Abstract
Abstract We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H 1. These spheres are conjectured to be the isoperimetric sets of H 1. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.
Highlights
We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H
We introduce in H two real parameters that can be used to deform H to the sub-Riemannian Heisenberg group, on the one hand, and to the Euclidean space, on the other hand
Even though we are not able to prove that these CMC spheres are isoperimetric sets, we obtain several partial results in this direction
Summary
We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H. The vector eld M is tangent to the round sphere of radius R > in the three-dimensional Euclidean space and its integral lines turn out to be the meridians from the north to the south pole. We prove a quantitative isoperimetric inequality for the CMC spheres ΣR with respect to compact perturbations in vertical cylinders, see Theorem 5.1.
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