We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the
L
∞
L^{\infty }
variational problem
\[
{
inf
|
|
∇
0
u
|
|
L
∞
(
Ω
)
,
u
=
g
∈
L
i
p
(
∂
Ω
)
on
∂
Ω
,
\begin {cases} \inf ||\nabla _0 u||_{L^{\infty }(\Omega )}, u=g\in Lip(\partial \Omega ) \text { on }\partial \Omega , \end {cases}
\]
where
Ω
⊂
G
\Omega \subset \mathbf {G}
is an open subset of a Carnot group,
∇
0
u
\nabla _0 u
denotes the horizontal gradient of
u
:
Ω
→
R
u:\Omega \to \mathbb {R}
, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more “regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to “free" systems of vector fields.