We study the isoperimetric problem for anisotropic left-invariant perimeter
measures on R3, endowed with the Heisenberg group structure. The
perimeter is associated with a left-invariant norm ϕ on the horizontal
distribution. We first prove a representation formula for the ϕ-perimeter
of regular sets and, assuming some regularity on ϕ and on its dual norm
ϕ∗, we deduce a foliation property by sub-Finsler geodesics of C2-smooth surfaces with constant ϕ-curvature. We then prove that the
characteristic set of C2-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 C2-smooth ϕ-isoperimetric sets as the
sub-Finsler analogue of Pansu's bubbles. We also show, under suitable
regularity properties on ϕ, that such sub-Finsler candidate isoperimetric
sets are indeed C2-smooth. By an approximation procedure, we finally
prove a conditional minimality property for the candidate solutions in the
general case (including the case where ϕ is crystalline)