Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group H¹

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In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to R³ are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensional Heisenberg group H¹, characterizing the isoperimetric sets and constants for a family of Riemannian adapted metrics. Using Γ-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in H¹ posed by P.Pansu.