The Isoperimetric Problem for Regular and Crystalline Norms in $${\mathbb {H}}^1$$

Abstract

We study the isoperimetric problem for anisotropic perimeter measures on \(\mathbb {R}^3\), endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm \(\phi \) on the horizontal distribution. In the case where \(\phi \) is the standard norm in the plane, such isoperimetric problem is the subject of Pansu’s conjecture, which is still unsolved. Assuming some regularity on \(\phi \) and on its dual norm \(\phi ^*\), we characterize \(\mathrm {C}^2\)-smooth isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. The argument is based on a fine study of the characteristic set of \(\phi \)-isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When \(\phi \) is a crystalline norm, we show the existence of a partial foliation for constant \(\phi \)-curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where \(\phi \) is crystalline).