*Published Paper*

**Inserted:** 10 jul 2002

**Last Updated:** 17 dec 2006

**Journal:** Annali di Mat. Pura Appl.

**Volume:** 184

**Pages:** 533-553

**Year:** 2005

**Abstract:**

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

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