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

Abstract

We consider an asymmetric norm $\| \cdot\|_K$ in $\mathbb{R}^2$ induced  by a convex body $K\subseteq \mathbb{R}^2$ containing the origin.  Associated to $\|\cdot\|_K$ there is a perimeter functional $P_K$ in the Heisenberg group $\mathbb{H}^1$ which coincides with the notion of  sub-Riemannian perimeter when $K$ is the unit disk centered at the  origin of $\mathbb{R}^2$. Under the assumption that $K$ has $C^2$  boundary with geodesic curvature strictly positive, we can define the  mean curvature $H_K$ for sets with $C^2$ boundary out of the singular  set by means of the first perimeter formula. The condition of $H_K$ be  constant implies that the boundary is foliated, out of the singular set,  by curves obtained lifting translations and dilations of $\partial K$.  This allows us to define spheres $\mathbb{S}_K$ with constant mean  curvature, the Pansu-Wulff spheres. We will see that the interior of the  Pansu-Wulff shapes are the only sets that are solution of the  sub-Finsler isoperimetric problem in a restricted class of sets. 

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