# harmonic functions on hyperbolic manifolds with finite volume is constant?

Consider a hyperbolic manifold $$M=H^n/\Gamma$$ with finite volume. Suppose that there exists a harmonic function $$u$$ defined on $$M$$. Then is $$u$$ a constant? If $$M$$ is compact, yes. So I want to know the non-compact case. Are there counterexamples?

• Did you check Compactifications os Symmetric Spaces by Guivarc'h, Ji and Taylor ? If I remember correctly, their main theorem state that the Martin boundary is the Satake-Furstenberg boundary. – M. Dus Dec 14 '19 at 6:37

However, any bounded harmonic function on a complete Riemannian manifold with finite volume is constant. In the book of Pigola and Setti [proposition 3.5 in Global divergence theorems in nonlinear PDEs and geometry. Ensaios Matematicos], you will more general result. But in the case of hyperbolic manifold $$M$$ with finite volume, I suspect that any harmonic function $$h$$ such that $$h(x)=\mathcal{O}\left(e^{\tau\ \mathrm{dist}(o,x)}\right)$$ (for some $$\tau<\dim M-1$$) is constant.