Pansu-Wulff shapes in $\mathbb{H}^1$

We consider an asymmetric left-invariant norm $||\cdot ||_K$ in the first Heisenberg group $\mathbb{H}^1$ induced by a convex body $K\subset\mathbb{R}^2$ containing the origin in its interior. Associated to $\|\cdot\|_K$ 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 $\mathbb{R}^2$. Under the assumption that $K$ has $C^2$ boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with $C^2$ 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 $H_K$ out of the singular set. In the case of non-vanishing mean curvature, the condition that $H_K$ be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of $\partial K$ dilated by a factor of $1/H_K$. Based on this we can defined a sphere $\mathbb{S}_K$ with constant mean curvature $1$ by considering the union of all horizontal liftings of $\partial K$ starting from $(0,0,0)$ 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.