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The Isoperimetric Problem for Regular and Crystalline Norms in \({\mathbb {H}}^1\)

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Abstract

We study the isoperimetric problem for anisotropic perimeter measures on \(\mathbb {R}^3\), endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm \(\phi \) on the horizontal distribution. In the case where \(\phi \) is the standard norm in the plane, such isoperimetric problem is the subject of Pansu’s conjecture, which is still unsolved. Assuming some regularity on \(\phi \) and on its dual norm \(\phi ^*\), we characterize \(\mathrm {C}^2\)-smooth isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. The argument is based on a fine study of the characteristic set of \(\phi \)-isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When \(\phi \) is a crystalline norm, we show the existence of a partial foliation for constant \(\phi \)-curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where \(\phi \) is crystalline).

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Acknowledgements

The authors thank M. Ritoré and C. Rosales for pointing out a gap in a preliminary version of the paper. The first and third authors acknowledge the support of ANR-15-CE40-0018 project SRGI - Sub-Riemannian Geometry and Interactions. The first author acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 794592, of the INdAM–GNAMPA project Problemi isoperimetrici con anisotropie, and of a public grant of the French National Research Agency (ANR) as part of the Investissement d’avenir program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.

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Authors and Affiliations

  1. Dipartimento di Matematica Tullio Levi Civita, Università di Padova, Padova, Italy

    Valentina Franceschi, Roberto Monti & Alberto Righini

  2. Laboratoire Jacques-Louis Lions, Sorbonne Université, Inria, CNRS, Université Paris Cité, Paris, France

    Mario Sigalotti

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  1. Valentina Franceschi
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  2. Roberto Monti
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Correspondence to Valentina Franceschi.

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Franceschi, V., Monti, R., Righini, A. et al. The Isoperimetric Problem for Regular and Crystalline Norms in \({\mathbb {H}}^1\). J Geom Anal 33, 8 (2023). https://doi.org/10.1007/s12220-022-01045-4

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  • DOI: https://doi.org/10.1007/s12220-022-01045-4

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