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Pansu isoperimetric conjecture on the Heisenberg group

Establish the validity of the Pansu conjecture concerning the isoperimetric problem on the Heisenberg group H_n, which would yield improved bounds (via the Faber–Krahn constant) used to deduce Pleijel’s theorem for H_n × R^k across all pairs (n, k).

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Background

In the Heisenberg setting studied by Frank and Helffer, Pleijel’s theorem follows unconditionally for most (n, k), while for the remaining cases it would follow assuming the Pansu conjecture on isoperimetric minimizers in the Heisenberg group. This conjecture implies a better Faber–Krahn constant than the one used in their general bound, strengthening the Pleijel conclusion.

The current note extends unconditional Pleijel results to H-type groups; validating Pansu’s conjecture would further solidify results in the Heisenberg × Euclidean product case by improving the key geometric constant in the analysis.

References

It was shown in Theorem~7.2 that Pleijel's theorem \gamma(H_n \times Rk) < 1 holds unconditionally for all but four pairs of (n, k) \in N \times N_0 as a consequence of gamma-bound and the sharp constant in the L2-Sobolev inequality for Heisenberg groups Corollary~C, see also Theorem~2.1, and otherwise holds for all pairs of (n, k) assuming the validity of the Pansu conjecture concerning the isoperimetric problem on the Heisenberg group and which gives a better bound than gamma-bound on (H_n \times Rk), see Proposition~11.1.

— A note on the Pleijel theorem for $H$-type groups (2510.19381 - Qiu, 22 Oct 2025) in Section 1 (Introduction and main result)