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arXiv · 1501.07357

Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\mathsf{H}^1$

Jih-Hsin Cheng·Andrea Malchiodi·Paul Yang
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Abstract

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 $Γ$-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in $\mathsf{H}^1$ posed by P.Pansu.

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Jih-Hsin Cheng, Andrea Malchiodi, Paul Yang. 2015-02-16. Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\mathsf{H}^1$. https://arxiv.org/abs/1501.07357

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