Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature

TL;DR

This paper explores the existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds with unbounded curvature, revealing complex boundary behaviors and challenging existing conjectures in geometric analysis.

Contribution

It provides examples of manifolds with unbounded curvature where the boundary at infinity behaves unexpectedly, complicating the understanding of harmonic functions on such spaces.

Findings

Bounded harmonic functions can be rich even with degenerate boundary limits

Full boundary at infinity requires non-trivial blow-up techniques

Challenges the Greene and Wu conjecture on harmonic functions

Abstract

The Liouville property of a complete Riemannian manifold (i.e., the question whether there exist non-trivial bounded harmonic functions) attracted a lot of attention. For Cartan-Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan-Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds is much more complicated than one might have expected.