Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groups

Donato, Daniela Di

doi:10.1007/s11118-019-09817-4

Cited by 7 publications

(17 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Combining some arguments leading to the proof of [13, Theorem 2.1] with results in [9] we obtain the following result. where J φ φ denotes the matrix corresponding to the intrinsic differential of φ.…”

Section: Building Approximationmentioning

confidence: 69%

“…Vice versa, many efforts have also been carried out to figure out which regularity has to be required (on top of continuity) to a function φ acting between complementary subgroups, to ensure that its intrinsic graph is a regular surface. This theme has been developed in various papers (among which [1,4,5,7,8,18,27]), in particular, many results have been developed for H-regular surfaces of codimension 1.…”

Section: Introductionmentioning

confidence: 99%

“…H-regular surfaces of codimension 1 [1], and successively, regular surfaces of low codimension in any Carnot groups [9] have been characterized as graphs of uniformly intrinsic differentiable functions acting between complementary subgroups, with horizontal, and hence commutative, target space (see Theorem 3.5).…”

Section: Introductionmentioning

confidence: 99%

“…Clearly the relative Lipschitz constant can change. If φ is an intrinsic Lipschitz function, D φ is a quasi-distance (see[8, Proposition 2.6.11].Definition 2.15. Let H n = M • H be the product of complementary subgroups.…”

mentioning

confidence: 99%

“…[9, Proposition 3.7] Let Ω be an open set and let φ : Ω ⊂ R 2n+1−k → R k be a function uniformly intrinsic differentiable on Ω, then the functionJ φ φ : Ω → M k,2n−k (R) is continuous.We now recall[8, Theorem 3.1.1]. Analogous result has been proved in[3,, in the proof of which, nevertheless, it was not explicitly stated relation (…”

mentioning

confidence: 99%

See 4 more Smart Citations

Intrinsic regular surfaces of low codimension in Heisenberg groups

Corni

2021

Ann. Fenn. Math.

View full text Add to dashboard Cite

In this paper we study intrinsic regular submanifolds of H n of low codimension in relation with the regularity of their intrinsic parametrization. We extend some results proved for H-regular surfaces of codimension 1 to H-regular surfaces of codimension k, with 1 ≤ k ≤ n. We characterize uniformly intrinsic differentiable functions, φ, acting between two complementary subgroups of the Heisenberg group H n , with target space horizontal of dimension k, in terms of the Euclidean regularity of their components with respect to a family of non linear vector fields ∇ φj . Moreover, we show how the area of the intrinsic graph of φ can be computed in terms of the components of the matrix representing the intrinsic differential of φ.

show abstract

Section: Building Approximationmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Intrinsic regular surfaces of low codimension in Heisenberg groups

Corni

2021

Ann. Fenn. Math.

View full text Add to dashboard Cite

show abstract

Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

2022

View full text Add to dashboard Cite

We prove that in arbitrary Carnot groups $\mathbb {G}$ G of step 2, with a splitting $\mathbb {G}=\mathbb {W}\cdot \mathbb {L}$ G = W ⋅ L with $\mathbb {L}$ L one-dimensional, the intrinsic graph of a continuous function $\varphi \colon U\subseteq \mathbb {W}\to \mathbb {L}$ φ : U ⊆ W → L is $C^{1}_{\mathrm {H}}$ C H 1 -regular precisely when φ satisfies, in the distributional sense, a Burgers’ type system Dφφ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution φ to a Burgers’ type system Dφφ = ω, with ω continuous, is actually a broad solution to Dφφ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to Dφφ = ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions.

show abstract

Lipschitz graphs and currents in Heisenberg groups

Vittone

2022

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

show abstract

Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groups

Cited by 7 publications

References 18 publications

Intrinsic regular surfaces of low codimension in Heisenberg groups

Intrinsic regular surfaces of low codimension in Heisenberg groups

Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

Lipschitz graphs and currents in Heisenberg groups

Contact Info

Product

Resources

About