Preprint

Inserted: 28 may 2026
Last Updated: 5 aug 2026

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

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 identify a new one-parameter family of rotationally invariant critical surfaces, which we call Pansu-Minkowski spheres. Among them, 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 Pansu-Minkowski spheres. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations.

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