  In this paper we prove that isoperimetric sets in three-dimensional
homogeneous spaces diffeomorphic to $\mathbb{R}^3$ 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 $\mathsf{H}^1$, characterizing the
isoperimetric sets and constants for a family of Riemannian adapted metrics.
Using $\Gamma$-convergence of the perimeter functionals, we also settle an
isoperimetric conjecture in $\mathsf{H}^1$ posed by P.Pansu.

Differential Geometry