  We say that a PDE in a Riemannian manifold $M$ is geometric if,$\ $whenever
$u$ is a solution of the PDE on a domain $\Omega$ of $M$, the composition
$u_{\phi}:=u\circ\phi$ is also solution on $\phi^{-1}\left( \Omega\right) $,
for any isometry $\phi$ of $M.$ We prove that if $u\in C^{1}\left(
\mathbb{H}^{n}\right) $ is a solution of a geometric PDE satisfying the
comparison principle, where $\mathbb{H}^{n}$ is the hyperbolic space of
constant sectional curvature $-1,$ $n\geq2,$ and if \[
\limsup_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla
u\right\Vert \right) =0, \] where $S_{R}$ is a geodesic sphere of
$\mathbb{H}^{n}$ centered at fixed point $o\in\mathbb{H}^{n}$ with radius $R,$
then $u$ is constant. Moreover, given $C>0,$ there is a bounded non-constant
harmonic function $v\in C^{\infty }\left( \mathbb{H}^{n}\right) $ such that \[
\lim_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla v\right\Vert
\right) =C. \] The first part of the above result is a consequence of a more
general theorem proved in the paper which asserts that if $G$ is a non compact
Lie group with a left invariant metric, $u\in C^{1}\left( G\right) $ a solution
of a left invariant PDE (that is, if $v$ is a solution of the PDE on a domain
$\Omega$ of $G$, the composition $v_{g}:=v\circ L_{g}$ of $v$ with a left
translation $L_{g}:G\rightarrow G,$ $L_{g}\left( h\right) =gh,$ is also
solution on $L_{g}^{-1}\left( \Omega\right) $ for any $g\in G),$ the PDE
satisfies the comparison principle and% \[ \limsup_{R\rightarrow\infty}\left(
\sup_{g\in B_{R}}\left\Vert \operatorname*{Ad}\nolimits_{g}\right\Vert
\sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0, \] where
$\operatorname*{Ad}\nolimits_{g}:\mathfrak{g}\rightarrow\mathfrak{g}$ is the
adjoint map of $G$ and $\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is
constant.
