Unexpected phenomena for mean curvature functionals in the Heisenberg group
Abstract
The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Korányi sphere is optimal for the variational problems associated with the Minkowski and Heintze-Karcher inequalities. Motivated by this phenomenon, we develop a variational theory for geometric problems driven by the horizontal mean curvature, focusing on the total mean curvature functional and the related Minkowski inequality, introducing suitable notions of non-characteristic stationarity and stability. We fully classify critical surfaces, identifying a new one-parameter family of rotationally invariant surfaces, which we call Pansu-Minkowski spheres. We show that a distinguished member, the optimal Pansu-Minkowski sphere, emerges as the unique critical point of the Minkowski quotient, and uniquely minimizes it among rotationally invariant competitors. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations. Finally, we show that the Minkowski quotient does not admit any positive lower bound: in the first sub-Riemannian Heisenberg group, no Minkowski inequality holds.
Related
- A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group
- Mean-convex horizontal mean curvature flow in the Heisenberg group
- Curvature distributions and the Steiner formula for characteristic surfaces in the Heisenberg group
- On the Bach tensor and quadratic curvature functionals
- Equilateral dimension of the Heisenberg group fHn
BibTeX
@misc{fogagnolo2026unexpected,
title = {{Unexpected phenomena for mean curvature functionals in the Heisenberg group}},
author = {Mattia Fogagnolo and Andrea Pinamonti and Simone Verzellesi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29912},
eprint = {2605.29912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29912},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unexpected phenomena for mean curvature functionals in the Heisenberg group
AU - Mattia Fogagnolo
AU - Andrea Pinamonti
AU - Simone Verzellesi
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.29912
UR - https://arxiv.org/abs/2605.29912
AB - The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Korányi sphere is optimal for the variational problems associated with the Minkowski and Heintze-Karcher inequalities. Motivated by this phenomenon, we develop a variational theory for geometric problems driven by the horizontal mean curvature, focusing on the total mean curvature functional and the related Minkowski inequality, introducing suitable notions of non-characteristic stationarity and stability. We fully classify critical surfaces, identifying a new one-parameter family of rotationally invariant surfaces, which we call Pansu-Minkowski spheres. We show that a distinguished member, the optimal Pansu-Minkowski sphere, emerges as the unique critical point of the Minkowski quotient, and uniquely minimizes it among rotationally invariant competitors. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations. Finally, we show that the Minkowski quotient does not admit any positive lower bound: in the first sub-Riemannian Heisenberg group, no Minkowski inequality holds.
ER -