On Liouville's theorem and the Strong Liouville Property

Abstract

We explore Liouville's theorem and the Strong Liouville Property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the case of manifolds with rotational symmetry), allowing us to recover and sharpen known results under minimal assumptions. We provide explicit estimates for the slowest-growing nonconstant harmonic functions on cones and surfaces, and construct examples where doubling fails but Liouville and SLP still hold. Finally, we prove a nonlinear Liouville theorem for $p$-subharmonic functions, $p\geq 2$, under curvature bounds, in complete Riemannian surfaces with a pole which simultaneously recover Milnor's and Cheng--Yau's theorems as particular cases. This result appears to be new and suggests a unified geometric perspective on linear and nonlinear Liouville phenomena.