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.