Pansu–Wulff shapes in ℍ1 Ritoré Cortés, Manuel María Pozuelo Domínguez, Julián Sub-Finsler perimeter Pansu–Wulff shapes Anisotropic variational problems Heisenberg group The authors have been supported byMEC-Feder grantMTM2017-84851-C2-1-P, Junta de Andalucía grant A-FQM-441-UGR18, MSCA GHAIA, and Research Unit MNat UCE-PP2017-3. This research was also funded by the 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. 2021-06-21T11:29:53Z 2021-06-21T11:29:53Z 2021-05-18 info:eu-repo/semantics/article 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] http://hdl.handle.net/10481/69324 10.1515/acv-2020-0093 eng http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess Atribución 3.0 España De Gruyter