Liouville property and quasi-isometries on negatively curved Riemannian surfaces

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai'...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2024-02, Vol.154 (1), p.131-153
Main Authors: Granados, Ana, Pestana, Domingo, Portilla, Ana, Rodríguez, José M., Tourís, Eva
Format: Article
Language:English
Subjects:
Euclidean space
Hypotheses
Riemann manifold
Riemann surfaces
Surface stability
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2022.93