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 decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $\mathbb{H}^1$. The main difference to this case arises from the fact that for $1\leq k<n$, the complementary vertical subgroups of $k$-dimensional horizontal subgroups in $\mathbb{H}^n$ are not commutative.
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}.
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 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 …
We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from t…
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…
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.
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…
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…
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…
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.
Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Korányi and Reimann using the so-called flow method. Here we supply further evidence of the flexible nature of this family, constructing quasiconformal mappings with extreme behavior on small sets. More precisely, we establish criteria to determine when a given logarithmic potential $Λ$ on $\mathbb{H}$ is such that there exists a quasiconformal mapping of $\mathbb{H}$ with Jacobian comparable to $e^{2Λ}$ (so that the Jaobian is zero or infinity at the…
Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of $\mathbb{H}$-regular surfaces. We prove that $\mathbb{H}$-regular surfaces in $\mathbb{H}^{n}$ with $α$-Hölder continuous horizo…
The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in \{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm 1},b_k^{\pm 1}\}$. The horizontal boundary of $Ω\subset \mathbb{H}_{\mathbb{Z}}^{2k+1}$, denoted $\partial_{h}Ω$, is the set of all $(x,y)\in Ω\times (\mathbb{H}_{\mathbb{Z}}^{2k+1}\setminus Ω)$ such that $x^{-1}y\in S$. The horizontal perimeter of $Ω$ is $|\partial_{h}Ω|$. Fo…
