2020/07/09 by Pozuelo, Julián, Ritoré, Manuel
#49Q20 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2007.04683
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 1/HK. Based on this we can defined a sphere \mathbbSK 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.