Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - S. Masnou

On the isoperimetric problem in the Heisenberg group ${\mathbf{H^n}}$

created on 10 Jul 2002
modified by leonardi on 17 Dec 2006


Published Paper

Inserted: 10 jul 2002
Last Updated: 17 dec 2006

Journal: Annali di Mat. Pura Appl.
Volume: 184
Pages: 533-553
Year: 2005


It has been recently conjectured that the isoperimetric sets in the Heisenberg group ${\mathbf{H}^n}$ could coincide with the solutions to a ``restricted'' isoperimetric problem within the class of sets having finite perimeter and smooth boundary with cylindrical symmetry. In this paper, we derive new properties of those restricted isoperimetric sets, that we call {\it $\mathbf{H}$-bubbles}. In particular, we show that their boundary has constant mean $\mathbf{H}$-curvature and, surprisingly, that it is the union of all Carnot-Carath{é}odory geodesics connecting two special points. We believe that this latter property could help for proving that $\mathbf{H}$-bubbles are the general isoperimetric sets of $\mathbf{H}^n$ and for better understanding the geometric structure of the group. Besides, we show that the classical method based on the Brunn-Minkowski inequality, which serves to prove that Euclidean balls are isoperimetric sets in $\mathbf{R}^n$, has no direct and easy counterpart in $\mathbf{H}^n$.

Keywords: Heisenberg group, isoperimetric problem, Carnot-Carath{é}odory metric, Brunn-Minkowski inequality