Abstract
Lyons and Sullivan conjectured in Lyons and Sullivan (J. Differential Geom. 19(2), 299–323, 1984) that if p : M → N is a normal Riemannian covering, with N closed, and M has exponential volume growth, then there are non-constant, positive harmonic functions on M. This was proved recently in Polymerakis (Adv. Math. 379, 107552–107558, 2021) exploiting the Lyons-Sullivan discretization and some sophisticated estimates on the green metric on groups. In this note, we provide a self-contained proof relying only on elementary properties of the Brownian motion.
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Acknowledgements
I would like to thank Werner Ballmann for some very fruitful discussions and remarks. I would also like to thank the Max Planck Institute for Mathematics in Bonn for its support and hospitality. I am also grateful to the referee for some helpful comments. Data sharing is not applicable to this article as no new data were created or analysed in this study.
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Polymerakis, P. On the Strong Liouville Property of Covering Spaces. Potential Anal 59, 1449–1455 (2023). https://doi.org/10.1007/s11118-022-10019-8
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DOI: https://doi.org/10.1007/s11118-022-10019-8
Keywords
- Positive harmonic functions
- Strong Liouville property
- Riemannian covering
- Volume growth
- Exponential growth