Intrinsic regular surfaces of low codimension in Heisenberg groups

Corni, Francesca

doi:10.5186/aasfm.2021.4605

Cited by 8 publications

(8 citation statements)

References 21 publications

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“…The answer to the previous question is affirmative in Heisenberg groups H n , see [SC16, Theorem 4.95], and [Cor19,Theorem 1.4]. In this paper we obtain a new result in this direction.…”

mentioning

confidence: 61%

“…A step towards obtaining analogous results in H n , in case L has higher dimension and ω is continuous, has been recently done by Corni in [Cor19]. In particular, the author proves that, if L is horizontal k-dimensional, the set graph( ϕ) is a co-horizontal C 1 H -surface if and only if ϕ is a broad* solution to D ϕ ϕ = ω for some ω ∈ C(U).…”

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 89%

“…The study of the relation between being a broad* solution to the system D ϕ ϕ = ω with a continuous ω and the intrinsic regularity of graph( ϕ) was first initiated, in H n for L one-dimensional in [ASCV06, Section 5], and then continued in [BSC10a,BSC10b,BCSC14]. For the case G = H n and L horizontal and k-dimensional see also [Cor19] and for the general case of Carnot groups of step 2 and L one-dimensional, see [DD18,DD19]. In Proposition 3.27 we give a sufficient condition for the map ϕ to be a broad* solution to the system D ϕ ϕ = ω, with ω continuous.…”

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 99%

“…We expect that the hypothesis on the vertically broad* hölder regularity in Corollary 4.7 can be dropped in general, see also the paragraph Geometric characterizations of intrinsic differentiability in the introduction. From the results proved in [BSC10b] and [Cor19], we know that the assumption on the vertically broad* hölder regularity in Corollary 4.7 is not necessary in the case of the Heisenberg groups H n , with L horizontal k-dimensional, see also the introduction to Section 5. We stress we obtain that we can remove the assumption on the vertically broad* hölder regularity in Corollary 4.7 also in the case of step-2 Carnot groups with L one-dimensional, see Section 6.…”

Section: Main Theorems In Arbitrary Carnot Groupsmentioning

confidence: 99%

“…Recently, in [Cor19, Proposition 4.10], the author has proved a generalization of [BSC10b, Theorem 3.2] that holds for arbitrary complemented subgroups in H n , and this statement is one of the key step in order to get the main theorem [Cor19,Theorem 1.4]. By using our reasoning, we are also able to recover [Cor19, Proposition 4.10], and thus [Cor19, Theorem 1.4], in the case H n = W • L with L horizontal and k-dimensional, with k < n. However, our argument does not apply to the remaining case k = n, while the argument in the reference does.…”

Section: Some Applicationsmentioning

confidence: 99%

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

Antonelli,

Di Donato,

Don

et al. 2020

Preprint

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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. Contents 0. Notation 3 1. Introduction 3 1.1. An historical account of the notion of C 1 H -surface in Carnot groups 3 1.2. Main theorems 5 1.3. Structure of the paper 9 2. Preliminaries 10 2.1. Carnot groups 10 2.2. Little Hölder continuous functions 12 2.3. Intrinsic surfaces, Intrinsically Lipschitz and Intrinsically differentiable functions 13 3. Intrinsic projected vector fields on subgroups 19 3.1. Definition of D ϕ and main properties 20 3.2. Invariance properties of D ϕ 24 3.3. Metric properties of integral curves of D ϕ 27 3.4. Definition of broad* 34 4. Main Theorems in arbitrary Carnot groups 36 4.1. From regularity of ϕ along integral curves of D ϕ to regularity of ϕ 38 4.2. Area formula for codimension-one graphs in terms of intrinsic derivatives 45 4.3. Relations between intrinsic differentiability and local approximability 49

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“…The answer to the previous question is affirmative in Heisenberg groups H n , see [SC16, Theorem 4.95], and [Cor19,Theorem 1.4]. In this paper we obtain a new result in this direction.…”

mentioning

confidence: 61%

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 89%

Section: Intrinsic Projected Vector Fields On Subgroupsmentioning

confidence: 99%

Section: Main Theorems In Arbitrary Carnot Groupsmentioning

confidence: 99%

Section: Some Applicationsmentioning

confidence: 99%

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

Antonelli,

Di Donato,

Don

et al. 2020

Preprint

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Area of intrinsic graphs and coarea formula in Carnot groups

2022

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We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.

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On rectifiable measures in Carnot groups: representation

Antonelli

Merlo

2021

Calc. Var.

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This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

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Intrinsic regular surfaces of low codimension in Heisenberg groups

Cited by 8 publications

References 21 publications

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

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

Area of intrinsic graphs and coarea formula in Carnot groups

On rectifiable measures in Carnot groups: representation

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