Abstract
We consider an asymmetric left-invariant norm
∥
⋅
∥
K
{\|\cdot\|_{K}}
in the first Heisenberg group
ℍ
1
{\mathbb{H}^{1}}
induced by a convex body
K
⊂
ℝ
2
{K\subset\mathbb{R}^{2}}
containing the origin in its interior. Associated to
∥
⋅
∥
K
{\|\cdot\|_{K}}
there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of
ℝ
2
{{\mathbb{R}}^{2}}
. Under the assumption that K has
C
2
{C^{2}}
boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with
C
2
{C^{2}}
boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function
H
K
{H_{K}}
out of the singular set. In the case of non-vanishing mean curvature, the condition that
H
K
{H_{K}}
be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of
∂
K
{\partial K}
dilated by a factor of
1
H
K
{\frac{1}{H_{K}}}
. Based on this we can define a sphere
𝕊
K
{\mathbb{S}_{K}}
with constant mean curvature 1 by considering the union of all horizontal liftings of
∂
K
{\partial K}
starting from
(
0
,
0
,
0
)
{(0,0,0)}
until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.
The hyperplane is the only stable, smooth solution to the Isoperimetric Problem in Gaussian space