A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition

Abstract

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 $u_{\varphi }:=u\circ \varphi$ is also solution on ${\varphi }^{-1}\left(\Omega \right)$, for any isometry $\varphi$ 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 $-1,$ $n\ge 2,$ 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 $L_g:G\to G,$ $L_g\left(h\right)=gh,$ 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 ${\mathrm{Ad}}_g:\mathfrak{g}\to \mathfrak{g}$ is the adjoint map of $G$ and $\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is constant.