@misc{10481/69324,
year = {2021},
month = {5},
url = {http://hdl.handle.net/10481/69324},
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.},
organization = {MEC-Feder grantMTM2017-84851-C2-1-P},
organization = {Junta de Andalucía
grant A-FQM-441-UGR18},
organization = {MSCA GHAIA},
organization = {Research Unit MNat UCE-PP2017-3},
organization = {Consejería de economía, conocimiento, empresas y universidad and European Regional
Development Fund (ERDF), ref. SOMM17/6109/UGR},
publisher = {De Gruyter},
keywords = {Sub-Finsler perimeter},
keywords = {Pansu–Wulff shapes},
keywords = {Anisotropic variational problems},
keywords = {Heisenberg group},
title = {Pansu–Wulff shapes in ℍ1},
doi = {10.1515/acv-2020-0093},
author = {Ritoré Cortés, Manuel María and Pozuelo Domínguez, Julián},
}