Antoine Julia, Sebastiano Nicolussi Golo, Davide Vittone
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}.
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Submanifolds with boundary in sub-Riemannian Heisenberg Groups
本文研究亚黎曼海森堡群中具有内在$C^1$正则性的带边界子流形，给出实例并建立涉及Rumin微分形式积分的斯托克斯定理。
Submanifolds with boundary and Stokes' Theorem in Heisenberg groups
本文在海森堡群中引入带边 $C^1_\mathbb{H}$-正则子流形，并结合 Rumin 微分形式复形证明了相应版本的斯托克斯定理。
We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is computed with respect to an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsic differentiable functions.
This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decomposi…
We show that the $β$--numbers of intrinsic Lipschitz graphs of Heisenberg groups $\mathbb{H}_n$ are locally Carleson integrable when $n \geq 2$. Our technique relies on a recent Dorronsoro inequality \cite{FO} as well as a novel slicing argument. A key ingredient in our proof is a Euclidean inequality bounding the $β$--number of a function on a cube of $\mathbb{R}^n$ using the $β$--number of the restriction of the function to codimension--1 slices of the cube.
We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On…
A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschi…
The Heisenberg group $\mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which \textit{subsets} of $\mathbb{H}$ bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant $L>0$ such that lines $L$-bi-Lipschitz embed into $\mathbb{R}^3$ and planes $L$-bi-Lipschitz embed into $\mathbb{R}^4$. Moreover, $C^{1,1}$ $2$-manifolds without characteristic points as well as all $C^{1,1}$ $1$-manifolds locally $L…
The ($n+1$)-Lipschitz homotopy group of the Heisenberg group $\mathbb{H}^n$
证明了对$n\geq 2$，Heisenberg群$\mathbb{H}^n$的Lipschitz同伦群$π_{n+1}^{\rm Lip}(\mathbb{H}^n)\neq 0$是非平凡的.
The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that …
Lipschitz extensions of maps between Heisenberg groups
设$\H^n$是拓扑维数为$2n+1$的海森堡群.证明了若$n$是奇数，则度量空间对$(\H^n, \H^n)$不具有Lipschitz扩张性质.
Lipschitz homotopy groups of the Heisenberg groups
从$n$维空间到$(2n+1)$维海森堡群$\H^n$的Lipschitz映射和水平映射是丰富的，而从高维空间的映射则受到更多的限制。DeJarnette-Hajmartasz-Lukyanenko-Tyson构造了从$S^k$到$\H^n$的水平映射，这些映射通过$n$ -球面进行因子分解，并表明这些映射没有平滑的水平填充。然而，在本文中，我们建立一个例子考夫曼表明，这些地图有时有Lipschitz填充。这表明空间的Lipschitz和光滑水平同伦群可以不同。相反，我们证明了任何Lipschitz映射$S^k\to \H^1$因子通过树，因此Lipschitz零同伦，如果$k\ge 2$。
Compression bounds for Lipschitz maps from the Heisenberg group to $L_1$
本文证明了具有Carnot-Carathéodory度量的Heisenberg群的一个定量的双Lipschitz非嵌入定理，并应用它给出了Spartens-Cut问题的Goemans-Linial半定松弛的积分间隙的一个下界.
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。
Volume preserving bi-Lipschitz homeomorphisms on the Heisenberg group
海森堡群H（n）上的初等次黎曼几何提供了一幅辛几何的紧凑图。利用生成函数，$R^{2n}$上的任何Hamilton同胚提升到H（n）的保体积双Lipschitz同胚.任何这样的同胚流的曲线偏离水平性的哈密尔顿流。从度量的观点来看，这意味着任何这样的曲线具有豪斯多夫维数2和等于哈密尔顿算子的$\mathcal{H}^{2}$（面积）密度。赫林距离的非简并性是这一事实的直接结果。
We prove that in the Heisenberg group $\mathbb{H}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.
The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$
本文定义并研究了万有 Lipschitz 路径空间，证明了纯 2-unrectifiable 空间（如 Heisenberg 群 $\mathbb{H}^1$）的该空间满足唯一提升、万有及 Lipschitz 单连通性质，为后续证明其 Lipschitz 基本群无挠奠定基础。
We focus our attention on the notion of intrinsic Lipschitz graphs, inside a subclass of Carnot groups of step 2 which includes a corank 1 Carnot groups (and so the Heisenberg groups), Free groups of step 2 and the complexified Heisenberg group. More precisely, we prove the equivalence between intrinsic Lipschitz map and a weak solution to a suitable non linear first order PDE system, which generalizes Lagrangian solution in the context of Heisenberg groups.
Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups
本文研究度量群的小子集（即测度为零且小于满维的子集）上的奇异积分。我们想到的群的主要例子是欧几里得空间和海森堡群。除了获得结果在一个非常一般的设置，这项工作的目的是双重的，我们将延长一些结果在欧几里得空间更一般的内核比以前考虑，我们将获得在海森堡群的一些应用调和（在海森堡意义上）功能的一些结果较早在欧几里得空间。
For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a function $f$. Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, we prove that their characteristic curves are of class $C^2$ and that this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1$.
Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces
我们证明了海森堡群在向下看时不是极小的。这就回答了David和Semmes在“破碎的分形和破碎的梦想”中的问题11.15，或者等价地回答了Heinonen和Semmes在“关于映射、测度和度量的33个是或否问题”中的问题22，因此也回答了问题24。海森堡群的非极小性是通过给出一个Ahlfors $4$ -正则度量空间$X$的例子来证明的，该空间具有自身的大片，使得从$X$的子集到海森堡群的Lipschitz映射都不具有正测度的像，并且通过提供一个从海森堡群到空间$X$的Lipschitz映射，该空间具有整个$X$的像。作为证明上述结果的一部分，我们在海森堡群上定义了一个新的距离，该距离以卡诺-卡拉西奥多里距离为界，保持了阿尔福斯正则性，并且使得卡诺-卡拉西奥多里距离和新距离在任何正测度集上都是biLipschitz等价的。这种构造在任何Ahlfors-正则度量空间中更普遍地起作用，在那里人们可以做适当的捷径。这样的空间包括例如所有雪花状Ahlfors正则度量空间。用同样的技巧，我们还提供了一个例子，证明海森堡群biLipschitz上的左不变距离到卡诺-卡拉迪奥里距离，对于这个距离，没有爆破允许非平凡的膨胀。
BV Minimizers of the area functional in the Heisenberg group under the bounded slope condition
本文研究了次黎曼海森堡群中t-图的面积泛函，并研究了相应的狄利克雷问题的极小解.证明了在边界数据上的有界斜率条件下，存在唯一的极小元，且此极小元是Lipschitz连续的。我们还提供了一个例子表明，在第一海森堡群，Lipschitz正则性是尖锐的，即使在有界斜率条件。
The strong geometric lemma in the Heisenberg group
本文证明第一Heisenberg群中内蕴Lipschitz图像强几何引理的最佳指数为4而非2，并结合既有工作完成该引理在Heisenberg群中的证明。
Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a cal…
Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups
本文研究了子黎曼结构中子流形的距离正则性，管状邻域体积渐进行为，以及海森堡群中曲线的韦伊不变性，证明了管体积仅依赖于Reeb角。
Several quantitative notions of rectifiability in the Heisenberg groups have emerged in the recent literature. In this paper we study the relationship between two of them, the big pieces of intrinsic Lipschitz graphs (BPiLG) condition and the bilateral weak geometric lemma (BWGL), and show that sets with BPiLG satisfy the BWGL.
Lipschitz non-extension theorems into jet space Carnot groups
我们证明了Lipschitz映射的非可拓性结果，目标空间是配备左不变黎曼距离的喷气空间，以及配备左不变子黎曼Carnot-Caratheodory距离的喷气空间。喷流空间给出了一类卡诺群的模型，特别是包括所有海森堡群。
Lipschitz extensions into Jet space Carnot groups
本文的目的是证明一个Lipschitz扩张定理，部分定义Lipschitz映射到喷流空间赋予左不变的子黎曼Carnot-Carathéodory距离。喷流空间给出了一类卡诺群的模型，特别是包括所有海森堡群。证明或我们的定理是相当初级的，他们的灵感来自于思想文件的格罗莫夫，杨，和朗Schlichenmaier。
This paper studies the geometry of bilipschitz maps $f \colon \mathbb{W} \to \mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W} \subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images $f(\mathbb{W})$ of such maps are called Rickman rugs in the Heisenberg group. The main theorem states that a Rickman rug in the Heisenberg group admits a corona decomposition by intrinsic bilipschitz graphs. As a corollary, Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs. Here, an intrinsic bilipschitz graph is an intrinsic Lipschitz graph, whi…
The coarea formula for projections of Lipschitz mappings on Carnot groups
本文证明了Carnot群上内在Lipschitz映射沿水平投影的余面积不等式，并在有限余畸变映射及Heisenberg群情形下取得等式。
In this paper we study intrinsic regular submanifolds of $\mathbb{H}^n$, of low co-dimension in relation with the regularity of their intrinsic parametrization. We extend some results proved for one co-dimensional $\mathbb{H}$-regular surfaces, characterizing uniformly intrinsic differentiable functions $ϕ$ acting between two complementary subgroups of the Heisenberg group $\mathbb{H}^n$, with target space horizontal of dimension $k$, with $1 \leq k \leq n$, in terms of the Euclidean regularity of its components with respect to a family of non linear vector fields $\nabla^{ϕ_j}$. Moreover, we …
Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity
我们给出了次黎曼海森堡群H^1中整面积最小化t-图的新例子。大多数的例子都是欧几里得意义上的局部Lipschitz。一些正则例子给出了由一条水平线或有限条水平半直线组成的奇异集。其中，一个大家庭的面积最小化锥。
Improved Lipschitz approximation of $H$-perimeter minimizing boundaries
本文在Heisenberg群中证明了$H$-周长极小边界的两个新的内蕴Lipschitz逼近结果，分别改进了Monti的结果并改编了De Lellis和Spadaro的极大函数方法。
On low-dimensional uniform rectifiability in Heisenberg groups
本文在Heisenberg群中提出新的Carleson型条件，用于研究低维集的Lipschitz映射大块性质，并探讨了不同量化可测性概念之间的关系。
On low-dimensional uniform rectifiability in Heisenberg groups - Part 2
该文在海森堡群中证明k维内禀Lipschitz图的几何引理，扩展了相关稳定性结果，为k正则集的corona分解建立必要条件，完善了低维一致可求长性的研究。
Tubular neighborhoods in the sub-Riemannian Heisenberg groups
研究了次黎曼海森堡群$\mathbb{H}^n$，$n\ge 1$中$δ_E$到闭集$E$的Carnot-Carathéodory距离.在较弱的条件下证明了$δ_E$的$\mathbb{H}$ -正则性，并引入了奇点的一般概念.当$E$是欧氏$C^k$子流形$k\ge 2$时，我们证明了$δ_E$是奇异集之外的$C^k$.当$E$的边界为类$C^2$时，得到了管状邻域的体积的显式表达式，在奇异集之外，根据$\partial E$的水平主曲率和函数$\langle N,T\rangle/|N_h|$及其切导数.
Invertible Carnot Groups
本文将允许1-拟共形度量反演的卡诺群刻画为李代数满足$J^2$ -条件的Heisenberg型李群，从而刻画了反演不变双Lipschitz齐性的一种特殊情况.对某些非分形度量空间，给出了反演不变双Lipschitz齐性的一个更一般的刻画.
Vertical curves and vertical fibers in the Heisenberg group
研究海森堡群中垂直曲线及相关纤维，通过证明连通垂直曲线性质、表明其与特定内在Lipschitz图交集类一致且维数可变，以及研究相关光滑函数纤维测度，揭示了海森堡群中这些对象的独特性质。
Isodiametric sets in the Heisenberg group
在配备有卡诺-卡拉西奥多利度量和哈尔测度的次黎曼海森堡群中，我们考虑等径集，即在给定直径的所有集合中最大化测度的集合。特别是，给定一个等径集，并可忽略的集，我们证明了它的边界是由两个局部Lipschitz函数的图形。此外，在旋转不变集的限制类中，我们给出了任何紧（旋转不变）等径集的一个相当完整的刻画。更具体地说，它的Steiner对称化相对于Cn-平面示出符合欧几里德凸包的CC-球。同时，我们也证明了相当意想不到的非唯一性结果。
Differentiating maps into L^1 and the geometry of BV functions
这是一个系列的文件审查之间的相互作用微分理论的Lipschitz地图，X->V，和双Lipschitz nonembeddability，其中X是一个度量测度空间和V是一个巴拿赫空间。在这里，我们考虑V=L^1的情况，其中可微性失败。我们建立了X的另一种可微性，包括R^n和H，海森堡群及其Carnot-Cartheodory度量。由此可见，H并不像J. Lee和A. Naor.当与他们的工作相结合时，这为理论计算机科学中的Goemans-Linial猜想提供了一个自然的反例;第一个这样的反例是由Khot-Vishnoi发现的。证明我们的主要定理的一个关键因素是到L^1的Lipschitz映射与有界变差函数之间的新联系，这使我们能够利用最近关于海森堡群上BV函数结构的工作。
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
令$\H$表示离散海森堡群，配备有与某个有限对称生成集相关联的字度量$d_W$。我们表明，如果$(X,\|\cdot\|)$是一个$p$凸Banach空间，那么对于任何Lipschitz函数$f:\H\to X$存在$x,y\in \H$与$d_W(x,y)$任意大和\begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation}我们还表明，任何嵌入到$X$的球半径$R\ge 4$在$\H$招致双Lipschitz失真增长至少作为一个常数倍的\begin{equation}\label{eq:dist abs} \left(\frac{\log R}{\log\log R}\right)^{1/p}. \end{equation}这两个~ \eqref{eq:comp abs}和~ \eqref{eq:dist abs}是尖锐的迭代对数项。当$X$是Hilbert空间时，我们得到了一个表示论证明，给出了~ \eqref{eq:comp abs}和~ \eqref{eq:dist abs}的上界，这两个上界都是一个普适常数.
Metric differentiation, monotonicity and maps to L^1
本文给出了L^1中Lipschitz映射的无穷小结构的一种新方法。作为第一个应用，我们给出了先前论文中主要定理的另一种证明，即海森堡群不允许在L^1中有双李普希茨嵌入。证明使用度量微分定理的保罗和削减度量分解减少非嵌入参数的分类单调子集的海森堡群。
Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a…
On uniform measures in the Heisenberg group
本文研究Heisenberg群中一致测度的分类，完整刻画1维和2维情形，并在支集为竖直直纹曲面时证明3维猜想，同时给出外蕴球测度渐近公式。
A solution of Gromov's Hölder equivalence problem for the Heisenberg group
证明了从一阶Lipschitz同调为零的准凸度量空间到次黎曼Heisenberg群的Hölder指数大于$1/2$的映射通过一棵树因子化.特别地，如果域包含圆盘，这样的映射不能是单射的。这给出了一个回答问题的格罗莫夫最简单的非平凡的情况下。同样的工具可以提高Borisov的结果，它表明，等距浸入类$C^{1,α}$的黎曼曲面正高斯曲率到$\mathbb{R}^3$有界的外在曲率，如果$α> 1/2$。
We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb{H}^n$ with its \cc metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger a…
Weak contact equations for mappings into Heisenberg groups
设k>n为正整数。我们考虑从k维欧几里得空间的子集R^k到海森堡群H^n的具有各种度量性质的映射，每一个度量性质都意味着所讨论的映射满足由海森堡群的次黎曼结构产生的接触方程的某种弱形式。我们说明了一个新的几何技术，直接显示如何弱接触方程极大地限制了映射的行为。特别地，我们提供了一个新的和初等的证明，证明了海森堡群H^n是纯k-不可求长的。证明了对R^k中的开集U，索伯列夫空间W^{1，1}_{loc}（U;R^{2n+1}）中弱切触映射的弱导数的秩几乎处处有界于$n$，回答了Magnani的一个问题.最后证明了如果从U到H^n的映射是s-Hölder连续的，s>1/2，并且当被认为是到R^{2n+1}的映射时是局部Lipschitz的，则该映射不可能是单射的。这个结果与Gromov的一个猜想有关。
L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry
我们调查双Lipschitz嵌入理论和组合优化中的稀疏割问题之间的联系。稀疏割问题的故事是一个引人注目的例子，一方面是分析、几何和概率之间的深刻相互作用，另一方面是离散数学的计算问题。我们解释了如何在过去的20年里的关键思想演变，强调与巴拿赫空间理论，几何测度论，几何群论的相互作用。作为一个重要的说明性例子，我们将考察最近建立的与海森堡群结构的联系，以及它的卡诺-卡拉西奥多里几何与勒贝格空间$L_1$几何的不相容性。
Besicovitch Covering Property for homogeneous distances in the Heisenberg groups
我们的主要结果是一个肯定的答案的问题是否可以找到齐次距离的海森堡群，有贝西科维奇覆盖性质（BCP）。这个性质是众所周知的测度论的基本工具之一，与测度微分理论有着密切的联系。证明了以原点为中心的单位球与欧氏球重合的齐次距离满足BCP。这样的齐次距离确实存在于任何卡诺群的结果Hebisch和Sikora。在海森堡群中，它们与Cygan-Koranyi（也称为Koranyi）距离有关。他们被认为特别是李和Naor提供了一个反例的戈曼-线性猜想在理论计算机科学。为了使我们的结果的角度来看，我们还证明了两个几何标准，暗示BCP的无效性，表明在某种意义上我们的例子是尖锐的。我们的第一个标准特别适用于海森堡群上常用的齐次距离，例如Cygan-Koranyi距离和Carnot-Caratheodory距离，这些距离已知不满足BCP。为了从不同的角度来看待这些结果，并为了完整性，我们也给出了D. Preiss等人的结论，在一般的度量空间中，总是可以构造一个不满足BCP的双Lipschitz等价距离。
Regularity properties of spheres in homogeneous groups
研究了存在单参数位似自同构族的李群上的左不变距离。主要的例子是卡诺群，特别是具有标准膨胀的海森堡群。我们感兴趣的标准意味着，当地和远离对角线，距离是欧几里得Lipschitz，因此，度量领域的边界Lipschitz域在欧几里得意义上。在本文的第一部分，我们考虑测地距离。在这种情况下，我们实际上证明了距离的正则性，在更一般的情况下，子芬斯勒流形没有异常测地线。其次，对于一般群，我们用扩张自同构给出了一个代数判据，例如，它使我们得出了Heisenberg群上齐次距离的正则性，在这样的群中，我们更详细地分析了度量球的几何。我们还提供了齐次群的例子，球提出尖点。
Vertical versus horizontal Poincaré inequalities on the Heisenberg group
令$\H= < a,b | a[a,b]=[a,b]a \wedge b[a,b]=[a,b]b>$为离散海森堡群，配备有与生成集${a,b,a^{-1},b^{-1}}$相关联的左不变字度量$d_W(\cdot,\cdot)$。设$B_n= {x\in \H: d_W(x,e_\H)\le n}$表示相应的半径为$n\in \N$的闭球，记为$c=[a,b]=aba^{-1}b^{-1}$，证明了若$(X,|\cdot|_X)$是一致凸模具有幂型$q\in [2,\infty)$的Banach空间，则存在$K\in (0,\infty)$使得每个$f:\H\to X$满足{multline*} \sum_{k=1}^{n^2} \sum_{x\in B_n} \frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}} \le K \sum_{x\in B_{21n}} \Big（|f（xa）-f（x）|^q_X+|f（xb）-f（x）||^q_X \Big）。{multline*}因此，对于每个$n\in \N$，每个$f:B_n\to X$的双Lipschitz失真至少是$(\log n)^{1/q}$的常数倍，这是$n\to\infty$的渐近最优估计。
Area of Hölder curves and coarea formula on the Heisenberg group
本文证明了从Heisenberg群到R^{2n}的Lipschitz映射的共面积公式，解决了子Riemannian几何中的开放问题，引入了Hölder曲线的积分定义，并给出了新的结果。
Disproof of the tree product conjecture via the Heisenberg group
研究通过嵌入简单图的乘积理解复杂图的问题，以离散三维海森堡群的凯莱图有限子图为反例，利用定量中心坍缩定理反驳d = 4时树积猜想。
Curvature-direction measures of self-similar sets
本文对R^d中的一大类自相似集F，得到了分形Lipschitz-Killing曲率方向测度.这样的措施共同描述的分布法向量和本地化的曲率类似的高阶中曲率的可微子流形。它们解耦为F上的单位Hausdorff测度和球面上的自相似纤维测度的独立乘积，可以通过积分公式计算。相应的局部密度方法使用遍历动力系统形成的扩展的代码空间移位的一个子群的正交群。然后，我们给出了一个非常简单的证明所产生的措施版本在最小的假设。
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a n…
Structure of Porous Sets in Carnot Groups
我们证明了任何卡诺群包含一个闭疏集，它的测度为零，但不是$σ$ -多孔的Carnot-Carathéodory（CC）距离。在第一海森堡群中，我们观察到存在关于CC距离而不是欧几里得距离是多孔的集合，反之亦然。在Carnot群中，我们构造了一个Lipschitz函数，它在给定的$σ$ -多孔集的任何点处都是Pansu可微的，并证明了开集在水平梯度下的原像远不是多孔的.
This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mat…
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