Pansu’s conjecture on the sharp isoperimetric constant on the Heisenberg group

Determine the exact isoperimetric constant I(Hn) for the Heisenberg group Hn by proving Pansu’s conjecture that the sharp constant equals the explicit quantity given in equation (11.6), i.e., I(Hn) = 2^{n+1}·(2n·(2n+2))^{(2n+2)/(2n+1)}·Γ((2n+3)/2)/Γ((2n+2)/2)·π^{−(2n+2)/(2n+1)}, where Γ denotes the Gamma function, and thereby identify the isoperimetric sets achieving equality.

Background

The isoperimetric inequality in the Heisenberg group Hn states that the (horizontal) perimeter of a measurable set is bounded below by a constant times a power of its measure. Pansu conjectured the identity of the extremal sets and the exact value of the sharp constant I(Hn).

In the paper, the authors present Pansu’s conjecture and show that, if it holds, their Pleijel-type bounds follow for several unresolved cases. They also provide bounds and partial results, but the full determination of I(Hn) remains a major open problem in sub-Riemannian geometry.