On the strong Liouville property for co-compact Riemannian covers. (English) Zbl 0848.31009
Let \(M\) be a Riemannian manifold and \(\Gamma\) be a discrete group of isometries of \(M\), isomorphic to some subgroup of the linear group \(GL (\mathbb{R}^d)\) and such that \(G/ \Gamma\) be a compact manifold. \(M\) is said to have the Liouville property when all the bounded harmonic functions (with respect to the Laplace-Beltrami operator) are constant and \(M\) is said to have the strong Liouville property when all the non-negative harmonic functions are constant. Then the following result holds:
\(M\) has the Liouville property if and only if the Laplace-Beltrami operator has no spectral gap.
\(M\) has the strong Liouville property if and only if \(M\) is not of exponential growth.
The paper presents a very short proof of the first assertion, using a recent result of F. Ledrappier and W. Ballmann, and a simplified proof of the “only if” part of the second assertion (already obtained by the same authors in a more general setting) since the “if” part of the second assertion is a result of Y. Guivarc’h, T. Lyons and D. Sullivan.
\(M\) has the Liouville property if and only if the Laplace-Beltrami operator has no spectral gap.
\(M\) has the strong Liouville property if and only if \(M\) is not of exponential growth.
The paper presents a very short proof of the first assertion, using a recent result of F. Ledrappier and W. Ballmann, and a simplified proof of the “only if” part of the second assertion (already obtained by the same authors in a more general setting) since the “if” part of the second assertion is a result of Y. Guivarc’h, T. Lyons and D. Sullivan.
Reviewer: J.Lacroix (Paris)
MSC:
| 31C12 | Potential theory on Riemannian manifolds and other spaces |
| 22D40 | Ergodic theory on groups |
| 22D05 | General properties and structure of locally compact groups |
| 31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |
| 30F15 | Harmonic functions on Riemann surfaces |
Keywords:
harmonic functions; Laplace Beltrami operator; Liouville property; strong Liouville propertyCitations:
Zbl 0801.53028References:
| [1] | Alexopoulos, G.,Fonctions harmoniques bornées sur les groupes résolubles, C. R. Acad. Sci. Paris, I,305 (1987), 777–779. · Zbl 0657.31014 |
| [2] | Ancona, A.,Théorie du potentiel sur les graphes et les variétés, Lecture Notes in Math.,1427, Springer Verlag, Berlin, Heidelberg, New York, (1990), 4–112. |
| [3] | Azencott, R.,Espaces de Poisson des groupes localement compacts, Lecture Notes in Math.,148, Springer Verlag, Berlin, Heidelberg, New York, (1970). · Zbl 0239.60008 |
| [4] | Babillot, M., Bougerol, Ph., andElie, L.,On the random difference equation X n=AnXn+Bn, the critical case, (1994), Preprint. |
| [5] | Ballman, W. andLedrappier, F.,Discretization of non-negative harmonic functions on Riemannian manifolds and Martin boundaries, Actes de la table ronde de géométrie differentielle en l’honneur de Marcel Berger, Séminaires et Congrès,1, Soc. Math. France. |
| [6] | Bougerol, Ph. andElie, L.,Existence of non-negative harmonic functions on groups and on covering manifolds, Ann. Ist. H. Poincaré (Probab. Stat.),31 (1995), 59–80. |
| [7] | Brooks, R.,The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv.,56 (1981), 501–508. · Zbl 0495.58029 · doi:10.1007/BF02566228 |
| [8] | Elie, L.,Comportement asymptotique du noyau potentiel sur les groupes de Lie, Ann. Scient. Ec. Norm. Sup.,15 (1982), 257–364. · Zbl 0509.60070 · doi:10.24033/asens.1428 |
| [9] | Furstenberg, H.,Random walks and discrete subgroups of Lie groups, Advances in Probability and Related topics,1, Editor P. Ney, Dekker, New York, (1971), 1–63. |
| [10] | Guivarc’h, Y.,Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France,101 (1973), 178–250. |
| [11] | Guivarc’h, Y.,Application d’un théorème limite local à la transience et à la récurrence de marches de Markov, in ”Théorie du potentiel”, Proceedings, Lecture Notes in Math.,1096, Springer Verlag, Berlin, Heidelberg, New York, (1984), 301–332. |
| [12] | Kaimanovich, V. A. andVershik, A. M.,Random walks on discrete groups: boundary and entropy, Ann. Probab., (1983), 457–490. · Zbl 0641.60009 |
| [13] | Kaimanovich, V. A.,Brownian motion and harmonic functions on covering manifolds: entropy approach, Soviet Math Dokl.,33 (1986), 812–816. · Zbl 0615.60074 |
| [14] | Kaimanovich, V. A.,Discretization of bounded harmonic functions on Riemannian manifolds and entropy, in ”Proceedings of the International Conference on Potential Theory”, Nagoya, (M. Kishi, ed.), de Gruyter, Berlin, (1992), 213–223. |
| [15] | Lin, M.,Conservative Markov processes on a topological space, Israel J. Math,8 (1970), 165–186. · Zbl 0219.60005 · doi:10.1007/BF02771312 |
| [16] | Lyons, T. andSullivan, D.,Function theory, random paths and covering spaces, J. Diff. Geom.,19 (1984), 299–323. · Zbl 0554.58022 · doi:10.4310/jdg/1214438681 |
| [17] | Milnor, J.,A note on curvature and the fundamental group, J. Diff. Geom.,2 (1968), 1–7. · Zbl 0162.25401 · doi:10.4310/jdg/1214501132 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
