On the isoperimetric problem in the Heisenberg group ℍn

Abstract

It has been recently conjectured that, in the context of the Heisenberg group ℍ endowed with its Carnot-Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a "restricted" isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of ℍ n, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn-Minkowski inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F ⊂ ℍn of class C2 can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed.