Pansu–Wulff shapes in ℍ1
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Sub-Finsler perimeterPansu–Wulff shapesAnisotropic variational problemsHeisenberg group
Pozuelo, Julián and Ritoré, Manuel. "Pansu–Wulff shapes in ℍ1" Advances in Calculus of Variations, vol. , no. , 2021, pp. 000010151520200093. [https://doi.org/10.1515/acv-2020-0093]
SponsorshipMEC-Feder grantMTM2017-84851-C2-1-P; Junta de Andalucía grant A-FQM-441-UGR18; MSCA GHAIA; Research Unit MNat UCE-PP2017-3; Consejería de economía, conocimiento, empresas y universidad and European Regional Development Fund (ERDF), ref. SOMM17/6109/UGR
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.