Positive harmonic functions on groups and covering spaces
Panagiotis Polymerakis

TL;DR
This paper proves that non-constant positive harmonic functions exist on certain Riemannian covering spaces with exponential volume growth, confirming a conjecture by Lyons and Sullivan.
Contribution
It establishes the existence of non-constant positive harmonic functions on Riemannian coverings with exponential volume growth, resolving a conjecture.
Findings
Existence of non-constant positive harmonic functions on specific covering spaces
Confirmation of Lyons and Sullivan's conjecture
Link between volume growth and harmonic functions
Abstract
We show that if is a normal Riemannian covering, with closed, and has exponential volume growth, then there are non-constant, positive harmonic functions on . This was conjectured by Lyons and Sullivan in \cite{LS}.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Dermatological and Skeletal Disorders
