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Harmonic functions on ℝ-covered foliations

Published online by Cambridge University Press:  01 August 2009

S. FENLEY ,
R. FERES  and
K. PARWANI
S. FENLEY
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA (email: [email protected])
R. FERES
Affiliation:
Department of Mathematics, Washington University, St Louis, MO 63130, USA (email: [email protected])
K. PARWANI
Affiliation:
Department of Mathematics, Eastern Illinois University, Charleston, IL 61920, USA (email: [email protected])
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Abstract

Let (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1141 - 1161
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Ancona, A.. Théorie du potentiel sur les graphes et les variétés. École d’été de Probabilités de Saint-Flour XVIII—1988 (Lecture Notes in Mathematics, 1427). Springer, Berlin, 1990, pp. 1–112.CrossRefGoogle Scholar
[2]Anderson, M. and Schoen, R.. Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. 121 (1985), 429–461.CrossRefGoogle Scholar
[3]Aronszajn, N.. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36 (1957), 235–249.Google Scholar
[4]Candel, A.. The harmonic measures of Lucy Garnett. Adv. Math. 176(2) (2003), 187–247.CrossRefGoogle Scholar
[5]Candel, A. and Conlon, L.. Foliations I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2000.Google Scholar
[6]Candel, A. and Conlon, L.. Foliations II (Graduate Studies in Mathematics, 60). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[7]Deroin, B. and Kleptsyn, V.. Random conformal dynamical systems. Geom. Funct. Anal. 17 (2007), 1043–1105.CrossRefGoogle Scholar
[8]Dippolito, P.. Codimension one foliations of closed manifolds. Ann. of Math. (2) 107 (1978), 403–453.CrossRefGoogle Scholar
[9]Emery, M.. Stochastic Calculus in Manifolds. Springer, Berlin, 1989.CrossRefGoogle Scholar
[10]Federer, H.. Geometric Measure Theory. Springer, Berlin, 1996.CrossRefGoogle Scholar
[11]Fenley, S.. Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows. Comment. Math. Helv. 77 (2002), 415–490.CrossRefGoogle Scholar
[12]Feres, R. and Zhegib, A.. Leafwise holomorphic functions. Proc. Amer. Math. Soc. 131(6) (2003), 1717–1725.CrossRefGoogle Scholar
[13]Feres, R. and Zhegib, A.. Dynamics on the space of harmonic functions and the foliated Liouville problem. Ergod. Th. & Dynam. Sys. 25(2) (2005), 503–516.CrossRefGoogle Scholar
[14]Garnett, L.. Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 (1983), 285–311.CrossRefGoogle Scholar
[15]Gabai, D. and Oertel, U.. Essential laminations in 3-manifolds. Ann. Math. 130 (1989), 41–73.CrossRefGoogle Scholar
[16]Hempel, J.. 3-manifolds (Annals of Mathematics Studies, 86). Princeton University Press, Princeton, NJ, 1976.Google Scholar
[17]Hsu, E.P.. Stochastic Analysis on Manifolds (Graduate Studies in Mathematics, 38). American Mathematical Society, Providence, RI, 2002.CrossRefGoogle Scholar
[18]Kifer, Y.. Brownian motion and positive harmonic functions on complete manifolds of non-positive curvature. From Local Times to Global Geometry, Control and Physics. Ed. K. D. Elworthy. Wiley, New York, 1986, pp. 187–232.Google Scholar
[19]Kifer, Y. and Ledrappier, F.. Hausdorff dimension of harmonic measures on negatively curved manifolds. Trans. Amer. Math. Soc. 318(2) (1990), 685–704.CrossRefGoogle Scholar
[20]Lyons, T. and Sullivan, D.. Function theory, random paths and covering spaces. J. Differential Geom. 19 (1984), 299–323.CrossRefGoogle Scholar
[21]Moser, J.. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577–591.CrossRefGoogle Scholar
[22]Plante, J.. Foliations with measure preserving holonomy. Ann. of Math. (2) 107 (1975), 327–361.CrossRefGoogle Scholar
[23]Rudin, W.. Real and Complex Analysis. McGraw-Hill, New York, 1987.Google Scholar
[24]Solodov, V. V.. Components of topological foliations. Math. USSR Sb. 47(2) (1984), 329–343.CrossRefGoogle Scholar
[25]Sullivan, D.. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36 (1976), 225–255.CrossRefGoogle Scholar
[26]Schoen, R. and Yau, S.-T.. Lectures on Differential Geometry (Conference Proceedings and Lecture Notes in Geometry and Topology, 1). International Press Incorporated, Boston.Google Scholar
[27]Thurston, W.. Three-manifolds, foliations and circles I. Preprint, 1997.Google Scholar