The isoperimetric problem for regular and crystalline norms in $\mathbb H^1$

We study the isoperimetric problem for anisotropic left-invariant 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. We first prove a representation formula for the $\phi$-perimeter of regular sets and, assuming some regularity on $\phi$ and on its dual norm $\phi^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $\phi$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $\mathrm C^2$-smooth $\phi$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $\phi$, that such sub-Finsler candidate isoperimetric sets are indeed $\mathrm C^2$-smooth. 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).