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 Kor\'anyi 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.