Pansu's isoperimetric conjecture in the Heisenberg group
On $\mathbb{R}^3$ with coordinates $(x, y, t)$ let $X = \partial_x - \frac{y}{2} \partial_t$ and $Y = \partial_y + \frac{x}{2} \partial_t$, a left-invariant horizontal frame of the first Heisenberg group. For a measurable set $E$ let $|E|$ be its Lebesgue measure and $$P_H(E) = \sup\Bigl\{ \int_E (Xa + Yb)\, dx\, dy\, dt : a, b \in C_c^1(\mathbb{R}^3),\ a^2 + b^2 \le 1 \Bigr\}$$ its horizontal perimeter. For $R > 0$ the bubble is $$B_R = \Bigl\{ r < R,\ |t| < \tfrac14 \bigl( r \sqrt{R^2 - r^2} + R^2 \arccos(r/R) \bigr) \Bigr\}, \qquad r = \sqrt{x^2 + y^2},$$ the set obtained by rotating about the $t$-axis a sub-Riemannian geodesic joining its two poles. Pansu proved an isoperimetric inequality $|E| \le C\, P_H(E)^{4/3}$ for smooth sets, and Leonardi and Rigot proved that isoperimetric sets exist. Ritoré and Rosales proved that every isoperimetric set with $C^2$ boundary is congruent to a bubble (2008); Monti and Rickly proved that every convex open isoperimetric set is a bubble up to left translation (2009); and Ritoré proved $P_H(E) \ge P_H(B_R)$ when $|E| = |B_R|$ for every set $E$ of finite perimeter lying between a closed horizontal disk centred at the origin and the vertical cylinder over it (2012).
Problem. Does every measurable $E \subset \mathbb{R}^3$ of finite positive volume and finite horizontal perimeter satisfy $P_H(E) \ge P_H(B_R)$ when $|E| = |B_R|$? No symmetry or boundary regularity of $E$ is assumed.
Source: Pansu (1983), who conjectured that the isoperimetric sets are the bubbles; see Monti and Rickly (2009) and Frank and Helffer (2025), §11.4. Stated in this normalization by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MG
- Keywords
- AIM Open Applied Problems, Heisenberg group, isoperimetric inequalities, sub-Riemannian geometry
References
@article{pansu1983,
author = {Pansu, Pierre},
title = {An isoperimetric inequality on the {H}eisenberg group},
journal = {Rend. Sem. Mat. Univ. Politec. Torino},
note = {Special issue, Conference on differential geometry on homogeneous spaces (Turin, 1983)},
year = {1983},
pages = {159--174},
}
@article{pansu1982,
author = {Pansu, Pierre},
title = {Une in{\'e}galit{\'e} isop{\'e}rim{\'e}trique sur le groupe de {H}eisenberg},
journal = {C. R. Acad. Sci. Paris S{\'e}r. I Math.},
volume = {295},
number = {2},
year = {1982},
pages = {127--130},
}
@article{leonardi2003,
author = {Leonardi, Gian Paolo and Rigot, S{\'e}verine},
title = {Isoperimetric sets on {C}arnot groups},
journal = {Houston J. Math.},
volume = {29},
number = {3},
year = {2003},
pages = {609--637},
}
@article{ritore2008,
author = {Ritor{\'e}, Manuel and Rosales, C{\'e}sar},
title = {Area-stationary surfaces in the {H}eisenberg group {$\mathbb{H}^1$}},
journal = {Adv. Math.},
volume = {219},
number = {2},
year = {2008},
pages = {633--671},
doi = {10.1016/j.aim.2008.05.011},
eprint = {math/0512547},
archivePrefix = {arXiv},
}
@article{monti2009,
author = {Monti, Roberto and Rickly, Matthieu},
title = {Convex isoperimetric sets in the {H}eisenberg group},
journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
volume = {8},
number = {2},
year = {2009},
pages = {391--415},
doi = {10.2422/2036-2145.2009.2.07},
eprint = {math/0607666},
archivePrefix = {arXiv},
}
@article{ritore2012,
author = {Ritor{\'e}, Manuel},
title = {A proof by calibration of an isoperimetric inequality in the {H}eisenberg group {$\mathbb{H}^n$}},
journal = {Calc. Var. Partial Differential Equations},
volume = {44},
number = {1-2},
year = {2012},
pages = {47--60},
doi = {10.1007/s00526-011-0425-8},
eprint = {0803.1313},
archivePrefix = {arXiv},
}
@book{capogna2007,
author = {Capogna, Luca and Danielli, Donatella and Pauls, Scott D. and Tyson, Jeremy T.},
title = {An Introduction to the {H}eisenberg Group and the Sub-{R}iemannian Isoperimetric Problem},
series = {Progress in Mathematics},
volume = {259},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {2007},
doi = {10.1007/978-3-7643-8133-2},
}
@article{frank2025,
author = {Frank, Rupert L. and Helffer, Bernard},
title = {On {C}ourant and {P}leijel theorems for sub-{R}iemannian {L}aplacians},
journal = {J. {\'E}c. polytech. Math.},
volume = {12},
year = {2025},
pages = {1083--1160},
doi = {10.5802/jep.307},
}
@misc{aim-pansu-heisenberg-isoperimetry,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Pansu's isoperimetric conjecture in the Heisenberg group}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/126-pansu-heisenberg-isoperimetry.md},
note = {Entry 126 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove the sharp horizontal isoperimetric inequality in the first Heisenberg group for all sets of finite horizontal perimeter and positive finite volume, with equality precisely for left translates and dilates of the Pansu ball.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.07632v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00493, "Pansu's isoperimetric conjecture in the Heisenberg group", clopen.solutions.