincaré inequality mean value property

Citation

Lee, Yong Hah. Polynomial growth harmonic functions on complete Riemannian manifolds. Rev. Mat. Iberoamericana 20 (2004), no. 2, 315--332. https://projecteuclid.org/euclid.rmi/1087482017


Export citation

References

  • Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213-230.
  • Cheeger, J. and Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature.} J. Differential Geom. 6 (1971), 119-128.
  • Cheeger, J., Colding, T. and Minicozzi II, W.: Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature.} Geom. Funct. Anal. 5 (1995), 948-954.
  • Cheeger, J., Gromov, M. and Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds.} J. Differential Geom. 17 (1982), 15-53.
  • Colding, T. and Minicozzi II, W.: On function theory on spaces with a lower Ricci curvature bound.} Math. Res. Lett. 3 (1996), 241-246.
  • Colding, T. and Minicozzi II, W.: Harmonic functions with polynomial growth.} J. Differential Geom. 46 (1997), 1-77.
  • Colding, T. and Minicozzi II, W.: Large scale behavior of kernels of Schrödinger operators.} Amer. J. Math. 117 (1997), 1355-1398.
  • Colding, T. and Minicozzi II, W.: Generalized Liouville properties for manifolds.} Math. Res. Lett. 3 (1996), 723-729.
  • Colding, T. and Minicozzi II, W.: Harmonic functions on manifolds.} Ann. of Math. 146 (1997), 725-747.
  • Colding, T. and Minicozzi II, W.: Liouville theorems for harmonic sections and applications manifolds.} Comm. Pure Appl. Math. 51 (1998), 113-138.
  • Colding, T. and Minicozzi II, W.: Weyl type bounds for harmonic functions.} Invent. Math. 131 (1998), 257-298.
  • Coulhon, Th. and Saloff-Coste, L.: Variétés riemanniennes isométriques á l'infini.} Rev. Mat. Iberoamericana 11 (1995), 687-726.
  • Grigor'yan, A.: Dimension of space of harmonic functions.} Mat. Zametki 48 (1990), 55-61. English translation in Math. Notes 48 (1990), 1114-1118.
  • Grigor'yan, A.: The heat equation on noncompact Riemannian manifolds.} Mat. Sb. 182 (1991) no.1, 55-87. English translation in Math. USSR-Sb. 72 (1992), no. 1, 47-77.
  • Kanai, M.: Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds.} J. Math. Soc. Japan 37 (1985), no. 3, 391-413.
  • Kim, S. W. and Lee, Y. H.: Polynomial growth harmonic functions on connected sums of complete Riemannian manifolds.} Math. Z. 233 (2000), 103-113.
  • Kim, S. W. and Lee, Y. H.: Generalized Liouville property for Schrödinger operator on Riemannian manifolds.} Math. Z. 238 (2001), no. 2, 355-387.
  • Li, P.: On the Sobolev constant and the $p$-spectrum of a compact Riemannian manifold.} Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 451-469.
  • Li, P.: Harmonic sections of polynomial growth.} Math. Res. Lett. 4 (1997), no. 1, 35-44.
  • Li, P. and Tam, L. F.: Positive harmonic functions on complete Riemannian manifolds with non-negative curvature outside a compact set.} Ann. of Math. (2) 125 (1987), no. 1, 171-207.
  • Li, P. and Tam, L. F.: Green's functions, harmonic functions, and volume comparison.} J. Differential Geom. 41 (1995), no. 2, 277-318.
  • Liu, Z-D.: Ball covering property and nonnegative Ricci curvature outside a compact set.} In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), 459-464. Proc. Sympos. Pure Math. 54, Part 3. Amer. Math. Soc., Providence, RI, 1993.
  • Saloff-Coste, L.: A note on Poincaré, Sobolev and Harnack inequalities.} Internat. Math. Res. Notices 1992, no. 2, 27-38.
  • Tam, L. F.: A note on harmonic forms on complete manifolds.} Proc. Amer. Math. Soc. 126 (1998), no. 10, 3097-3108.
  • Wang, J.: Linear growth harmonic functions on complete manifolds.} Comm. Anal. Geom. 3 (1995), no. 3-4, 683-698.
  • Yau, S. T.: Harmonic functions on complete Riemannian manifolds.} Comm. Pure Appl. Math. 28 (1975), 201-228.