首页/论文/2004.02520

arXiv 2004.025202020-04-06math.CAmath.DGmath.MG

Area of intrinsic graphs and coarea formula in Carnot Groups

Antoine Julia, Sebastiano Nicolussi Golo, Davide Vittone

英文摘要：

We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $C^1$ regularity ($C^1_H$). Our first main result is an area formula for $C^1_H$ 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 $C^1_H$ submanifolds into level sets of a $C^1_H$ function.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

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

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 inte…

2020-05-22math.APmath.CAmath.DG

Whitney延拓定理与Carnot-Carathéodory空间中跳跃集的内在可求长性

Whitney extension theorem and intrinsic rectifiability of jump sets in Carnot-Carathéodory spaces

ETH Zürich（苏黎世联邦理工学院）

证明等正则Carnot-Carathéodory空间中局部可积函数的内在跳跃集具有内在可数可求长性，关键新工具是内在C^1函数的局部标量Whitney延拓定理。

2026-09-10math.CAmath.FAmath.MG

Carnot-Carathéodory空间中SBV函数的内蕴周长、紧性和Poincaré不等式

Intrinsic perimeter, compactness and Poincaré inequality for SBV functions in Carnot-Carathéodory spaces

ETH Zürich（苏黎世联邦理工学院）；Università di Padova（帕多瓦大学）

通过引入内蕴可数可求积集的内蕴周长测度，在满足性质$\mathcal R$的等则Carnot-Carathéodory空间中证明了有界变差特殊函数的紧性结果和Poincaré不等式。

2026-06-17math.CAmath.FAmath.MG

Intrinsically Lipschitz functions with normal target in Carnot groups

We provide a Rademacher theorem for intrinsically Lipschitz functions $ϕ:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a Carnot group, where we require that $\mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.

2020-09-29math.CAmath.FAmath.MG

Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

We prove that in arbitrary Carnot groups $\mathbb G$ of step 2, with a splitting $\mathbb G=\mathbb W\cdot\mathbb L$ with $\mathbb L$ one-dimensional, the graph of a continuous function $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $φ$ satisfies, in the distributional sense, a Burgers' type system $D^φφ=ω$, with a continuous $ω$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $φ$ to a Burgers' type system $D^φφ=ω…

2020-08-02math.APmath.CAmath.DG

Regularity of sets with constant intrinsic normal in a class of Carnot groups

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called ``groups of type $\star$'' throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $\star$ is rectifiable in the intrinsic sense (De Giorgi's rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.

2013-12-27math.CAmath.FAmath.MG

更多相关论文收起更多论文

Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups $\mathbb{H}^n$ are $C^1_{\mathbb{H}^n}$-rectifiable, providing the first non-Euclidean extension of Preiss's rectifiability theorem and a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups.

2020-08-23math.CAmath.MG

2步Carnot群中$C^{1,α}$内蕴图上的奇异积分

Singular integrals on $C^{1,α}$ intrinsic graphs in step 2 Carnot groups

该研究针对2步Carnot群中Calderón-Zygmund核诱导的奇异积分算子，证明满足自然抵消条件时其在特定$C^{1,α}$内蕴图上$L^2$有界，可应用于Riesz算子，还推得相关内蕴图闭子集对Lipschitz调和函数不可去。

2025-09-02math.CA

Lipschitz functions on submanifolds in Heisenberg groups

We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on $\HH$-rectifiable sets, and a coarea formula on $\HH$-rectifiable sets that completes the program started in~\cite{JNGV}.

2021-07-01math.CAmath.MG

Blow-up of regular submanifolds in Heisenberg groups and applications

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mapp…

2005-06-22math.CAmath.DG

Nowhere differentiable intrinsic Lipschitz graphs

We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.

2021-01-08math.CAmath.MG

The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Caratheodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group H^n. In particular we will positively answer to the Bernstein problem in the case n=1 and we will provide counterexamples when n>=5.

2006-08-03math.CAmath.MG

An intrinsic measure for submanifolds in stratified groups

For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the Carnot-Caratheodory distance, respectively. Our main technical tool is an intrinsic blow-up at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations i…

2006-08-03math.CAmath.DG

海森伯格群中的Lipschitz图与流形

Lipschitz graphs and currents in Heisenberg groups

本文证明子黎曼海森伯格群中余维数k≤n的内蕴Lipschitz图的Rademacher型定理，并给出延拓、逼近、常值定理及Lusin型结果、可修正性等价和面积公式等应用。

2020-07-28math.CAmath.DGmath.FA