Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

TL;DR

This paper studies Semmes surfaces in the Heisenberg group, showing they are Ahlfors-regular and contain large parts of intrinsic Lipschitz graphs, with implications for geometric measure theory and analysis in sub-Riemannian spaces.

Contribution

It establishes that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular and contain big pieces of intrinsic Lipschitz graphs, extending Euclidean results to a sub-Riemannian setting.

Findings

Semmes surfaces are lower Ahlfors-regular in the Heisenberg group.

Semmes surfaces contain big pieces of intrinsic Lipschitz graphs.

Results apply to boundaries of chord-arc domains and reduced isoperimetric sets.

Abstract

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 Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets.

Equations

$${\mathcal{H}}^s(B(p,r)\cap E)\le C{r}^s,p\in {\mathbb{H}}^k,r>0.$$

$${\mathcal{H}}^s(B(p,r)\cap E)\ge c{r}^s,p\in E,0<r<\mathrm{diam}E.$$

$$q_1,q_2\in B(p,r)\text{and}\min \{\mathrm{dist}(q_1,S),\mathrm{dist}(q_2,S)\}>cr.$$

$${\mathcal{H}}^{2k+1}(B(p,r)\cap S\cap \Gamma )\ge c{r}^{2k+1}.$$

$$\underset{p\in H}{\sup }{\mathcal{H}}^{2k+1}(B(0,1)\cap p\cdot H)<+\infty .$$

$${\mathcal{H}}^{2k+1}(B(0,1)\cap P){\sim }_k{\int }_{B(0,1)\cap P}|C(q)n|d{\mathcal{H}}_{\mathrm{Euc}}^{2k}(q)$$

$${\mathrm{Per}}_{\mathbb{H}}(E,{\mathbb{H}}^k)\le {\mathrm{Per}}_{\mathbb{H}}(F,{\mathbb{H}}^k)$$

$$\text{BVP}+\text{WGL for vertical hyperplanes}⟹\text{BPiLG}$$

$$(v,t)\cdot (v^{\prime },t^{\prime }):=\left(v+v^{\prime },t+t^{\prime }+\omega (v,v^{\prime })/2\right),(v,t),(v^{\prime },t^{\prime })\in {\mathbb{H}}^k,$$

$$\Vert (v,t)\Vert :=\sqrt[4]{|v{|}^4+16t^2}\text{and}d(p,q):=\Vert q^{-1}\cdot p\Vert ,$$

$${\delta }_r:{\mathbb{H}}^k\to {\mathbb{H}}^k,{\delta }_r(v,t):=(rv,r^{2}t),$$

$${\mathbb{L}}_{\nu }:=\mathbb{R}\nu \times \{0\}\text{and}{\mathbb{W}}_{\nu }:={\nu }^{\perp }\times \mathbb{R}.$$

$${\pi }_{{\mathbb{W}}_{\nu }}:{\mathbb{H}}^k\to {\mathbb{W}}_{\nu },$$

$${\pi }_{{\mathbb{L}}_{\nu }}:{\mathbb{H}}^k\to {\mathbb{L}}_{\nu },$$

$${\pi }_{{\mathbb{W}}_{\nu }}(p)$$

$${\pi }_{{\mathbb{L}}_{\nu }}(p)$$

$$C_{\gamma }(\nu ):=\{p\in {\mathbb{H}}^k:\Vert {\pi }_{{\mathbb{W}}_{\nu }}(p)\Vert \le \gamma \Vert {\pi }_{{\mathbb{L}}_{\nu }}(p)\Vert \}.$$

$$(p\cdot C_{1/L}(\nu ))\cap \Gamma =\{p\}\text{for all }p\in \Gamma .$$

$${\tilde{C}}_{\lambda }(\nu ):=\{p\in {\mathbb{H}}^k:\lambda d_{SR}(p,0)<d_{SR}({\pi }_{{\mathbb{L}}_{\nu }}(p),0)\},$$

$$(p\cdot {\tilde{C}}_{\lambda }(\nu ))\cap \Gamma =\varnothing \text{for all }p\in \Gamma .$$

$$\Vert p{\Vert }^4\ge \Vert {\pi }_{{\mathbb{L}}_{\nu }}(p){\Vert }^4+ϵ\Vert {\pi }_{{\mathbb{W}}_{\nu }}(p){\Vert }^4\text{for all }p\in {\mathbb{H}}^k.$$

$${\tilde{C}}_{\lambda }(\nu )\subset C_{\gamma }(\nu )\setminus \{0\}$$

$$\Vert p{\Vert }^4$$

$$d_{SR}(p,0{)}^4\ge \Vert p{\Vert }^4\ge \left(1+ϵ{\gamma }^4\right)\Vert {\pi }_{{\mathbb{L}}_{\nu }}(p){\Vert }^4=\left(1+ϵ{\gamma }^4\right)d_{SR}({\pi }_{{\mathbb{L}}_{\nu }}(p),0{)}^4,$$

$$\mathfrak{h}(\mathcal{A}):={\int }_{{\mathbb{S}}^{2k-1}}{\mathcal{H}}^{2k+1}(\{p\in {\mathbb{W}}_{\nu }:p\cdot {\mathbb{L}}_{\nu }\in \mathcal{A}\})d\nu$$

$$\mathfrak{h}({\delta }_r(\mathcal{A}))=r^{2k+1}\mathfrak{h}(\mathcal{A}),\text{for every Borel set }\mathcal{A}\subset \mathcal{L},r>0.$$

$$\mathfrak{h}(\mathcal{L}(A))={\int }_{{\mathbb{S}}^{2k-1}}{\mathcal{H}}^{2k+1}({\pi }_{{\mathbb{W}}_{\nu }}(A))d\nu$$

$$\mathfrak{h}(\mathcal{A})={\int }_{{\mathbb{S}}^{2k-1}}{\mathcal{H}}^{2k+1}(\{p\in {\mathbb{W}}_{\nu }:p\cdot {\mathbb{L}}_{\nu }\in \mathcal{A}\})d\nu ={\int }_{{\mathbb{S}}^{2k-1}}{\mathcal{H}}^{2k+1}(A_{\nu })d\nu ,$$

$${\Phi }_{p,\nu }(q):={\pi }_{{\mathbb{W}}_{\nu }}(p\cdot q)=p\cdot q\cdot {\pi }_{{\mathbb{L}}_{\nu }}(p{)}^{-1},q\in {\mathbb{W}}_{\nu }.$$

$${\Phi }_{p^{-1},\nu }=({\Phi }_{p,\nu }{)}^{-1}.$$

$${\mathcal{H}}^{2k+1}({\Phi }_{p,\nu }(A_{\nu }))={\mathcal{H}}^{2k+1}(A_{\nu }),p\in {\mathbb{H}}^k,\nu \in {\mathbb{S}}^{2k-1}.$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 The Heisenberg group

4. 2.2 Subgroups, projections, and intrinsic Lipschitz graphs

5. 2.3 Horizontal lines

6. 2.4 Notations

7. 3 Approximating Semmes surfaces with small width by hyperplanes

8. 3.1 Horizontal width

9. 3.2 The approximation

10. 4 Lower Ahlfors-regularity and big vertical projections

11. 5 The bilateral weak geometric lemma

12. 5.1 Width is integrable

13. 5.2 Bilateral weak geometric lemma for arbitrary hyperplanes

14. 5.3 Bilateral weak geometric lemma for vertical hyperplanes

15. 6 Semmes surfaces have big pieces of intrinsic Lipschitz graphs

16. 7 BVP + WGL for vertical hyperplanes imply BPiLG

17. 8 Comments

18. 8.1 Regular triples and corona decompositions

19. 8.2 Semmes surfaces in Euclidean space

Acknowledgements

K.F. is supported by Swiss National Science Foundation via the project Intrinsic rectifiability and mapping theory on the Heisenberg group, grant no. $161299$. T.O. is supported by the Academy of Finland via the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, grant no. $309365$, and by the University of Helsinki via the project Quantitative rectifiability of sets and measures in Euclidean spaces and Heisenberg groups, grant no. $75160012$. S.R. is partially supported by the French National Research Agency, Sub-Riemannian Geometry and Interactions ANR-15-CE40-0018 project. Part of the research was conducted while K.F. and S.R. were visiting the University of Helsinki. The hospitality of the university is gratefully acknowledged. We would like to thank Bruno Franchi, Vasilis Chousionis, and Sean Li for numerous fruitful discussions, and Sylvester Eriksson-Bique for pointing out that the lower Ahlfors regularity of Semmes surfaces follows from relative isoperimetric inequalities. We are also grateful to the reviewer for pointing out relevant references.

1. Introduction

A study of quantitative rectifiability has recently been initialised in the Heisenberg groups $\mathbb{H}^{k}$, modelled after the Euclidean theory of uniformly rectifiable sets, developed by David and Semmes [15, 16] in the 90’s. A starting point for the theory in $\mathbb{H}^{k}$ is the notion of intrinsic Lipschitz graphs introduced by Franchi, Serapioni, and Serra Cassano [21, 22], see also [19]. Some evidence already suggests that intrinsic Lipschitz graphs are suitable building blocks for developing a theory of quantitative rectifiability, see for instance the papers of Naor and Young [30, 31] and of the first two authors with Chousionis [11] (we note that the class of intrinsic Lipschitz graphs studied in [21, 11] coincides with the class considered in [30, 31], see Remark 2.5). The present paper continues this line of research and provides further evidence in this direction.

We focus on Semmes surfaces in $\mathbb{H}^{k}$. These sets were introduced in Euclidean spaces by Semmes [32] in connection with the study of the boundedness of certain singular integrals. David [17] proved that Semmes surfaces in $\mathbb{R}^{n}$ are uniformly rectifiable, and they even satisfy the stronger property of having big pieces of Lipschitz graphs. The main result in the present paper is the Heisenberg analogue of this theorem.

Before discussing the results more thoroughly, we recall some definitions. We refer to Section 2.1 for more details about the Heisenberg group $\mathbb{H}^{k}$, which agrees with $\mathbb{R}^{2k}\times\mathbb{R}$ as a set and which can be viewed as a metric space of Hausdorff dimension $2k+2$ when equipped with the Korányi metric.

Definition 1.1** (Upper and lower Ahlfors-regular sets).**

Let $k\in\mathbb{N}$ and $0\leq s\leq 2k+2$. A closed set $E\subset\mathbb{H}^{k}$ is upper Ahlfors-regular with dimension $s$ if there is a constant $C>0$ such that

[TABLE]

Similarly, a closed set $E\subset\mathbb{H}^{k}$ is lower Ahlfors-regular with dimension $s$ if there is a constant $c>0$ such that

[TABLE]

If $E$ is both upper and lower Ahlfors-regular with dimension $s$, then $E$ is called Ahlfors-regular with dimension $s$.

We will mostly be concerned with Ahlfors-regular sets with dimension $2k+1$. So we will shorten the terminology and say that a closed subset of $\mathbb{H}^{k}$ is upper/lower Ahlfors-regular, without specifying the dimension, to mean that it is upper/lower Ahlfors-regular with dimension $2k+1$.

We denote by $A^{c}:=\mathbb{H}^{k}\setminus A$ the complement of a set $A\subset\mathbb{H}^{k}$.

Definition 1.4** (Condition B).**

A closed set $S$ satisfies Condition B if there is a constant $c>0$ such that the following holds. For all $p\in S$ and $0<r<\operatorname{diam}S$, there exist $q_{1},q_{2}$ in different connected components of $S^{c}$ such that

[TABLE]

Semmes surfaces are then defined as follows.

Definition 1.6** (Semmes surfaces).**

A closed set $S\subset\mathbb{H}^{k}$ is called a Semmes surface if $S$ is upper Ahlfors-regular and satisfies Condition B.

We refer to Definition 2.3 for the definition of intrinsic $L$-Lipschitz graphs in $\mathbb{H}^{k}$ and give below the definition of a set with big pieces of intrinsic Lipschitz graphs.

Definition 1.7** (BPiLG).**

A closed set $S\subset\mathbb{H}^{k}$ has big pieces of intrinsic Lipschitz graphs (or BPiLG in short) if there exist constants $L>0$ and $c>0$ such that the following holds. For all $p\in S$ and $0<r<\operatorname{diam}S$, there exists an intrinsic $L$-Lipschitz graph $\Gamma$ such that

[TABLE]

As in the Euclidean setting, BPiLG is a scale-invariant and quantitative notion of rectifiability. It implies $(2k+1)$-dimensional $\mathbb{H}$-rectifiability, or equivalently $\mathbb{H}_{L}$-rectifiability, in the sense of [33, Definition 4.101]. Namely, up to a set of null $\mathcal{H}^{2k+1}$ measure, a closed Ahlfors-regular set with BPiLG can be covered by countably many intrinsic Lipschitz graphs, but the BPiLG condition is significantly stronger than $\mathbb{H}$-rectifiability.

Our main result reads as follows.

Theorem 1.8**.**

Semmes surfaces in $\mathbb{H}^{k}$ are lower Ahlfors-regular and have big pieces of intrinsic Lipschitz graphs.

The constants $L$ and $c$ in the BPiLG property can be chosen depending only on $k$, and on the upper Ahlfors-regularity and Condition B constants for the Semmes surface. The lower Ahlfors-regularity constant only depends on $k$, and the Condition B constant. The lower Ahlfors-regularity of closed sets satisfying Condition B actually holds in much more general metric measure spaces than $\mathbb{H}^{k}$, see Remark 4.3.

Before we give an outline of the proof of Theorem 1.8, we give some applications. We first recall the corkscrew condition for open sets.

Definition 1.9** (Corkscrew condition).**

An open set $\Omega\subset\mathbb{H}^{k}$ satisfies the corkscrew condition if for every $p\in\partial\Omega$ and $0<r<\operatorname{diam}\partial\Omega$, there exist points $q_{1}\in\Omega$ and $q_{2}\in\overline{\Omega}^{c}$ satisfying (1.5) with $S:=\partial\Omega$.

It is clear that the boundary of an open set satisfying the corkscrew condition satisfies Condition B. However, there are Semmes surfaces that do not arise as the boundary of some open set satisfying the corkscrew condition, see Section 8.1 for more comments in this direction. Going back to the corkscrew condition, Theorem 1.8 has the following immediate consequence.

Corollary 1.10**.**

Let $\Omega\subset\mathbb{H}^{k}$ be an open set satisfying the corkscrew condition whose boundary is upper Ahlfors-regular. Then $\partial\Omega$ is lower Ahlfors-regular and has BPiLG.

Corollary 1.10 should be viewed as a quantitative counterpart of the following qualitative statement. If $\Omega\subset\mathbb{H}^{k}$ is an open set satisfying the corkscrew condition whose boundary has locally finite $\mathcal{H}^{2k+1}$ measure, then $\partial\Omega$ is $(2k+1)$-dimensional $\mathbb{H}$-rectifiable, or equivalently $\mathbb{H}_{L}$-rectifiable, in the sense of [33, Definition 4.101]. This follows from the theory of finite $\mathbb{H}$-perimeter sets in $\mathbb{H}^{k}$, developed by Franchi, Serapioni, and Serra Cassano in [20], by noting that the topological boundary of such a set coincides with its measure-theoretic boundary.

We recall that NTA domains are examples of domains satisfying the corkscrew condition together with an additional (interior) Harnack chain condition. In Euclidean spaces, NTA domains were introduced in 1982 by Jerison and Kenig [24] and have found numerous applications, in particular in the study of harmonic functions and in the theory of elliptic equations and free boundary problems. The boundary of an arbitrary NTA domain need not be rectifiable, let alone have big pieces of Lipschitz graphs. Imposing the additional assumption that the boundary is Ahlfors-regular leads to the notion of a chord-arc domain, as stated in Definition 1.11 below in the Heisenberg setting. For more information and references, we refer for instance to the recent papers [2] of Azzam, Hofmann, Martell, Nyström, and Toro, [1] of Azzam, and [3] of Badger in the Euclidean setting. NTA domains in $\mathbb{H}^{k}$ have been investigated by Capogna and Tang [8], and in more general Carnot groups by Capogna and Garofalo [5] and Monti and Morbidelli [29]. See also the survey [7] by Capogna, Garofalo and Nhieu.

Definition 1.11** (Chord-arc domains).**

An NTA domain with Ahlfors-regular boundary in $\mathbb{H}^{k}$ is called a chord-arc domain.

Corollary 1.10 applies in particular to chord-arc domains.

Corollary 1.12**.**

The boundary of a chord-arc domain in $\mathbb{H}^{k}$ has BPiLG.

To illustrate Corollary 1.12, we next discuss a simple example of a chord-arc domain in $\mathbb{H}^{k}$ that is not an intrinsic Lipschitz domain, even though it has smooth boundary in the Euclidean sense. The same example was presented in [5, Proposition 15] as a motivation for studying NTA domains in $\mathbb{H}^{1}$.

Example 1.13**.**

Let $\Omega:=\{(v,t)\in\mathbb{H}^{k}:t>0\}$ be the upper half-space whose boundary is $H:=\mathbb{R}^{2k}\times\{0\}$. It is easy to see that $H\cap B(0,r)$ is not an intrinsic Lipschitz graph for any $r>0$. Nevertheless, we now check that $\Omega$ is a corkscrew domain with upper Ahlfors-regular boundary, so $H$ has BPiLG by Corollary 1.10. First, $\Omega$ is actually an NTA domain. This follows for instance from a result of Monti and Morbidelli [29] since $H$ is smooth. Alternatively, one can check that $\Omega$ satisfies the corkscrew condition via direct geometric arguments. Next, the upper Ahlfors-regularity of $H$ follows directly from [6, Theorem 1.2], or can be deduced from [20, Corollary 7.7(i)]. Indeed, using left translations and dilations, the upper Ahlfors-regularity of $H$ is equivalent to

[TABLE]

If $P:=p\cdot H$ for some $p\in H$, then $P$ is the boundary of a $C^{1}$ domain, and it follows from [20, Corollary 7.7(i)] that

[TABLE]

where $n$ denotes the unit Euclidean normal to $P$, the expression of $|C(p)n|$ is given in (4.11), and $\mathcal{H}^{2k}_{\mathrm{Euc}}$ denotes the $2k$-dimensional Hausdorff measure with respect to the Euclidean distance. Since $B(0,1)$ is contained in a Euclidean ball with radius $2$, it follows that $\mathcal{H}^{2k+1}(B(0,1)\cap P)\lesssim_{k}1$ which proves the upper Ahlfors-regularity of $H$. The lower Ahlfors-regularity of $H$ will be proven in (4.12), or follows again from [6, Theorem 1.2]. This last remark together with the previous ones hence show that $\Omega$ is a chord-arc domain.

Other examples of Semmes surfaces in $\mathbb{H}^{k}$ are given by the boundaries of reduced isoperimetric sets. We recall that a measurable set $E\subset\mathbb{H}^{k}$ is said to be isoperimetric if

[TABLE]

for every measurable set $F\subset\mathbb{H}^{k}$ such that $\mathcal{H}^{2k+2}(F)=\mathcal{H}^{2k+2}(E)$, where $\operatorname{Per}_{\mathbb{H}}$ denotes the $\mathbb{H}$-perimeter. We refer to [20] for the definition of $\mathbb{H}$-perimeter. The existence of isoperimetric sets in $\mathbb{H}^{k}$, and more generally in any Carnot group, has been proven by the third author with Leonardi [26, Theorem 3.2]. It is still an open question to find explicitly which sets are isoperimetric in this setting. One of the reasons why this question remains open is the difficulty of proving a priori regularity estimates for isoperimetric sets. It was however shown in [26] that every isoperimetric set $E$ is equivalent to a unique open set $\Omega$, meaning that $\mathcal{H}^{2k+2}(E\bigtriangleup\Omega)=0$, which satisfies the corkscrew condition and whose boundary is Ahlfors-regular [26, Theorem 3.3]. Such a set $\Omega$ is still isoperimetric and is called a reduced isoperimetric set. Corollary 1.10 can hence be applied to boundaries of reduced isoperimetric sets.

Corollary 1.14**.**

The boundary of a reduced isoperimetric set in $\mathbb{H}^{k}$ has BPiLG.

We give now an outline of the proof of Theorem 1.8, the details of which can be found in Sections 3 to 7. As already mentioned, Theorem 1.8 is the Heisenberg analogue of David’s Euclidean result [17, Proposition 2]. Several different proofs for the Euclidean result, besides [17], are available in the literature and can be found in [14] and [18]. The general strategy of our proof is inspired by the strategy in [18]. However, as far as we can tell, the arguments to make this strategy work in the Euclidean setting do not easily translate to $\mathbb{H}^{k}$, and neither do the other known proofs of the Euclidean analogue of Theorem 1.8. Our arguments to implement the strategy in the Heisenberg setting will hence be quite different from Euclidean ones, and are inspired by the concept of non-monotonicity introduced by Cheeger, Kleiner and Naor in [10], and by some of the techniques developed by Naor and Young in [31].

Following the general strategy of [18], we prove that a Semmes surface $S$ is lower Ahlfors-regular and has BPiLG by showing that $S$ has the following properties. First, $S$ is lower Ahlfors-regular by Proposition 4.1. This will be obtained as a consequence of the relative isoperimetric inequality in $\mathbb{H}^{k}$. Next, $S$ has big vertical projections (BVP), see Definition 4.4 and Proposition 4.6. Roughly speaking, BVP is a uniform and scale invariant way of requiring that, for every ball $B$ centred on $S$, there is a vertical subgroup onto which $B\cap S$ has a large projection in $\mathcal{H}^{2k+1}$ measure. Finally, $S$ satisfies the weak geometric lemma (WGL) for vertical hyperplanes, see Definition 6.1 and Proposition 6.3. Informally, WGL for vertical hyperplanes states that $S$ can be well-approximated by vertical hyperplanes at most points and scales. For closed Ahlfors-regular sets in $\mathbb{H}^{k}$, the implication

[TABLE]

is the Heisenberg analogue of a Euclidean result due to David and Semmes [18, Theorem 1.14]. It has been proven in $\mathbb{H}^{1}$ by the first two authors with Chousionis in [11]. The extension of this result to higher dimensional Heisenberg groups will be given in Section 7, see Theorem 7.1, and this will allow us to conclude the proof of Theorem 1.8.

A challenge in $\mathbb{H}^{k}$ is to deduce from Condition B, which does not include any reference to the horizontal distribution of the Heisenberg group, information about the family of vertical projections and their fibers, which are horizontal lines. This challenge is not present in $\mathbb{R}^{n}$, where lines in all directions appear as fibers of orthogonal projections. Moreover, unlike orthogonal projections in $\mathbb{R}^{n}$, group projections onto a vertical subgroup are not Lipschitz maps when both $\mathbb{H}^{k}$ and the vertical subgroup are equipped with the Korányi distance. Consequently, while showing that a Semmes surface in $\mathbb{R}^{n}$ has big projections is a rather immediate consequence of Condition B, proving the corresponding results in $\mathbb{H}^{k}$ is not so simple. Complications related to the fact that we are dealing with a restricted family of lines also appear in the proof of WGL for vertical hyperplanes.

We introduce in Section 3.1 the notion of horizontal width as a central tool in the proof of both BVP and WGL for vertical hyperplanes. This notion is inspired by the concept of non-monotonicity, introduced by Cheeger, Kleiner, and Naor in [10]. The non-monotonicity of a given set is indeed controlled by the horizontal width of its boundary, see Lemma 3.4 and Proposition 3.7. The novelty here is that horizontal width turns out to be a suitable tool to deal with Semmes surfaces that do not necessarily arise as boundaries of sets satisfying the corkscrew condition.

We show that if the horizontal width of a Semmes surface $S$ is small inside a ball, then $S$ is well-approximated, in a bilateral way, by a hyperplane inside a slightly smaller ball, Corollary 3.24. Our proof goes through a first step, Corollary 3.13, which combines the control of the non-monotonicity of every connected components of the complement of $S$ by the horizontal width of $S$ with a result due to Naor and Young [31, Proposition 66]. Naor and Young’s result is itself inspired by a deep stability theorem for monotone sets due to Cheeger, Kleiner and Naor. Condition B first appears in the second step of the proof of Corollary 3.24 and allows us to prove that if $S$ satisfies the conclusion of Corollary 3.13, namely, if all connected components of its complement are measure-theoretically close to half-spaces inside a ball, then $S$ is rather flat in a slightly smaller ball, see Proposition 3.15.

With Corollary 3.24 in hand, we can then proceed to the proof of BVP and WGL for vertical hyperplanes. The proof of BVP is given in Section 4. Then Section 5 is devoted to the proof of the validity of the bilateral weak geometric lemma (BWGL) for vertical hyperplanes, see Definition 5.8 and Proposition 5.11. BWGL is stronger than WGL and obviously implies WGL for vertical hyperplanes, see Section 6. Our proof of the validity of BWGL for vertical hyperplanes relies on the following facts. First, we prove that, for a closed upper Ahlfors-regular set, balls with large horizontal width satisfy a Carleson packing condition, Proposition 5.1. Combined with Corollary 3.24, this implies that a Semmes surface is well-approximated, in a bilateral way, by arbitrary hyperplanes at most points and scales, Proposition 5.7. Then we upgrade this approximation by arbitrary hyperplanes to a similar approximation by vertical hyperplanes, Proposition 5.11. We explicitly note in Section 5.3 that the validity of BWGL for vertical hyperplanes for a closed Ahlfors-regular set is equivalent to the validity of BWGL for arbitrary hyperplanes, Theorem 5.10. The proof of this latter fact is already essentially contained in the work of Naor and Young [31].

We end the paper with a section where we first comment on connections as well as differences, between Naor and Young’s work [31, Section 9] and the setting and results of the present paper. This suggests further questions to better understand the theory of “uniform rectifiability” that is now emerging in the Heisenberg setting. In the final Section 8.2, we go back to Euclidean spaces and note that the method of the current paper applies to the Euclidean setting, too, and gives a direct proof of the validity of the bilateral weak geometric lemma for Semmes surfaces in $\mathbb{R}^{n}$. As a consequence, it also gives a new path for the proof of the big pieces of Lipschitz graphs property for Semmes surfaces in Euclidean spaces.

2. Preliminaries

In this section, we introduce some of the concepts frequently used in the rest of the paper and state notational conventions.

2.1. The Heisenberg group

For a fixed integer $k\geq 1$, we identify the Heisenberg group $\mathbb{H}^{k}$ with $\mathbb{R}^{2k}\times\mathbb{R}$ equipped with the group law

[TABLE]

where $\omega(v,v^{\prime}):=\sum_{j=1}^{k}v_{j}v^{\prime}_{j+k}-v_{j+k}v^{\prime}_{j}$ for $v=(v_{1},\dots,v_{2k})$, $v^{\prime}=(v^{\prime}_{1},\dots,v^{\prime}_{2k})\in\mathbb{R}^{2k}$.

We equip $\mathbb{H}^{k}$ with the Korányi norm $\|\cdot\|$ and Korányi metric $d$ defined respectively by

[TABLE]

where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^{2k}$.

The metric $d$ is homogeneous, meaning that it is left-invariant, that is, $d(p\cdot q,p\cdot q^{\prime})=d(q,q^{\prime})$ for all $p,q,q^{\prime}\in\mathbb{H}^{k}$, and one-homogeneous with respect to the Heisenberg dilations $(\delta_{r})_{r>0}$ given by

[TABLE]

that is, $d(\delta_{r}(p),\delta_{r}(q))=rd(p,q)$ for all $p,q\in\mathbb{H}^{k}$ and $r>0$. We recall that the family $(\delta_{r})_{r>0}$ of dilations is a one-parameter group of group automorphisms of $\mathbb{H}^{k}$. We also recall that any two homogeneous distances on $\mathbb{H}^{k}$ are biLipschitz equivalent.

For $s\geq 0$, we denote by $\mathcal{H}^{s}$ the Hausdorff measure of dimension $s$ with respect to the Korányi metric. We recall that $\mathcal{H}^{2k+2}$ is a Haar measure on $\mathbb{H}^{k}$ and is $(2k+2)$-uniform. In particular $(\mathbb{H}^{k},d)$ is a metric space of Hausdorff dimension $2k+2$. A thorough introduction to $(\mathbb{H}^{k},d)$ can be found for instance to the monograph [4].

Unless otherwise explicitly stated, all metric notions in $\mathbb{H}^{k}$ will always be defined with respect the Korányi metric. In particular, we write $B(p,r):=\{q\in\mathbb{H}^{k}:d(p,q)\leq r\}$ for a closed ball in $(\mathbb{H}^{k},d)$ centred at $p\in\mathbb{H}^{k}$ with radius $r>0$. By default, a ball will refer to a closed ball.

2.2. Subgroups, projections, and intrinsic Lipschitz graphs

We denote by $\mathbb{S}^{2k-1}$ the Euclidean unit sphere in $\mathbb{R}^{2k}$. Given $A\subset\mathbb{R}^{2k}$, we denote by $A^{\perp}$ the linear subspace orthogonal to $A$ in the Euclidean sense in $\mathbb{R}^{2k}$.

Given $\nu\in\mathbb{S}^{2k-1}$, we set

[TABLE]

These sets are complementary homogeneous subgroups of $\mathbb{H}^{k}$ which means that both $\mathbb{L}_{\nu}$ and $\mathbb{W}_{\nu}$ are subgroups of $\mathbb{H}^{k}$ closed under the family of dilations $(\delta_{r})_{r>0}$ and every point $p\in\mathbb{H}^{k}$ can be uniquely written as $p=p_{\mathbb{W}_{\nu}}\cdot p_{\mathbb{L}_{\nu}}$