Area formula for regular submanifolds of low codimension in Heisenberg groups

Corni, Francesca; Magnani, Valentino

doi:10.1515/acv-2021-0049

Advances in Calculus of Variations

2022

DOI: 10.1515/acv-2021-0049

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Area formula for regular submanifolds of low codimension in Heisenberg groups

Francesca Corni

Valentino Magnani

Abstract: We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is constructed by an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsically differentiable functions.

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2024

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Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Antonelli,

Di Donato,

Don

et al. 2024

Annales De l'Institut Fourier

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In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are C 1 H regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph.We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group.As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.Résumé. -Dans les groupes de Carnot, nous étudions les graphes intrinsèques des fonctions avec codomaine horizontal. Ces graphes sont C 1 H réguliers quand la fonction est uniformément intrinsèquement différentiable. Notre premier résultat est une caractérisation de la différentiabilité intrinsèque en termes de régularité Hölder des projections sur le graphe des champs de vecteurs invariants à gauche.Nous améliorons le résultat dans les groupes de Carnot de rang 2 pour les fonctions avec codomaine unidimensionnel: dans ce cas, la régularité horizontale suffit pour obtenir le résultat. Nous remarquons que cette amélioration n'est pas vraie dans le groupe de Carnot de rang 3 le plus simple. Enfin on montre une formule de l'aire pour les fonctions uniformément intrinsèquement différentiables avec codomaine unidimensionnel et on donne une expression explicite de l'élément de surface en fonctions des dérivées intrinsèques de la fonction.

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Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Antonelli,

Di Donato,

Don

et al. 2024

Annales De l'Institut Fourier

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Area formula for regular submanifolds of low codimension in Heisenberg groups

Cited by 1 publication

References 33 publications

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

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