Pansu–Wulff shapes in ℍ1

Abstract

We consider an asymmetric left-invariant norm ∥⋅∥K in the first Heisenberg group H1 induced by a convex body K⊂R2 containing the origin in its interior. Associated to ∥⋅∥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 R2. Under the assumption that K has C2 boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C2 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 HK out of the singular set. In the case of non-vanishing mean curvature, the condition that HK be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂K dilated by a factor of 1HK. Based on this we can define a sphere SK with constant mean curvature 1 by considering the union of all horizontal liftings of ∂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.