Preprint

Inserted: 28 may 2026
Last Updated: 28 may 2026

Year: 2026

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}. To investigate this phenomenon, we establish first and second variation formulas for general mean curvature functionals in arbitrary Riemannian manifolds, and then obtain corresponding formulas in Heisenberg groups through a Riemannian approximation scheme. We subsequently specialize to the optimization of total mean curvature under area constraint in the first Heisenberg group, introducing suitable notions of \emph{non-characteristic} stationarity and stability. We identify a new one-parameter family of rotationally invariant critical surfaces, which we call \emph{Pansu-Minkowski spheres}. Among them, 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 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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