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

TL;DR

This paper proves a Moser/Bernstein type theorem for solutions of geometric PDEs on Lie groups with a left invariant metric, showing under certain decay conditions that solutions must be constant, and constructs non-constant harmonic functions with specific gradient growth.

Contribution

It extends classical Liouville theorems to Lie groups with left invariant metrics under gradient decay conditions, generalizing results from hyperbolic space.

Findings

Solutions satisfying decay conditions are constant.

Existence of non-constant harmonic functions with prescribed gradient growth.

Generalization of Moser/Bernstein theorems to Lie groups.

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 $u_{\phi}:=u\circ\phi$ 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.