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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.29912 |
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| _version_ | 1866917543874134016 |
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| author | Fogagnolo, Mattia Pinamonti, Andrea Verzellesi, Simone |
| author_facet | Fogagnolo, Mattia Pinamonti, Andrea Verzellesi, Simone |
| contents | 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. 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unexpected phenomena for mean curvature functionals in the Heisenberg group Fogagnolo, Mattia Pinamonti, Andrea Verzellesi, Simone Differential Geometry 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. 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 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. |
| title | Unexpected phenomena for mean curvature functionals in the Heisenberg group |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.29912 |