Preprint

Inserted: 28 may 2026
Last Updated: 29 sep 2026

Year: 2026
Doi: https://doi.org/10.48550/arXiv.2605.29912

Abstract:

The Euclidean paradigm that \emph{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 \emph{total mean curvature} functional and the related \emph{Minkowski inequality}, introducing suitable notions of \emph{non-characteristic} stationarity and stability. We fully classify critical surfaces, identifying a new one-parameter family of rotationally invariant surfaces, which we call \emph{Pansu-Minkowski spheres}. We show that a distinguished member, the \emph{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.

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