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 on the hyperbolic space ${\mathbb {H}}^{n}$ of constant sectional curvature $-1,$ $n\ge 2$, is geometric if, whenever u is a solution of the PDE on a domain $\Omega$ of ${\mathbb {H}}^{n}$, the composition $u_{\phi }:=u\circ \phi$ is also a solution on $\phi ^{-1}\left( \Omega \right)$ for any isometry $\phi$ of ${\mathbb {H}}^{n}$. We prove that if $u\in C^{1}\left( {\mathbb {H}}^{n}\right)$ is a solution of a geometric PDE satisfying the comparison principle and if

$$\begin{aligned} \limsup _{r\rightarrow \infty }\left( e^{2r}\sup _{S_{r}}\left\| \nabla u\right\| \right) =0, \end{aligned}$$

(1)

where $S_{r}$ is a geodesic sphere of ${\mathbb {H}}^{n}$ centered at a fixed point $o\in {\mathbb {H}}^{n}$ with radius r,  then u is constant. However, given $C>0,$ there exists a bounded non-constant harmonic function $v\in C^{\infty }\left( {\mathbb {H}}^{n}\right)$ such that

$$\begin{aligned} \lim _{r\rightarrow \infty }\left( e^{r}\sup _{S_{r}}\left\| \nabla v\right\| \right) =C. \end{aligned}$$

(2)

We prove (1) by showing a similar result for left invariant PDEs on a Lie group and by endowing ${\mathbb {H}}^{n}$ with a Lie group structure.