On the isoperimetric problem in the Heisenberg group ℍ n

Abstract

It has been recently conjectured that, in the context of the Heisenberg groupHn endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetricsets (i.e., minimizers of the H-perimeter among sets of constant Haar measure) couldcoincide with the solutions to a “restricted” isoperimetric problem within the class ofsets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper,we derive new properties of these restricted isoperimetric sets, which we call Heisenbergbubbles. In particular, we show that their boundary has constant mean H-curvature and, quitesurprisingly, that it is foliated by the family of minimal geodesics connecting two specialpoints. In view of a possible strategy for proving that Heisenberg bubbles are actuallyisoperimetric among the whole class of measurable subsets of Hn, we turn our attentionto the relationship between volume, perimeter, and -enlargements. In particular, we provea Brunn–Minkowski inequality with topological exponent as well as the fact that the Hperimeterof a bounded, open set F ⊂ Hn of class C2 can be computed via a generalizedMinkowski content, defined by means of any bounded set whose horizontal projection is the2n-dimensional unit disc. Some consequences of these properties are discussed

Regularity of C 1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group

“…Motivated by the isoperimetric problem in the Heisenberg group, one also studied nonzero constant p-mean curvature surfaces and the regularity problem (e.g., [4,[13][14][15][16]18,20]). Starting from the work [6] (see also [5]), we studied the subject from the viewpoint of partial differential equations and that of differential geometry.…”

“…13) We remark that in case H ≡ 0 or constant, we can prove Theorem D directly from the explicit parametric expression of x or y for the characteristic curves. The situation H ≡ a nonzero constant arises from considering the boundary of a C 2 isoperimetric set.…”

Constant mean curvature surfaces in sub-Riemannian geometry

“…Further in this direction of the isoperimetric problem, Leonardi and Rigot, [15], independently showed the existence of isoperimetric sets in any Carnot group and investigated some of their properties. Leonardi and Masnou, [14], investigated the geometry of the isoperimetric minimizers in the Heisenberg group and also showed a version of the result in [7] showing the among sets with a cylindrical symmetry, Pansu's set is the isoperimetric minimizer.…”

Heisenberg isoperimetric problem. The axial case

“…Recently, Pansu's conjecture appeared again in [14]. We prove the natural generalization of Pansu's conjecture in H n for any n 1 under an additional symmetry assumption on the sets.…”