Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

TL;DR

This paper explores which subsets of the Heisenberg group can be bi-Lipschitz embedded into Euclidean spaces, establishing universal embedding constants for various manifolds and providing counterexamples.

Contribution

It identifies classes of subsets of the Heisenberg group that admit bi-Lipschitz embeddings into Euclidean spaces with uniform constants, and constructs examples that do not embed.

Findings

Lines embed into R^3 with a universal constant

Planes embed into R^4 with a universal constant

Certain manifolds embed into R^4 with a universal constant

Abstract

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 $L$-bi-Lipschitz embed into $\mathbb{R}^4$ where the constant $L$ is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Kor\'{a}nyi spheres bi-Lipschitz embed into $\mathbb{R}^4$ with a uniform constant. Finally, we show that there exists a compact, porous subset of $\mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.

Figures

Equations

$$L^{-1}{d}_X(a,b)\le d_Y(f(a),f(b))\le L{d}_X(a,b)\text{for any }a,b\in X.$$

$$\mathrm{Lip}(X,Y):=\inf \{L:\text{ there exists L-bi-Lipschitz }f:X\to Y\}\in [1,\infty ]$$

$$L_{\Phi }^{-1}|x-y{|}^{1/2}\le |\Phi (x)-\Phi (y)|\le L_{\Phi }|x-y{|}^{1/2},$$

$$\mathrm{dist}(\Phi (t),\partial [0,1{]}^3)\le L_{\Phi }^{-1}\mathrm{dist}(\Phi (t),\{(1/2,1/2,0),(1/2,1/2,1)\}).$$

$$L_{\Phi }^{-1}|x-y{|}^{1/2}\le |\varphi (x)-\varphi (y)|\le L_{\Phi }|x-y{|}^{1/2}.$$

$$X={\partial }_x-\frac{1}{2}y{\partial }_z\text{and}Y={\partial }_y+\frac{1}{2}x{\partial }_z$$

$$g(aX+bY,cX+dY)=ac+bd.$$

$$(x,y,z)\cdot (x^{\prime },y^{\prime },z^{\prime })=\left(x+x^{\prime },y+y^{\prime },z+z^{\prime }+\frac{1}{2}(x{y}^{\prime }-x^{\prime }y)\right).$$

$$z^{\prime }(t)+\frac{1}{2}{x}^{\prime }(t)y(t)-\frac{1}{2}x(t)y^{\prime }(t)=0\text{for almost all }t\in [0,T].$$

$$d_{cc}(p,q)=\underset{\gamma }{\inf }{\int }_0^1\sqrt{g(\dot{\gamma }(t),\dot{\gamma }(t))}dt$$

$$\left\{\left(t,bt+c,-\frac{1}{2}ct+C_1\right):t\in \mathbb{R}\right\},C_1\in \mathbb{R}$$

$$\{(-2b,2a)+ts:t\in \mathbb{R}\},s\in {\mathbb{S}}^1,$$

$$\{(-2b,2a)+ts:s\in {\mathbb{S}}^1\},t>0$$

$$d_K(p,q)={\left({\left((x-x^{\prime }{)}^2+(y-y^{\prime }{)}^2\right)}^2+16(z-z^{\prime }+\frac{1}{2}(x{y}^{\prime }-x^{\prime }y){)}^2\right)}^{1/4}$$

$$C^{-1}{d}_K(p,q)\le d_{cc}(p,q)\le C{d}_K(p,q)\text{for all }p,q\in \mathbb{H}$$

$$d_{\mathbb{H}}(p,q)=|x-x^{\prime }|+|y-y^{\prime }|+{\left|z-z^{\prime }+\frac{1}{2}(x{y}^{\prime }-x^{\prime }y)\right|}^{\frac{1}{2}}.$$

$$\min \{R^{1/2},R^{-1}\}|w-w^{\prime }|\lesssim d_{\mathbb{H}}(w,w^{\prime })\lesssim \max \{1,R^{1/2}\}|w-w^{\prime }{|}^{1/2}.$$

$$d_{\mathbb{H}}(w,w^{\prime })=|x-x^{\prime }|+|y-y^{\prime }|+|z-z^{\prime }+\frac{1}{2}{y}^{\prime }(x-x^{\prime })-\frac{1}{2}{x}^{\prime }(y-y^{\prime }){|}^{1/2}.$$

$$d_{\mathbb{H}}(w,w^{\prime })$$

$$\lesssim (1+R^{1/2})|w-w^{\prime }{|}^{1/2}.$$

$$d_{\mathbb{H}}(w,w^{\prime })\ge (|z-z^{\prime }|-\frac{1}{2}R|x-x^{\prime }|-\frac{1}{2}R|y-y^{\prime }|{)}^{1/2}$$

$$\gtrsim \min \{1,R^{1/2}\}|w-w^{\prime }{|}^{1/2}$$

$$\ge \min \{R^{-1/2},R^{1/2}\}|w-w^{\prime }|.$$

$$d_{\mathbb{H}}(w,w^{\prime })\ge |x-x^{\prime }|+|y-y^{\prime }|\gtrsim \min \{1,R^{-1}\}|w-w^{\prime }|.$$

$$d_{\mathbb{H}}(w(t_1),w(t_2))\lesssim \left(\Vert x^{\prime }(t){\Vert }_{\infty }+\Vert y^{\prime }(t){\Vert }_{\infty }\right)|t_1-t_2|.$$

$$|x(t_1)-x(t_2)|+|y(t_1)-y(t_2)|=\left|{\int }_{t_1}^{t_2}{x}^{\prime }(s)ds\right|+\left|{\int }_{t_1}^{t_2}{y}^{\prime }(s)ds\right|\le \left(\Vert x^{\prime }(t){\Vert }_{\infty }+\Vert y^{\prime }(t){\Vert }_{\infty }\right)|t_1-t_2|.$$

$$\displaystyle\Big{|}z(t_{1})$$

$$=2^{-1/2}{\left|{\int }_{t_1}^{t_2}[x^{\prime }(t)y(t)-x(t)y^{\prime }(t)]dt+x(t_1)y(t_2)-x(t_2)y(t_1)\right|}^{1/2}$$

$$=2^{-1/2}{\left|{\int }_{t_1}^{t_2}[x^{\prime }(t)y(t)-x(t)y^{\prime }(t)]dt-y(t_2)(x(t_2)-x(t_1))+x(t_2)(y(t_2)-y(t_1))\right|}^{1/2}$$

$$=2^{-1/2}{\left|{\int }_{t_1}^{t_2}[x^{\prime }(t)(y(t)-y(t_2))dt-{\int }_{t_1}^{t_2}{y}^{\prime }(t)(x(t)-x(t_2))dt\right|}^{1/2}$$

$$\le {\left({\int }_{t_1}^{t_2}|x^{\prime }(t)||y(t)-y(t_2)|dt+{\int }_{t_1}^{t_2}|y^{\prime }(t)||x(t)-x(t_2)|dt\right)}^{1/2}$$

$$\le \Vert x^{\prime }(t){\Vert }_{\infty }^{1/2}\Vert y^{\prime }(t){\Vert }_{\infty }^{1/2}|t_1-t_2|.$$

$$w(t)=(a_{1}t+c_1,a_{2}t+c_2,a_{3}t+c_3),t\in \mathbb{R}$$

$$d_{\mathbb{H}}(w(\tau ),w({\tau }^{\prime }))=|\tau -{\tau }^{\prime }{|}^{1/2}\text{and}|F(w(\tau ))-F(w({\tau }^{\prime }))|=|\Phi (\tau )-\Phi ({\tau }^{\prime })|.$$

Full text

V.C. is supported by the Simons Collaboration grant no.  521845. S.L. is supported by the NSF grant DMS-1600804. V.V. is supported by the NSF grant DMS-1800731.

1. Introduction

Given two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, we say that a mapping $f:X\to Y$ is $L$-bi-Lipschitz for $L\geq 1$ if

[TABLE]

In this case we say that $X$ bi-Lipschitz embeds in $Y$ or that $X$ admits a bi-Lipschitz embedding into $Y$. For two metric spaces $X$ and $Y$, we define the quantity

[TABLE]

so that $X$ admits a bi-Lipschitz embedding into $Y$ if and only if $\operatorname{Lip}(X,Y)<\infty$.

When does a metric space admit a bi-Lipschitz embedding into some (finite-dimensional) Euclidean space? Spaces admitting such embeddings can roughly be thought to live inside a Euclidean space, and this is very important for a comprehensive study of their geometry. The embedding problem has attracted considerable attention over the years due to its applications in theoretical computer science and, more specifically, in graphic imaging and storage and access issues for large data sets [KV15, Nao10].

It is well know that a metric space bi-Lipschitz embeds into some Euclidean space only if it is doubling. Recall that a space is doubling if every ball of radius $r$ can be covered by at most $N$ balls of radii $r/2$ for some fixed $N>1$. Moreover Assouad [Ass77, Ass79, Ass83] showed that, if a metric space $(X,d)$ is doubling, then, for any $\epsilon>0$, the snowflaked space $(X,d^{\epsilon})$ bi-Lipschitz embeds into some Euclidean space. Nevertheless, the doubling condition is not sufficient for the existence of a bi-Lipschitz embedding into a Euclidean space. The Heisenberg group endowed with a sub-Riemannian metric is probably the most well known example of a doubling space which does not admit bi-Lipschitz embedding into any Euclidean space. This follows from a deep theorem of Pansu [Pan89] and an observation of Semmes [Sem96]. In fact, the second author showed that $\mathbb{H}$ does not bi-Lipschitz embed in any Hilbert space [Li16]. We also record that bi-Lipschitz embeddability of sub-Riemannian manifolds, and especially of Carnot groups, has been studied by various authors [Sem96, Wu15a, RV17, Wu15b, Rom16].

Here, we are concerned with the following question: which subsets of $\mathbb{H}$ admit a bi-Lipschitz embedding into some Euclidean space? Clearly, every finite subset of the Heisenberg group admits such an embedding. On the other hand, following the arguments of Pansu and Semmes, it is easy to see that no open subset of the Heisenberg group admits such a bi-Lipschitz embedding. We are particularly interested in the bi-Lipschitz embeddability of submanifolds of $\mathbb{H}$, which have been studied extensively in the past 20 years in connection to geometric measure theory and geometric analysis, see e.g. [SC16, CDPT07].

Our first theorem focuses on affine subsets of $\mathbb{R}^{3}$. Here and in what follows, $d_{K}$ is the Korányi metric defined in Section 2.

Theorem 1.1**.**

There exists a universal $L>1$ with the following properties.

1. (1)

If $\ell\subset\mathbb{R}^{3}$ is a line, then $\operatorname{Lip}((\ell,d_{K}),\mathbb{R}^{3})\leq L$. 2. (2)

If $P\subset\mathbb{R}^{3}$ is a plane, then $\operatorname{Lip}((P,d_{K}),\mathbb{R}^{4})\leq L$.

We note that the Euclidean dimensions in the preceding theorem are sharp. Indeed, the $z$-axis in the Heisenberg group is bi-Lipschitz homeomorphic to the snowflaked space $(\mathbb{R},|\cdot|^{1/2})$ which cannot be embedded in $\mathbb{R}^{2}$. On the other hand, the plane $\{x=0\}$ is bi-Lipschitz homeomorphic to the product space $(\mathbb{R},|\cdot|)\times(\mathbb{R},|\cdot|^{1/2})$ which can not be embedded in $\mathbb{R}^{3}$.

Using a bi-Lipschitz welding theorem [MM16], we obtain the following result as a corollary.

Theorem 1.2**.**

If $M$ is a compact piecewise linear $2$-manifold in $\mathbb{R}^{3}$, then there exists $N\in\mathbb{N}$ such that $\operatorname{Lip}((M,d_{K}),\mathbb{R}^{N})<\infty$.

It is natural now to ask if Theorem 1.2 holds if “piecewise linear” is replaced by some degree of differentiability. Our next theorem asserts that any $C^{1,1}$ $1$-manifold can be locally bi-Lipschitz embedded in $\mathbb{R}^{4}$ with a uniform constant. Moreover we prove that the same holds true for $C^{1,1}$ $2$-manifolds around non-characteristic points. Recall that a point $p$ on a differentiable manifold in $\mathbb{R}^{3}$ is called characteristic if the horizontal distribution at that point is the same as the tangent space. Otherwise, the point is called $\mathbb{H}$-regular. By well known results of Balogh [Bal03], almost all points of a $C^{1,1}$ smooth $2$-manifold in $\mathbb{R}^{3}$ are $\mathbb{H}$-regular. A $2$-manifold is called $\mathbb{H}$-regular if all of its points are $\mathbb{H}$-regular.

Theorem 1.3**.**

There exists universal $L>1$ with the following properties.

1. (1)

If $\gamma\subset\mathbb{R}^{3}$ is a $C^{1,1}$ $1$-manifold and $x\in\gamma$, then there exists a neighborhood $U\subset\gamma$ of $x$ such that $\operatorname{Lip}((U,d_{K}),\mathbb{R}^{4})\leq L$. 2. (2)

If $M\subset\mathbb{R}^{3}$ is a $C^{1,1}$ $2$-manifold and $x\in M$ is $\mathbb{H}$-regular, then there exists a neighborhood $U\subset M$ of $x$ such that $\operatorname{Lip}((U,d_{K}),\mathbb{R}^{4})\leq L$.

The dimension for the second part is sharp. Moreover, the embeddings are only local as there exists a $C^{1,1}$ curve $\gamma$ such that $(\gamma,d_{\mathbb{H}})$ does not bi-Lipschitz embed into any Euclidean space (simply take a $C^{1,1}$ curve that contains $\mathbb{Z}^{3}$).

The case of 2-manifolds with characteristic points is more complicated, and we only give partial results. In §5, we show that every $C^{1,1}$ surface obtained by revolving a curve around the $z$-axis admits local bi-Lipschitz embeddings into $\mathbb{R}^{4}$. In particular, we show that all Korányi spheres bi-Lipschitz embed in $\mathbb{R}^{4}$ with a uniform constant. Such surfaces contain at most 2 characteristic points. In §5.3 we show that the surface $z=\frac{1}{2}xy$, which contains infinitely many characteristic points, bi-Lipschitz embeds in $\mathbb{R}^{19}$.

Recall that a subset $E$ of a metric space $X$ is porous if there exists $c\geq 1$ such that, for every $x\in E$ and every $r>0$, the set $B(x,r)\setminus E$ contains a ball of radius $r/c$. Following the techniques of Pansu and Semmes and using an Arzelá-Ascoli argument, one can show that a non-porous subset of $\mathbb{H}$ admits no bi-Lipschitz embedding in any Euclidean or Hilbert space. As all the curves and surfaces considered above are porous sets in $\mathbb{H}$, it is natural to ask if all compact porous subsets of $\mathbb{H}$ admit a bi-Lipschitz embedding into some Euclidean space. In §6 we answer this question negatively.

Theorem 1.4**.**

There exists a compact porous subset $X\subset\mathbb{H}$ that does not admit a bi-Lipschitz embedding into $\ell_{2}$.

The paper is organized as follows. In Section 2 we will introduce the Heisenberg group and we will prove some basic estimates for the Korányi metric which will be used repeatedly in the following. In Section 3 we will prove Theorem 1.1, while in Section 4 we consider smooth curves and smooth regular surfaces and provide the proof of Theorem 1.3. In Section 5 we study several examples of smooth surfaces with characteristic points, and, in particular, we prove that Korányi spheres bi-Lipschitz embed into $\mathbb{R}^{4}$ with a uniform constant. Finally in Section 6 we use Laakso graphs in order to construct a porous compact subset of $\mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.

2. Preliminaries

Given numbers $x,y\geq 0$ and parameters $a_{1},\dots,a_{n}$, we write $x\lesssim_{a_{1},\dots,a_{n}}y$ if there exists a positive and finite constant $C$ depending on at most $a_{1},\dots,a_{n}$ such that $x\leq Cy$. We write $x\simeq_{a_{1},\dots,a_{n}}y$ when $x\lesssim_{a_{1},\dots,a_{n}}y$ and $y\lesssim_{a_{1},\dots,a_{n}}x$. Similarly, we write $x\lesssim y$ or $x\simeq y$ to denote that the implicit constants are universal.

Recall (see [BH04, Wu15a]) that there exists $L_{\Phi}>1$ and an embedding $\Phi:\mathbb{R}\to\mathbb{R}^{3}$ such that

1. (1)

for all $x,y\in\mathbb{R}$,

[TABLE] 2. (2)

$\Phi([0,1])\subset[0,1]^{3}$ with $\Phi(0)=(1/2,1/2,0)$ and $\Phi(1)=(1/2,1/2,1)$, 3. (3)

for all $t\in[0,1]$,

[TABLE]

That is, $\Phi$ is a bi-Lipschitz embedding of $(\mathbb{R},|\cdot|^{1/2})$ in $\mathbb{R}^{3}$.

Similarly, there exists $\phi:\mathbb{S}^{1}\to\mathbb{R}^{3}$ such that, for all $x,y\in\mathbb{S}^{1}$,

[TABLE]

2.1. Heisenberg group

The Heisenberg group is the sub-Riemannian manifold $(\mathbb{R}^{3},H,g)$ where $H$ is the horizontal distribution generated by the vector fields

[TABLE]

and with $g$ given by

[TABLE]

The Heisenberg group can also be thought as a group $\mathbb{H}=(\mathbb{R}^{3},\cdot)$ with the group law

[TABLE]

An absolutely continuous curve $\gamma(t)=(x(t),y(t),z(t))$ with $t\in[0,T]$ is horizontal if $\gamma^{\prime}(t)\in H_{\gamma(t)}$ for almost all $t\in[0,T]$. That is,

[TABLE]

The Carnot-Carathéodory metric on $\mathbb{H}$ is defined by

[TABLE]

where the infimum is taken over all horizontal curves $\gamma:[0,1]\to\mathbb{R}^{3}$ such that $\gamma(0)=p$ and $\gamma(1)=q$.

The following example describes the horizontal curves on planes.

Example 2.1*.*

In vertical planes $y=bx+c$, the horizontal curves are exactly the lines

[TABLE]

while the vertical lines $\{(0,0,t+C_{2}):t\in\mathbb{R}\}$, $C_{2}\in\mathbb{R}$ are bi-Lipschitz homeomorphic to the snowflake $(\mathbb{R},|\cdot|^{1/2})$.

On the other hand, in the plane $z=ax+by+c$, the horizontal curves are exactly the lines which project to

[TABLE]

in the $xy$-plane, while the curves which project to the concentric circles

[TABLE]

are bi-Lipschitz homeomorphic to re-scaled copies of the snowflake $(\mathbb{S}^{1},|\cdot|^{1/2})$.

Given points $p=(x,y,z)$ and $q=(x^{\prime},y^{\prime},z^{\prime})$ in $\mathbb{H}$, define the Korányi metric

[TABLE]

It is well known that $d_{K}$ is a metric, and it is also bi-Lipschitz equivalent to $d_{cc}$ in the sense that

[TABLE]

for some constant $C\geq 1$; see [CDPT07]. More often, we consider the following distance $d_{\mathbb{H}}$ (which is not a true metric since there is a sub-additive constant present in the triangle inequality):

[TABLE]

Though $d_{\mathbb{H}}$ is not a metric, it is bi-Lipschitz equivalent to $d_{K}$ in the above sense. Thus, it suffices to construct bi-Lipschitz embeddings of sets $A\subset\mathbb{R}^{3}$ endowed with $d_{\mathbb{H}}$. This provides a bi-Lipschitz embedding of the set $A$ when considered as a subset of the metric space $(\mathbb{H},d_{K})$. The distance $d_{\mathbb{H}}$ is related to the Euclidean metric in the following way.

Lemma 2.2**.**

Suppose that $R>0$ and $w,w^{\prime}\in\mathbb{R}^{3}$ with $\max\{|w|,|w^{\prime}|\}\leq R$. Then

[TABLE]

Proof.

Note that, if $w=(x,y,z)$ and $w^{\prime}=(x^{\prime},y^{\prime},z^{\prime})$, then

[TABLE]

Therefore, by (2.2) and the triangle inequality, we have

[TABLE]

If $\max\{|x-x^{\prime}|,|y-y^{\prime}|\}\leq(2R)^{-1}|z-z^{\prime}|$, then by (2.2)

[TABLE]

If $\max\{|x-x^{\prime}|,|y-y^{\prime}|\}>(2R)^{-1}|z-z^{\prime}|$, then by (2.2)

[TABLE]

The proof completes by noting that $\min\{1,R^{-1},R^{1/2},R^{-1/2}\}=\min\{R^{-1},R^{1/2}\}$. ∎

The length of a horizontal curve between two points can be estimated by the following lemma.

Lemma 2.3**.**

Let $w:[t_{1},t_{2}]\to\mathbb{H}$ be a horizontal curve with $w(t)=(x(t),y(t),z(t))$. Then,

[TABLE]

Proof.

We have from the absolute continuity of $x$ and $y$ that

[TABLE]

Moreover,

[TABLE]

The proof completes by noting that $\|x^{\prime}(t)\|_{\infty}^{1/2}\|y^{\prime}(t)\|_{\infty}^{1/2}\lesssim\|x^{\prime}(t)\|_{\infty}+\|y^{\prime}(t)\|_{\infty}$. ∎

Given a $C^{1,1}$ $2$-manifold $M\subset\mathbb{R}^{3}$ and a point $p\in M$, we denote by $T_{p}M$ (resp. $H_{p}$) the tangent plane of $M$ (resp. the horizontal distribution) at $p$.

Definition 2.1**.**

Let $M$ be a $C^{1,1}$ $2$-manifold. A point $p\in M$ is characteristic if $T_{p}M=H_{p}$. A point $p\in M$ is regular if it is not characteristic.

In the example of vertical planes, all points are regular. In the example of the plane $z=0$, all points are regular except for the origin which is characteristic.

Theorem 2.4**.**

[Bal03, Theorem 1.2]** Let $\mathscr{C}(M)$ be the set of characteristic points of a $2$-manifold $M$ in $\mathbb{R}^{3}$.

1. (1)

If $M$ is $C^{1,1}$ then $\mathscr{C}$(M) has (Euclidean) Hausdorff dimension less than $2$. 2. (2)

If $M$ is $C^{2}$ then $\mathscr{C}$(M) has (Euclidean) Hausdorff dimension less or equal to $1$.

3. Lines, planes and PL complexes

In §3.1 we show the first part of Theorem 1.1 and in §3.2 the second part. In §3.3 we show Theorem 1.2.

3.1. Lines in the Heisenberg group

Here we show the first part of Theorem 1.1 which we restate in the following proposition.

Proposition 3.1**.**

There exists universal $L>1$ such that, for any straight line $\ell\subset\mathbb{R}^{3}$, the space $(\ell,d_{\mathbb{H}})$ $L$-bi-Lipschitz embeds into $\mathbb{R}^{3}$.

Proof.

Suppose that $\ell$ is given by the formula

[TABLE]

with $|a_{1}|^{2}+|a_{2}|^{2}+|a_{3}|^{2}=1$. Without loss of generality we may assume that $|a_{2}|\leq|a_{1}|$. Moreover, since $d_{\mathbb{H}}$ is invariant under vertical translations, we may assume that $c_{3}=0$. Consider the following cases.

Case 1. Suppose that $a_{1}=a_{2}=0$. Define the map $F:(\ell,d_{\mathbb{H}})\to\mathbb{R}^{3}$ given by $F(w(t))=\Phi(t)$. Since $|a_{3}|=1$, we have for any $\tau,\tau^{\prime}\in\mathbb{R}$,

[TABLE]

Thus $F$ is $L_{\Phi}$-bi-Lipschitz where $L_{\Phi}$ is the bi-Lipschitz constant of $\Phi$.

Case 2. Suppose that $2a_{3}+c_{2}a_{1}-c_{1}a_{2}=0$. That is, the line is a horizontal curve. Define $F:(\ell,d_{\mathbb{H}})\to\mathbb{R}$ by $F(w(t))=(|a_{1}|+|a_{2}|)t$. Let $\tau,\tau^{\prime}\in\mathbb{R}$. Then

[TABLE]

so $F$ is in fact an isometry.

Case 3. Suppose that none of the above cases applies. Set

[TABLE]

and define $F:(\ell,d_{\mathbb{H}})\to\mathbb{R}^{3}$ by

[TABLE]

where $[a]$ denotes the integer part of $a$.

Let $\tau,\tau^{\prime}\in\mathbb{R}$, $w=w(\tau)$ and $w^{\prime}=w(\tau^{\prime})$. Without loss of generality, assume that $\tau^{\prime}>\tau$. We have that

[TABLE]

We now divide Case 3 into three sub-cases.

Case 3.1. Suppose that there exist at least two integers contained in the interval $(\mu\tau,\mu\tau^{\prime})$. Let

[TABLE]

Then, $|\tau-\tau^{\prime}|\geq\mu^{-1}$ which yields $\lambda\leq\kappa|\tau-\tau^{\prime}|^{1/2}$, and thus

[TABLE]

On the other hand, since $|\mu\tau-\mu\tau^{\prime}|>1$,

[TABLE]

Notice that the third component of $\Phi$ lies between 0 and 1. Thus we have

[TABLE]

since $3(n-m)\geq(n-m)+2>|\mu\tau-\mu\tau^{\prime}|$. Hence, in this case, $|F(w)-F(w^{\prime})|\simeq\kappa|\tau-\tau^{\prime}|\simeq d_{\mathbb{H}}(w,w^{\prime})$.

Case 3.2. Suppose that there exists no integer in $(\mu\tau,\mu\tau^{\prime})$. Then, $|\tau-\tau^{\prime}|<\mu^{-1}$ which yields

[TABLE]

Moreover, $[\mu\tau]=[\mu\tau^{\prime}]$, so

[TABLE]

Case 3.3. Suppose that there exists exactly one integer $n$ in $(\mu\tau,\mu\tau^{\prime})$. Then, $|\tau^{\prime}-\tau|<2\lambda^{2}/\kappa^{2}$ which yields

[TABLE]

Set $\tau^{\prime\prime}=n/\mu\in(\tau,\tau^{\prime})$ and $w^{\prime\prime}=w(\tau^{\prime\prime})$. Before estimating $|F(w)-F(w^{\prime})|$, we make two observations. Firstly, by (3) of the definition of $\Phi$, the assumptions of this case, and the choice of $\tau^{\prime\prime}$, it is easy to see that

[TABLE]

Also, $t\mapsto\Phi(\mu t-[\mu t])$ is periodic with period $1/\mu$; in particular,

[TABLE]

Combining these two observations, we have

[TABLE]

3.2. Planes in the Heisenberg group

We now show the second part of Theorem 1.1 which we restate in the following proposition.

Proposition 3.2**.**

There exists a universal $L>1$ such that, for any plane $P$ in $\mathbb{R}^{3}$, there exists an $L$-bi-Lipschitz embedding of $(P,d_{\mathbb{H}})$ into $\mathbb{R}^{4}$.

For the proof of Proposition 3.2 we distinguish two cases.

3.2.1. Planes parallel to $z$-axis

Suppose that

[TABLE]

for some $b,c\in\mathbb{R}$. Without loss of generality we may assume that $|b|<1$. Otherwise we consider the plane $P^{\prime}=\{(x,y,z)\in\mathbb{R}^{3}:x=b^{-1}y+b^{-1}c\}$ and the proof is similar. Define the map $f:P\to\mathbb{R}^{4}$ with

[TABLE]

Then, if $w=(x,bx+c,z)$ and $w^{\prime}=(x^{\prime},bx^{\prime}+c,z^{\prime})$ we have

[TABLE]

Similarly we obtain

[TABLE]

3.2.2. Planes not parallel to $z$-axis

Suppose that

[TABLE]

for some $a,b,c\in\mathbb{R}$.

Define first $P_{0}=\{(x,y,z)\in\mathbb{R}^{3}:z=0\}$ and $g:P_{0}\to P$ with

[TABLE]

Note that $g$ is an isometry with respect to $d_{\mathbb{H}}$. Indeed, if $w=(x,y,0)$ and $w^{\prime}=(x^{\prime},y^{\prime},0)$ then

[TABLE]

Therefore, it suffices to prove Proposition 3.2 for the plane $P_{0}$.

Given a point $w\in\mathbb{R}^{2}$ there exist $t\geq 0$ and $s\in\mathbb{S}^{1}$ such that $w=ts$; if $w\neq(0,0)$, then $t>0$ and $s$ are unique. Define now the embedding $F:P_{0}\to\mathbb{R}^{4}$ by $F(ts,0)=(t\,\phi(s),t)$ where $\phi$ is defined as in (2.1). The following two simple lemmas imply that $F$ is bi-Lipschitz.

Lemma 3.3**.**

For each $i\in\{1,2\}$ let $s_{i}\in\mathbb{S}^{1}$, $t_{i}\geq 0$ and $w_{i}=(s_{i}t_{i},0)\in\mathbb{R}^{3}$. Then,

[TABLE]

Proof.

Without loss of generality assume that $t_{1}\leq t_{2}$. For $i=1,2$ write $s_{i}=(\cos(\theta_{i}),\sin(\theta_{i}))$ for some $\theta_{i}\in[0,2\pi)$. Then,

[TABLE]

If $t_{2}\geq 2t_{1}$, then $t_{1}\leq(t_{2}-t_{1})$, so $(t_{1}t_{2})^{1/2}\lesssim(t_{2}-t_{1})$ and the claim holds.

If $t_{2}<2t_{1}$ and $(\theta_{1}-\theta_{2})>\pi/2$, then $|s_{1}-s_{2}|\simeq 1$, $t_{1}\simeq t_{2}$ and the claim holds.

If $t_{2}<2t_{1}$ and $(\theta_{1}-\theta_{2})\leq\pi/2$, then $|s_{1}-s_{2}|\simeq\sin(\theta_{1}-\theta_{2})$, $t_{1}\simeq t_{2}$, and $|s_{1}-s_{2}|\lesssim|s_{1}-s_{2}|^{1/2}$, and so the claim holds. ∎

Lemma 3.4**.**

If $t_{1},t_{2}\geq 0$ and $s_{1},s_{2}\in\mathbb{S}^{1}$ then

[TABLE]

Proof.

Indeed,

[TABLE]

3.3. Simplicial complexes

Recall that a [math]-simplex is a point of $\mathbb{R}^{3}$, a $1$-simplex $E\subset\mathbb{R}^{3}$ is the convex hull of two distinct points and a $2$-simplex $E\subset\mathbb{R}^{3}$ is the convex hull of three distinct points in $\mathbb{R}^{3}$ that do not lie on the same line. The boundary of a $k$-simplex consists of $j$-faces ($j\leq k-1$) which are $j$-simplices.

A (compact) simplicial $2$-complex $K\subset\mathbb{R}^{3}$ is a finite union of simplices $S_{1}\cup\dots\cup S_{k}$ with the following properties

1. (1)

each $S_{i}$ is a $k_{i}$-simplex for some $k_{1}\in\{0,1,2\}$; 2. (2)

for each $i,j\in\{1,\dots,n\}$, either $S_{i}\cap S_{j}$ is empty, or $S_{i}\cap S_{j}$ is a common edge of $S_{i}$ and $S_{j}$, or $S_{i}=S_{j}$.

Theorem 3.5**.**

Let $K\subset\mathbb{R}^{3}$ be a finite union $K=D_{1}\cup\cdots\cup D_{k}$ where each $D_{i}$ is a simplicial $2$-complex. Then, there exists $N\in\mathbb{N}$ and a bi-Lipschitz embedding of $(K,d_{\mathbb{H}})$ into $\mathbb{R}^{N}$.

For the proof of Theorem 3.5 we use the following bi-Lipschitz welding theorem.

Lemma 3.6** ([MM16, Theorem 1.1]).**

Let $X$ be a metric space and let $X_{1},X_{2}\subset X$ be closed subsets such that $X=X_{1}\cup X_{2}$. If $X_{1}$ bi-Lipschitz embeds in $\mathbb{R}^{n}$ and $X_{2}$ bi-Lipschitz embeds in $\mathbb{R}^{m}$ then $X$ bi-Lipschitz embeds in $\mathbb{R}^{n+m+1}$.

We now show Theorem 3.5.

Proof of Theorem 3.5.

By Lemma 3.6 we only need to show that a simplicial $2$-complex admits a bi-Lipschitz embedding into some $\mathbb{R}^{N}$. Let $K\subset\mathbb{R}^{3}$ be a simplicial $2$-complex. Then, $K=\bigcup_{i=1}^{n}S_{i}$ where each $S_{i}$ is a $k_{i}$-simplex for some $k_{i}\in\{0,1,2\}$. By Proposition 3.2, each $(S_{i},d_{\mathbb{H}})$ $L$-bi-Lipschitz embeds into $\mathbb{R}^{4}$ for some universal $L>1$. Applying Lemma 3.6 $n$ times we obtain a bi-Lipschitz embedding of $(K,d_{\mathbb{H}})$ into $\mathbb{R}^{5n-1}$. ∎

4. Smooth curves and $\mathbb{H}$-regular surfaces

In §4.1 we show the first part of Theorem 1.3, and in §4.2 we prove the second part. For a set $A\subset\mathbb{R}^{3}$, we will say that $(A,d_{\mathbb{H}})$ locally $L$-bi-Lipschitz embeds in $\mathbb{R}^{N}$ for some $L>1$ if, for any $p\in A$, there is a neighborhood $U$ of $p$ such that $(U\cap A,d_{\mathbb{H}})$ $L$-bi-Lipschitz embeds in $\mathbb{R}^{N}$.

4.1. Smooth curves

For the rest of §4.1, we denote by $I$ a non-degenerate interval of $\mathbb{R}$. We prove the following proposition.

Proposition 4.1**.**

There is a universal constant $L>1$ such that, for any $C^{1,1}$ curve $w:I\to\mathbb{R}^{3}$ with unit speed, $(w(I),d_{\mathbb{H}})$ locally $L$-bi-Lipschitz embeds into $\mathbb{R}^{4}$.

The idea of the proof is to decompose the curve $\gamma=w(I)$ into two types of sub-arcs and embed each one in $\mathbb{R}^{4}$. In the following lemma we consider sub-arcs of $\gamma$ that are “vertical” enough.

Lemma 4.2**.**

Let $w:[0,T]\to\mathbb{R}^{3}$ be a $C^{1,1}$ smooth curve with unit speed such that $|2z^{\prime}(t)+x^{\prime}(t)y(t)-y^{\prime}(t)x(t)|\geq 1$ for all $t\in[0,T]$. Then, $(w([0,T]),d_{\mathbb{H}})$ locally $L$-bi-Lipschitz embeds into $\mathbb{R}^{3}$ for some universal $L>1$.

Proof.

Let $M>0$ be such that, for all $t\in[0,T]$,

[TABLE]

By compactness of $[0,T]$ and uniform continuity of $w^{\prime}(t)$, there exists $\epsilon\in(0,1)$ such that if $t_{0},\dots,t_{5}\in[0,T]$ are such that $|t_{i}-t_{j}|\leq\epsilon$, then

[TABLE]

We claim that, for any $t,t^{\prime}\in[0,T]$ with $|t-t^{\prime}|<\epsilon$, there is an $L$-bi-Lipschitz map between $(w([t,t^{\prime}]),d_{\mathbb{H}})$ and $([t,t^{\prime}],\lambda|\cdot|^{1/2})$ for some constant $\lambda\geq 1$ and a universal constant $L>1$. Assuming the claim, we see that $(w([0,T]),d_{\mathbb{H}})$ locally $L$-bi-Lipschitz embeds into $\mathbb{R}^{3}$ for some universal $L$.

To prove the claim, fix $\tau_{1},\tau_{2}\in[0,T]$ such that $0<\tau_{2}-\tau_{1}\leq\epsilon$. Fix also $t_{0}\in[\tau_{1},\tau_{2}]$ and set

[TABLE]

By the Mean Value Theorem, (4.1), and the fact that $\epsilon<1\leq\lambda$, if $t_{1},t_{2}\in[\tau_{1},\tau_{2}]$, there exist $c_{1},\dots,c_{5}\in(t_{1},t_{2})$ such that

[TABLE]

Hence, $(w([\tau_{1},\tau_{2}]),d_{\mathbb{H}})$ is $L$-bi-Lipschitz equivalent to $([\tau_{1},\tau_{2}],\lambda|\cdot|^{1/2})$ and the claim follows. ∎

Proof of Proposition 4.1.

Fix $t_{1},t_{2}\in I$ with $t_{1}<t_{2}$. We show that the space $(w([t_{1},t_{2}]),d_{\mathbb{H}})$ locally bi-Lipschitz embeds in $\mathbb{R}^{4}$.

Let $M>1$ be such that

[TABLE]

Let $\epsilon_{0}=(4M)^{-2}$. Note that if $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}\leq\epsilon_{0}$, then, since $w$ has unit speed in $\mathbb{R}^{3}$, we have

[TABLE]

By uniform continuity of $w^{\prime}|_{[t_{1},t_{2}]}$ and compactness of $[t_{1},t_{2}]$, there exist finitely many closed intervals $I_{1},\dots,I_{k}\subset[t_{1},t_{2}]$ such that

1. (1)

$\bigcup_{j=1}^{k}I_{j}=[t_{1},t_{2}]$, 2. (2)

for each $j\in\{1,\dots,k\}$, either $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}\leq\epsilon_{0}$ for all $t\in I_{j}$ or $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}\geq\epsilon_{0}/2$ for all $t\in I_{j}$.

If $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}\leq\epsilon_{0}$ for all $t\in I_{j}$, then (4.2) and Lemma 4.2 imply that $(w(I_{j}),d_{\mathbb{H}})$ locally bi-Lipschitz embeds in $\mathbb{R}^{3}$.

If $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}\geq\epsilon_{0}/2$ for all $t\in I_{j}$, then the projection of $w(I_{j})$ on $\mathbb{R}^{2}\times\{0\}$ is a $C^{1,1}$ smooth curve which can be re-parameterized so that $|x^{\prime}(t)|^{2}+|y^{\prime}(t)|^{2}=1$. The surface $(w(I_{j})\times\mathbb{R},d_{\mathbb{H}})$ is $C^{1,1}$ and consists only of $\mathbb{H}$-regular points. Hence, we will see in Proposition 4.3 that $(w(I_{j})\times\mathbb{R},d_{\mathbb{H}})$ locally bi-Lipschitz embeds in $\mathbb{R}^{4}$ with a universal constant. In particular, this means that $(w(I_{j}),d_{\mathbb{H}})$ locally bi-Lipschitz embeds in $\mathbb{R}^{4}$. ∎

4.2. $\mathbb{H}$-regular surfaces

Here we show the second part of Theorem 1.3 which we restate in the following proposition.

Proposition 4.3**.**

There exists a universal constant $L>1$ satisfying the following. Let $M\subset\mathbb{R}^{3}$ be a $C^{1,1}$ $2$-manifold in $\mathbb{R}^{3}$ and let $p_{0}$ be an $\mathbb{H}$-regular point of $M$. Then there exist a neighborhood $U\subset\mathbb{R}^{3}$ of $p_{0}$ and an $L$-bi-Lipschitz embedding of $(U\cap M,d_{\mathbb{H}})$ in $\mathbb{R}^{4}$.

The proof of this proposition is comprised by the following two lemmas. In Lemma 4.4, we show that $M$ can be foliated near $p_{0}$ by horizontal curves with $C^{1}$ dependence on parameters. In Lemma 4.5, we construct a local embedding of any surface which possesses such a foliation. In fact, Lemma 4.5 will be used in several other arguments throughout the rest of this paper when a foliation by horizontal curves can be constructed.

Lemma 4.4**.**

There exists a universal constant $L^{\prime}>1$ satisfying the following. Let $M$ be a $C^{1,1}$ $2$-manifold in $\mathbb{R}^{3}$ and let $p_{0}$ be an $\mathbb{H}$-regular point of $M$. Then there exist $\epsilon>0$ and an $L^{\prime}$-bi-Lipschitz map $G:[-\epsilon,\epsilon]^{2}\to(M,|\cdot|)$ such that

1. (1)

$G(0,0)=p_{0}$** 2. (2)

for all $v\in[-\epsilon,\epsilon]$, the curve $u\mapsto G(u,v)$ is a horizontal curve.

Proof.

In a neighborhood of $p_{0}=(x_{0},y_{0},z_{0})$, the manifold $M$ can be written as the level set $H(x,y,z)=0$ for some $C^{1,1}$ function $H$ with $|\nabla H(x_{0},y_{0},z_{0})|>0$. Without loss of generality, we assume that

[TABLE]

The other two cases are similar and are left to the reader.

Then, in a sufficiently small neighborhood $U$ of $p_{0}$ we can write

[TABLE]

for some open $V\subset\mathbb{R}^{2}$ where $F$ is a $C^{1,1}$ function with $F(x_{0},y_{0})=z_{0}$ and

[TABLE]

Since $p_{0}$ is $\mathbb{H}$-regular, either $x_{0}-2F_{y}(x_{0},y_{0})\neq 0$ or $y_{0}+2F_{x}(x_{0},y_{0})\neq 0$. Without loss of generality, we assume that $x_{0}-2F_{y}(x_{0},y_{0})\neq 0$ and that

[TABLE]

There exists a neighborhood $V_{1}\subset V$ of $(x_{0},y_{0})$ such that $x-2F_{y}\neq 0$ in $V_{1}$. Thus the horizontal curves in a neighborhood of $p_{0}$ in $M$ are given by the ODE

[TABLE]

By standard existence and uniqueness theorems (e.g. [CL55, Theorem 1.2] and [CL55, Theorem 2.2]), there is some $\epsilon_{1}>0$ such that, for each $v\in[-\epsilon_{1},\epsilon_{1}]$, there exists a unique solution $y=g_{v}(x)$ of (4.4) defined on $[x_{0}-\epsilon_{1},x_{0}+\epsilon_{1}]$ such that $g_{v}(x_{0})=y_{0}+v$. Moreover, the function

[TABLE]

is $C^{1}$ [CL55, Theorem 7.1]. By existence and uniqueness of solutions, there is a neighborhood of $(x_{0},y_{0})$ in which the graphs of $g_{v}$ do not intersect, and every point in this neighborhood is contained in the graph of some $g_{v}$. Moreover, the derivative of $f$ at $(0,0)$ satisfies

[TABLE]

Thus, for some $\epsilon_{2}>0$, $G$ is a $C^{1}$ diffeomorphism in $[-\epsilon_{2},\epsilon_{2}]^{2}$, and, by (4.3), there exists a universal constant $L_{1}>1$ such that

[TABLE]

for all $(u,v),(u^{\prime},v^{\prime})\in[-\epsilon_{2},\epsilon_{2}]^{2}$.

Define now $G:[-\epsilon_{2},\epsilon_{2}]^{2}\to M$ by $G(u,v)=(f(u,v),F(f(u,v)))$ which has been defined so that the curves $u\mapsto G(u,v)$ are horizontal curves contained in $M$ for any $v\in[-\epsilon_{2},\epsilon_{2}]$. Moreover

[TABLE]

and hence $G$ is $L^{\prime}$-bi-Lipschitz for some universal $L^{\prime}>1$. ∎

Lemma 4.5**.**

There exists a universal constant $L>1$ satisfying the following. Fix $M\subset\mathbb{R}^{3}$. Suppose there exist $\epsilon>0$, $L^{\prime}>1$, and an $L^{\prime}$-bi-Lipschitz map $G:[-\epsilon,\epsilon]^{2}\to M$ such that

1. (1)

$G(0,0)=p_{0}$** 2. (2)

for all $v\in[-\epsilon,\epsilon]$, the curve $u\mapsto G(u,v)$ is a horizontal curve.

Write $G=(g_{1},g_{2},g_{3})$, and set

[TABLE]

Define the map

[TABLE]

Then there is a neighborhood $U\subset\mathbb{R}^{3}$ of $p_{0}$ such that $\Psi:(U\cap M,d_{\mathbb{H}})\to\mathbb{R}^{4}$ is $L$-bi-Lipschitz.

Note that, while $L$ may be found independent of the bi-Lipschitz constant $L^{\prime}$, the size of the neighborhood $U$ certainly depends on the value of $L^{\prime}$.

Proof.

We first note that $\lambda>0$ since the curve $\gamma_{1}(u)=G(u,0)$ is horizontal. Moreover, since $G$ is $L^{\prime}$-bi-Lipschitz, $\lambda$ is bounded from above by a constant $C$ depending only on $L^{\prime}$. On the other hand, the curve $\gamma_{2}(v)=G(0,v)$ intersects $\gamma_{1}$ transversely at $p_{0}$ and hence cannot be horizontal. Thus, $\kappa>0$. Fix $(u_{1},v_{1}),(u_{2},v_{2})\in[-\epsilon,\epsilon]^{2}$ and set

[TABLE]

That is, $p_{2}$ and $p_{3}$ lie along the same horizontal curve $u\mapsto G(u,v_{2})$.

Choosing $\epsilon$ possibly smaller, we have for some $c_{1},c_{2},c_{3}\in[-\epsilon,\epsilon]$

[TABLE]

Additionally, by Lemma 2.3 we have

[TABLE]

assuming that $\epsilon$ is sufficiently small. By the triangle inequality,

[TABLE]

We now show that $d_{\mathbb{H}}(p_{1},p_{2})\gtrsim|\Psi(p_{1})-\Psi(p_{2})|$. By (4.6), there exists a universal constant $C_{0}$ such that

[TABLE]

Consider the following two cases

Case 1. Suppose that $C_{0}\lambda|u_{1}-u_{2}|\geq(\kappa/16)^{1/2}|v_{1}-v_{2}|^{1/2}$.

[TABLE]

Case 2. Suppose that $C_{0}\lambda|u_{1}-u_{2}|<(\kappa/16)^{1/2}|v_{1}-v_{2}|^{1/2}$. Then,

[TABLE]

Now,

[TABLE]

Again assuming that $\epsilon$ is sufficiently small,

[TABLE]

Thus, by (4.7)

[TABLE]

Therefore, $\Psi$ is bi-Lipschitz in a neighborhood of $p_{0}$. ∎

4.3. The smooth torus

Although the set of characteristic points of a given smooth manifold is very small, it rarely is empty. By the Hairy-Ball Theorem and the classification of $2$-manifolds, if a compact differentiable manifold $M$ has no characteristic points, then it must be homeomorphic to the torus $\mathbb{S}^{1}\times\mathbb{S}^{1}$. On the other hand, a topological torus could be $\mathbb{H}$-regular. Given numbers $0<r<R$, define the torus

[TABLE]

We claim that $\mathbb{T}_{r,R}$ has no characteristic points. Recall first that both the horizontal distribution and the tangent space of $\mathbb{T}_{r,R}$ are invariant under rotations with respect to the $z$-axis in the following sense. If $P_{w}$ (resp. $\hat{P}_{w}$) is the affine space containing $w\in\mathbb{T}_{r,R}$ and generated by $T_{w}\mathbb{T}_{r,R}$ (resp. generated by $H_{w}$) and if $\zeta:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a rotation with respect to $z$-axis, then $\zeta(P_{w})=P_{\zeta(w)}$ and $\zeta(\hat{P}_{w})=\hat{P}_{\zeta(w)}$. To show the claim, suppose that a point $w\in\mathbb{T}_{r,R}$ is characteristic and write $w=(r_{0},\theta_{0},z_{0})$ in cylindrical coordinates. By the rotation invariance of the tangent space and the horizontal distribution, every point of the circle $S=\{(r_{0},\theta,z_{0}):\theta\in[0,2\pi)\}$ is characteristic. Therefore, $S$ is horizontal. However, by §3.2, $(S,d_{\mathbb{H}})$ is bi-Lipschitz equivalent to $(\mathbb{S}^{1},|\cdot|^{1/2})$, which is a contradiction.

Let $V$ be a smooth unitary vector field on $\mathbb{T}_{r,R}$ such that $V(w)\in T_{w}\mathbb{T}_{r,R}\cap H_{w}$ for all $w\in\mathbb{T}_{r,R}$. By the Frobenius Theorem, $V$ is integrable and $\mathbb{T}_{r,R}$ is foliated by the (closed) integral curves of $V$ which are obtained by rotating one of them with respect to the $z$-axis. Specifically, there exists a diffeomorphism $G:\mathbb{S}^{1}\times\mathbb{S}^{1}\to\mathbb{T}_{r,R}$ such that $s\mapsto G(s,\eta)$ is a horizontal curve for each $\eta\in\mathbb{S}^{1}$.

We now claim that $(\mathbb{T}_{r,R},d_{\mathbb{H}})$ bi-Lipschitz embeds in $\mathbb{R}^{4}$. Indeed, say $\tau$ is an identification of $\mathbb{R}^{2}/\mathbb{Z}^{2}$ with $\mathbb{S}^{1}\times\mathbb{S}^{1}$ and define $\Psi:\mathbb{T}_{r,R}\to\mathbb{R}^{4}$ so that

[TABLE]

According to Lemma 4.5, there is a constant $L>1$ such that, for any $p\in\mathbb{T}_{r,R}$, there exists a neighborhood $U_{p}$ of $p$ in $\mathbb{T}_{r,R}$ wherein $\Psi|_{(U_{p},d_{\mathbb{H}})}$ is $L$-bi-Lipschitz. By the compactness of $\mathbb{T}_{r,R}$, there exists $\epsilon>0$ such that $\Psi$ is bi-Lipschitz on all compact sets of $\mathbb{T}_{r,R}$ with (Euclidean) diameter at most $\epsilon$. On the other hand, by Lemma 2.2, for all $w,w^{\prime}\in\mathbb{T}_{r,R}$ with $|w-w^{\prime}|\geq\epsilon$ we have

[TABLE]

Therefore, $\Psi$ is bi-Lipschitz.

5. Some smooth surfaces with characteristic points

Here we examine the bi-Lipschitz embeddability of some special classes of smooth surfaces that contain characteristic points.

5.1. Surfaces obtained by revolution about the $z$-axis

Let $F:[0,\infty)\to\mathbb{R}$ be a $C^{1,1}$ function. Let $\Sigma$ be the surface of revolution generated by $F$:

[TABLE]

Below we denote by 0 the origins $(0,0)$ and $(0,0,0)$, depending on the context.

Theorem 5.1**.**

Suppose that $F$ is $C^{1,1}$ with $F^{\prime}(0)=0$, and set

[TABLE]

*There exist $\epsilon>0$, $L=L(M)>1$, and a neighborhood $U\subset\Sigma$ of $(0,0,F(0))$ such that $(U,d_{\mathbb{H}})$ $L$-bi-Lipschitz embeds in $\mathbb{R}^{4}$. *

Before proving this theorem, we observe an application of this embedding. Combined with Lemma 3.6 and the $\mathbb{H}$-regularity of $\Sigma\setminus\{(0,0,F(0))\}$, Theorem 5.1 yields the following corollary.

Corollary 5.2**.**

Suppose that $F$ is $C^{1,1}$ with $F^{\prime}(0)=0$. Any compact subset of $(\Sigma,d_{\mathbb{H}})$ bi-Lipschitz embeds in some Euclidean space $\mathbb{R}^{n}$.

Proof.

We first show that all points of $\Sigma\setminus\{(0,0,F(0))\}$ are regular points. Indeed, suppose that $(x,y)\neq(0,0)$ is a characteristic point of $\Sigma$. Then,

[TABLE]

If $x=0$ or $y=0$, then both must be equal to 0, and this is a contradiction. However, if $x\neq 0$ and $y\neq 0$, then $y/x=x/y$ which implies that $x^{2}+y^{2}=0$, and this is again a contradiction. Thus the only characteristic point of $\Sigma$ is $(0,0,F(0))$.

Let $S$ be a compact subset of $\Sigma$. By Theorem 5.1, Proposition 4.3, and the compactness of $S$, there exist open sets $D_{1},\dots,D_{k}$ such that each $D_{i}$ bi-Lipschitz embeds into $\mathbb{R}^{4}$ and $S\subset\bigcup_{i=1}^{k}D_{i}$. By Lemma 3.6, we obtain a bi-Lipschitz embedding of $\bigcup_{i=1}^{k}D_{i}$ (and thus of $S$) into $\mathbb{R}^{5k-1}$. ∎

We now prove Theorem 5.1. The idea of the proof is as follows. In Lemma 5.3, we construct a bi-Lipschitz homeomorpism of a disc centered at the origin in $\mathbb{R}^{2}$ such that any radial segment in the disc is mapped to a curve which is the projection of a horizontal curve in $\Sigma$. Since $\Sigma$ is a surface of revolution, rotations of this horizontal curve foliate the surface. We then complete the proof of Theorem 5.1 by re-parameterizing $\Sigma$ in terms of the horizontal curve (which embeds into $\mathbb{R}$) and the snowflaked unit circle (which embeds into $\mathbb{R}^{3}$).

Lemma 5.3**.**

Let $F:[0,+\infty)\to\mathbb{R}$ be as in Theorem 5.1. There exist $T>0$, $L=L(M)>1$, and an $L$-bi-Lipschitz map $G:B^{2}(0,T)\to B^{2}(0,T)$ with $G(0)=0$, such that for each $s\in\mathbb{S}^{1}$, the curve

[TABLE]

is horizontal. Moreover, $|G(st)|=|G(s^{\prime}t)|$ for any $s,s^{\prime}\in\mathbb{S}^{1}$ and $t\in[0,T)$.

Proof.

For each $t\geq 0$ consider the point $w_{t}=(t,0,F(t))$. Then

[TABLE]

Consider now the vector field $V:\mathbb{R}^{2}\setminus\{0\}\to\mathbb{R}^{2}$ given by

[TABLE]

where $t>0$, $s,\hat{s}\in\mathbb{S}^{1}$ and $\hat{s}$ is the rotation of $s$ by an angle of $\pi/2$. Note that $V$ has been defined in such a way that, when any integral curve $\gamma$ of $V$ is lifted to a curve on $\Sigma$, this lifted curve is horizontal. That is, the curve $t\mapsto(\gamma(t),F(\gamma(t)))$ is a horizontal curve.

Fix $T>0$ such that $|F^{\prime}(t)|/t\leq 2M$ for every $t\in(0,T)$. That is,

[TABLE]

Then, for all $p\in B^{2}(\textbf{0},T)\setminus\{\textbf{0}\}$ we have

[TABLE]

From (5.1) we have that if $\gamma$ is an integral curve of $V$, then

[TABLE]

which implies that $(|\gamma|)^{\prime}=1/2$. Therefore, given $0<t_{1}\leq t_{2}$ in the domain of $\gamma$,

[TABLE]

We construct a particular integral curve in $B(\textbf{0},T)$ as follows. For each $i\in\mathbb{N}$ let $S_{i}$ be the circle $\partial B^{2}(\textbf{0},2^{1-i}T)$, and let $A_{i}$ be the closed annulus $A(\textbf{0};2^{-i}T,2^{1-i}T)$. By (5.2), every integral curve of $V|_{A_{i}}$ is a $C^{1}$ curve that joins $S_{i}$ with $S_{i+1}$ defined on an interval of length $2^{1-i}T$.

For each $i\in\mathbb{N}$, let $\tilde{\gamma}^{(i)}$ be the integral curve of $V|_{A_{i}}$ joining $S_{i+1}$ to the point $(2^{1-i}T,0)$ defined on the interval $[2^{1-i}T,2^{2-i}T]$. The vector field $V$ is defined in such a way that rotating $\tilde{\gamma}^{(i)}$ produces all integral curves of $V|_{A_{i}}$. Appropriately rotating each $\tilde{\gamma}^{(i)}$, we obtain a curve $\gamma_{0}$ that joins 0 with the point $(T,0)$ such that $\tilde{\gamma}|_{A_{i}}$ is an integral curve of $V|_{A_{i}}$. Note that, by a limiting argument, (5.2) also holds if one or both of the $t_{i}$ is zero. Hence the domain of $\gamma_{0}$ is $[0,2T]$.

For each $s\in\mathbb{S}^{1}$, let $\gamma_{s}(t):[0,2T]\to\mathbb{R}^{2}$ be the curve obtained by rotating $\gamma_{0}$ in such a way that $\gamma_{s}(2T)=Ts\in\partial B^{2}(\mathbf{0},T)$. By the rotation invariance of $V$, the integral curves of $V$ are exactly the curves $\{\gamma_{s}(t):s\in\mathbb{S}^{1}\}$ which join 0 with $\partial B^{2}(\textbf{0},T)$. As discussed above, the lift of each $\gamma_{s}$ to $\Sigma$ is a horizontal curve since $\gamma_{s}^{\prime}(t)=V(\gamma_{s}(t))$.

Define now $G:B^{2}(0,T)\to B^{2}(0,T)$ with $G(ts)=\gamma_{s}(2t)$. To show that $G$ is bi-Lipschitz, fix $s_{1},s_{2}\in\mathbb{S}^{1}$ and $0\leq t_{1}\leq t_{2}\leq T$. By the triangle inequality, the Mean Value Theorem, and (5.2), for some $c\in[2t_{1},2t_{2}]$,

[TABLE]

For the other inequality we consider two cases.

Case 1. Suppose that $10M|t_{1}-t_{2}|\geq t_{1}|s_{1}-s_{2}|$. By (5.2),

[TABLE]

Case 2. Suppose that $10M|t_{1}-t_{2}|<t_{1}|s_{1}-s_{2}|$. By (5.2) and the Mean Value Theorem, for some $c\in[2t_{1},2t_{2}]$,

[TABLE]

We can now prove Theorem 5.1.

Proof of Theorem 5.1.

For a point $w\in\mathbb{R}^{3}$, we denote by $\pi(w)$ the projection of $w$ on $\{z=0\}$. Since $d_{\mathbb{H}}$ is invariant under vertical translations, we may assume that $F(0)=0$. Define

[TABLE]

and choose $\epsilon>0$ such that $B^{3}(\textbf{0},\epsilon)\cap\Sigma\subset\Sigma_{0}$. Define now the homeomorphism

[TABLE]

where $s\in\mathbb{S}^{1}$ and $t\in[0,T]$. We show that, if $T$ is chosen small enough, then the map $\Psi$ is $L$-bi-Lipschitz for some $L=L(M)>1$.

Fix $s_{1},s_{2}\in\mathbb{S}^{1}$, $0<t_{1}\leq t_{2}\leq T$ and points $w_{1}=(s_{1}t_{1},0)$ and $w_{2}=(s_{2}t_{2},0)$. By Lemma 3.3 it suffices to show that

[TABLE]

For the rest of the proof denote $w_{1}^{\prime}=\Psi(w_{1})$ and $w_{2}^{\prime}=\Psi(w_{2})$. We write $G(st)=(x(st),y(st))$. Recall that $G(st)=G(s^{\prime}t)$ lie on the same circle centered at the origin. Therefore, since $d_{\mathbb{H}}$ and $G(st)$ are both invariant under rotations, we may assume that $x(s_{1}t_{1})=x(s_{2}t_{1})$. Then

[TABLE]

Since the points $(G(s_{2}t_{1}),F(|G(s_{2}t_{1})|))$ and $(G(s_{2}t_{2}),F(|G(s_{2}t_{2})|))$ lie on the horizontal curve $t\mapsto(G(s_{2}t),F(|G(s_{2}t)|))$, Lemma 2.3 gives a constant $C_{1}=C_{1}(M)>0$ such that

[TABLE]

Set $\epsilon=\min\{2^{-3/2},(2C_{1})^{-1},(12L^{2})^{-1}\}$ where $L$ is the bi-Lipschitz constant of $G$. (Note that $\epsilon$ depends only on $M$.) Also, we may choose $\delta>0$ sufficiently small so that $|x(s_{2}t_{1})|>|y(s_{2}t_{1})|$ whenever $|s_{1}-s_{2}|<\delta$. Indeed, we assumed earlier that $x(s_{1}t_{1})=x(s_{2}t_{1})$ so that the points $G(s_{1}t_{1})$ and $G(s_{2}t_{1})$ are reflections of one another across the $x$-axis. Since $G$ is $L(M)$-bi-Lipschitz, such a $\delta$ can be chosen depending only on $M$. Therefore, $|x(s_{2}t_{1})|\geq\tfrac{1}{2}|G(s_{2}t_{1})|$ when $|s_{1}-s_{2}|<\delta$. We consider the following cases.

Case 1: $|s_{1}-s_{2}|\geq\delta$. By Lemma 3.3, $d_{\mathbb{H}}(w_{1},w_{2})\simeq_{\delta}t_{2}$. On the other hand,

[TABLE]

Moreover, there exist $0<\xi_{1}<|G(s_{1}t_{1})|$ and $0<\xi_{2}<|G(s_{2}t_{2})|$ such that

[TABLE]

Assuming now that $T$ is sufficiently small we have

[TABLE]

Therefore, $d_{\mathbb{H}}(w_{1}^{\prime},w_{2}^{\prime})\simeq_{M,\delta}t_{2}\simeq_{M,\delta}|w_{1}-w_{2}|$.

Case 2: $|s_{1}-s_{2}|<\delta$ and $|t_{1}-t_{2}|\geq\epsilon t_{1}^{1/2}t_{2}^{1/2}|s_{1}-s_{2}|^{1/2}$. Note that

[TABLE]

Hence by (5.3), (5.4), and Lemma 3.3

[TABLE]

Moreover,

[TABLE]

Case 3: $|s_{1}-s_{2}|<\delta$ and $|t_{1}-t_{2}|<\epsilon t_{1}^{1/2}t_{2}^{1/2}|s_{1}-s_{2}|^{1/2}$. By (5.3), (5.4), and the choice of $\epsilon$,

[TABLE]

while

[TABLE]

For the other direction, note that, $|s_{1}-s_{2}|<\delta$ gives $|x(s_{2}t_{1})|\geq\tfrac{1}{2}|G(s_{2}t_{1})|$. Thus,

[TABLE]

since $\epsilon\leq 1/(12L^{2})$. Therefore,

[TABLE]

and the proof is complete. ∎

5.2. Korányi spheres

Following the ideas in §5.1, we show directly that the Korányi sphere

[TABLE]

admits a bi-Lipschitz embedding into $\mathbb{R}^{4}$. (Note that this is the sphere in the metric $d_{K}$ rather than $d_{\mathbb{H}}$.) Since the metric is left invariant on $\mathbb{H}$ and commutes with dilations, this shows that any Korányi sphere admits a bi-Lipschitz embedding into $\mathbb{R}^{4}$.

Proposition 5.4**.**

There exists a bi-Lipschitz embedding of $(\mathbb{S}_{K},d_{\mathbb{H}})$ into $\mathbb{R}^{4}$.

Since the proof is similar to that of Theorem 5.1, we only sketch the steps and leave the details to the reader.

Proof.

Given $s\in\mathbb{S}^{1}$ and $t\in[-1,1]$, we define

[TABLE]

The arc $\ell_{s}([-1,1])$ is the longitude of $\mathcal{C}$ in the direction of $s$ with $\ell_{s}(-1)$ (resp. $\ell_{s}(1)$) being the south pole $\mathcal{S}=(0,0,-1)$ (resp. north pole $\mathcal{N}=(0,0,1)$) for all $s\in\mathbb{S}^{1}$. Given $\epsilon\in(0,1)$, define the sets

[TABLE]

Define also the two poles of $\mathbb{S}_{K}$, $\mathcal{N}^{*}=(0,0,1/4)$ and $\mathcal{S}^{*}=(0,0,-1/4)$.

Working as in Lemma 5.3, we can show that there exists a bi-Lipschitz homeomorphism $G:\mathcal{C}\to\mathbb{S}_{K}$ such that $G(\mathcal{S})=\mathcal{S}^{*}$, $G(\mathcal{N})=\mathcal{N}^{*}$ and every longitude $\ell_{s}([-1,1])$ is mapped to a horizontal curve. In fact every curve $t\mapsto G(\ell_{s}(t))$ is an integral curve of the vector field

[TABLE]

which lies in the intersection of $T_{p}\mathbb{S}_{K}$ and $H_{p}$ at any $p=(x,y,z)\in\mathbb{S}_{K}$.

Define now $\Psi:(\mathbb{S}_{K},d_{\mathbb{H}})\to\mathbb{R}^{4}$ by

[TABLE]

where $\phi$ is the bi-Lipschitz embedding of $(\mathbb{S}^{1},|\cdot|^{1/2})$ into $\mathbb{R}^{3}$. By §3.2.2 and Theorem 5.1, we know that $\Psi$ is bi-Lipschitz on a neighborhood of $\mathcal{N}$ and on a neighborhood of $\mathcal{S}$. Moreover, for all $w\in\mathbb{S}_{K}\setminus\{\mathcal{N}^{*},\mathcal{S}^{*}\}$, we can use the result of Lemma 4.5 to see that there exists a neighborhood of $w$ on $\mathbb{S}_{K}$ on which $\Psi$ is bi-Lipschitz with a universal constant. By the compactness of $\mathbb{S}_{K}$, there exist $\epsilon>0$ and $L>1$ such that $\Psi$ is $L$-bi-Lipschitz on each $B(p,\epsilon)\cap\mathbb{S}_{K}$ with $p\in\mathbb{S}_{K}$. On the other hand, by Lemma 2.2, if $w,w^{\prime}\in\mathbb{S}_{K}$ with $|w-w^{\prime}|\geq\epsilon$, then

[TABLE]

We may follow very similar arguments to prove the following result.

Proposition 5.5**.**

There exists a bi-Lipschitz embedding of $(\mathbb{S}^{2},d_{\mathbb{H}})$ into $\mathbb{R}^{4}$.

As with Proposition 5.4, one needs to construct a bi-Lipschitz mapping $G:\mathbb{S}^{2}\to\mathbb{S}^{2}$ such that every longitude $\ell_{s}([-1,1])$ is mapped to a horizontal curve. Here, the vector field $V$ is defined as

[TABLE]

The rest of the proof follows in the same way as above.

5.3. The surface $z=\frac{1}{2}xy$

The manifolds considered in §5.1 and in §5.2 are manifolds that contain at most two characteristic points. Now we consider the surface $z=xy/2$ which has infinitely many characteristic points, namely all points along the $x$-axis. Solving the ODE for $dx/dy$ in (4.4), we obtain the horizontal curves

[TABLE]

On the other hand, for any $a\in\mathbb{R}$ and $c\in\mathbb{R}$, the curve

[TABLE]

is a snowflake; that is, it is bi-Lipschitz homeomorphic to a rescaled copy of $\Phi([0,1])$ where $\Phi:\mathbb{R}\to\mathbb{R}^{3}$ is the $\frac{1}{2}$-snowflaking map from §2. The construction of the following embedding resembles a general construction by Seo [Seo11].

Proposition 5.6**.**

The metric space $(\{z=\frac{1}{2}xy\},d_{\mathbb{H}})$ bi-Lipschitz embeds in $\mathbb{R}^{19}$.

Proof.

For each $n,m\in\mathbb{Z}$ let

[TABLE]

Firstly, we show that $(Q_{0,0}^{+},d_{\mathbb{H}})$ bi-Lipschitz embeds in $[1,2]\times[0,1]^{3}\subset\mathbb{R}^{4}$. Indeed, consider the map

[TABLE]

Then, for $w=(x,y,\frac{1}{2}xy)\in Q_{0,0}^{+}$ and $w^{\prime}=(x^{\prime},y^{\prime},\frac{1}{2}x^{\prime}y^{\prime})\in Q_{0,0}^{+}$, we have

[TABLE]

Secondly, we remark that the maps $\zeta_{n,m}^{\pm}:(Q_{0,0}^{+},d_{\mathbb{H}})\to(Q_{n,m}^{\pm},d_{\mathbb{H}})$ given by

[TABLE]

are $2^{-n}$-similarities i.e. $d_{\mathbb{H}}(w,w^{\prime})=2^{-n}d_{\mathbb{H}}(\zeta_{n,m}^{\pm}(w),\zeta_{n,m}^{\pm}(w^{\prime}))$. Let also $\sigma_{n,m}^{\pm}:\mathbb{R}^{4}\to\mathbb{R}^{4}$ be the similarities given by

[TABLE]

Thirdly, we decompose $(\{z=\frac{1}{2}xy\},d_{\mathbb{H}})$ into 4 pieces, we embed each piece bi-Lipschitz into $\mathbb{R}^{4}$, and we glue these 4 embeddings via Lemma 3.6. Define the following four families of squares

[TABLE]

Fix an $i\in\{1,2,3,4\}$ and define $G_{i}:(\{y=z=0\}\cup\bigcup\mathscr{Q}_{i},d_{\mathbb{H}})\to\mathbb{R}^{4}$ by

[TABLE]

By the first two steps, for each $Q\in\mathscr{Q}_{i}$, the map $G_{i}|_{Q}$ is bi-Lipschitz with bi-Lipschitz constant independent of the size of $Q$. It remains to show that each $G_{i}$ is bi-Lipschitz. Assuming this is true, $(\{z=\frac{1}{2}xy\},d_{\mathbb{H}})$ bi-Lipschitz embeds into $\mathbb{R}^{19}$ by Lemma 3.6.

Fix $Q,Q^{\prime}\in\mathscr{Q}_{i}$ with $Q\neq Q^{\prime}$ and fix $w=(x,y,\frac{1}{2}xy)\in Q$ and $w^{\prime}=(x^{\prime},y^{\prime},\frac{1}{2}x^{\prime}y^{\prime})\in Q^{\prime}$. (The case that one of them is on the $x$-axis follows by a limit argument). Recall that

[TABLE]

We claim that $d_{\mathbb{H}}(w,w^{\prime})\simeq|x-x^{\prime}|+|y-y^{\prime}|$. To see that, consider the following two cases.

Case 1. Suppose that $Q=Q_{n,m}^{+}$ and $Q^{\prime}=Q_{n,m^{\prime}}^{+}$ for $m\neq m^{\prime}$. Then,

[TABLE]

Hence, $|y+y^{\prime}|^{1/2}|x-x^{\prime}|^{1/2}\lesssim|x-x^{\prime}|$. The same conclusion holds if $Q=Q_{n,m}^{-}$ and $Q^{\prime}=Q_{n,m^{\prime}}^{-}$.

Case 2. Suppose that Case 1 does not hold. Then $Q=Q_{n,m}^{\pm}$ and $Q^{\prime}=Q_{n^{\prime},m^{\prime}}^{\pm}$ for $n\neq n^{\prime}$, and so $|y-y^{\prime}|\simeq|y+y^{\prime}|$. The claim follows.

With similar reasoning, we can show that, for any $Q,Q^{\prime}\in\mathscr{Q}_{i}$ with $Q\neq Q^{\prime}$ and for any $w=(x,y,\frac{1}{2}xy)\in Q$ and $w^{\prime}=(x^{\prime},y^{\prime},\frac{1}{2}x^{\prime}y^{\prime})\in Q^{\prime}$, we have

[TABLE]

Indeed, notice that $(g\circ(\zeta_{n,m}^{\pm})^{-1})(Q)\subset[1,2]\times[0,1]^{3}$ for any $Q=Q_{n,m}^{\pm}\in\mathscr{Q}_{i}$, and thus

[TABLE]

According to the definition of $\mathscr{Q}_{i}$ and arguments as above, then, (5.5) holds for any $Q,Q^{\prime}\in\mathscr{Q}_{i}$ with $Q\neq Q^{\prime}$. That concludes the proof of Proposition 5.6. ∎

6. Nonembedabbility of Heisenberg porous sets

The focus of this section is the proof of Theorem 1.4. For the rest of §6, we set $\pi:\mathbb{H}\to\mathbb{R}^{2}$ to be the 1-Lipschitz abelianization map. Given $\alpha>0$, we define a $\alpha$-cone in $\mathbb{H}$ as the set

[TABLE]

It is easily seen that $C_{\alpha}$ is dilation and rotation invariant and $C_{\alpha}\subseteq C_{\beta}$ when $\alpha\leq\beta$. In §6.1 we define the set $X$ and explain why there is no bi-Lipschitz embedding of $X$ in $\ell^{2}$, and in §6.2 we show that $X$ is indeed porous.

6.1. Laakso graphs

Let $\{G_{n}\}_{n=1}^{\infty}$ denote the sequence of Laakso graphs defined in Section 2.2 of [Li16] and let $f_{n}:G_{n}\to\mathbb{H}$ denote the embedding defined starting from the bottom of p. 1625 (there it was just denoted $f$ by abuse of notation). (Note in particular that each $G_{n}$ consists only of the vertices in the graph and does not include the edges.) Recall that $f_{n}$ is specified with a sequence of angles $\theta_{j}=\left(\sqrt{M+j}\log(M+j)\right)^{-1}$ for some $M$ sufficiently large to be fixed. Theorem 1.2 of [Li16] showed that $f_{n}$ on $G_{n}$ has distortion $O((\log|G_{n}|)^{1/4}\sqrt{\log\log|G_{n}|})$.

One of the main goals of [Li16] was to use $f_{n}$ to show that the image $f_{n}(G_{n})$ cannot uniformly bi-Lipschitz embed into Markov $p$-convex metric spaces when $p<4$. This is explicitly done in the proof of Corollary 1.4 of [Li16] (although one needs to remove the intermediary $\mathbb{H}(\mathbb{Z})$ that appears there). Since Hilbert space is Markov 2-convex, we get that $f_{n}(G_{n})$ do not embed into $\ell_{2}$ with uniform bi-Lipschitz distortion.

We now require that the diameter of each $G_{n}$ is 1. Thus, the length of edges in $G_{n}$ become $6^{-n}$. This allows us to define a limiting compact object $G_{\infty}$ also of diameter 1. It is not hard to show that $G_{\infty}$ contains each $G_{n}$ isometrically.

One also has that $G_{\infty}$ is self-similar and is composed as a union of 10 scaled copies of $G_{\infty}$. Call these level 1 copies. We use $H_{1}$ to denote any one of these copies, and they are all isometric. (Note that $H_{1}$ could refer to any level 1 copy of $G_{\infty}$.) Each of these in turn are composed of 10 scaled copies of $G_{\infty}$. Call these level 2 copies, and use $H_{2}$ to denote each of these copies. We can continue and identify level $n$ copies for all $n\in\mathbb{N}$ each denoted $H_{n}$. If a copy $H_{i}$ does not contain a terminal point of some $H_{i-1}$ we say it is forked. Note that each $H_{i}$ contains 8 forked and 2 non-forked copies of $H_{i+1}$.

Each $H_{i}$ contains a unique scaled copy of $G_{k}$ that is maximal in diameter. We call this $H_{i,k}$. Again, $H_{i,k}$ could refer to any one of many isometric objects.

We can define a Lipschitz map $f:G_{\infty}\to\mathbb{H}$ so that the restriction of $f$ on each embedded copy of $G_{n}$ in $G_{\infty}$ is a re-scaled (and possibly translated and rotated) copy of $f_{n}(G_{n})$. Thus, $f(G_{\infty})$ is compact and cannot embed into $\ell_{2}$ as it contains copies of $f_{n}(G_{n})$. Define $X=f(G_{\infty})$.

6.2. Proof of Theorem 1.4

From the discussion above, Theorem 1.4 immediately follows from the following proposition.

Proposition 6.1**.**

The set $X=f(G_{\infty})$ is porous in $\mathbb{H}$.

For the rest of §6.2, given a subset $A\subset G_{\infty}$, we let $\tilde{A}:=\pi(f(A))\subset\mathbb{R}^{2}$, and if $x\in G_{\infty}$, set $\tilde{x}:=\pi(f(x))$. We require two lemmas from [Li16]. The first one follows from Lemma 2.6 of [Li16].

Lemma 6.2**.**

For $M$ sufficiently large, if $s,t$ are terminal points of a level $n$ copy $H_{n}$ of $G_{\infty}$, then $\frac{6^{-n}}{2}\leq|\pi(f(s))-\pi(f(t))|\leq d_{\mathbb{H}}(f(s),f(t))$.

The second is Lemma 2.7 of [Li16].

Lemma 6.3**.**

For any $i$, the set $\tilde{H}_{i}$ is contained in the closed convex hull of the corresponding $\tilde{H}_{i,1}$.

Remark 6.4*.*

Let $C$ and $C^{\prime}$ be convex hulls of $\tilde{H}_{i+1,1}$ for copies of $H_{i+1}$ in $H_{i}$ that are in parallel and intersect at a fork point $a$. Lemma 6.3 says that there exists a cone based at $a$ of aperture at $\theta_{i}-\frac{\theta_{i+1}}{2}\geq\frac{\theta_{i}}{2}$ between $C$ and $C^{\prime}$.

Corollary 6.5**.**

For any $H_{n}$, we have $\frac{6^{-n}}{2}\leq\operatorname{diam}_{d_{\mathbb{H}}}{f(H_{n})}\leq 6^{-n}$.

Proof.

The upper bound comes from 1-Lipschitzness of $f$. The lower bound comes from Lemma 6.2. ∎

Proposition 6.6**.**

There exists $M>0$ sufficiently large so that the following holds. Let $H_{i}\supset H_{i+1}\supset...\supset H_{k}$ be a nested sequence of sub-graphs so that $H_{i+1}$ is forked in $H_{i}$. Let $S\subseteq\{i+2,...,k\}$ be the indices $j$ so that $H_{j}$ is forked in $H_{j-1}$. Suppose there exists another forked copy $H_{i+1}^{\prime}\subset H_{i}$ that is in parallel with $H_{i+1}$ so that there are points $x\in H_{k}$ and $y\in H_{i+1}^{\prime}$ for which $6^{-k}\leq|\tilde{x}-\tilde{y}|\leq 6^{-k+1}$. Then $\sum_{j\in S}\theta_{j}\leq 1/100$.

Proof.

We suppose for contradiction that $\sum_{j\in S}\theta_{j}>\frac{1}{100}$. In particular, $S$ is nonempty so let $\ell=\min S$ and $a,b,c\in\mathbb{R}^{2}$ be the points of $\tilde{H}_{i,1}\subset\tilde{H}_{i}$ with degree at least 3. Then the distance from $\tilde{x}$ to $a,b,c$ is at least $6^{-\ell}/2$ by Lemma 6.2. By Remark 6.4, the distance from $x$ to any point in $\tilde{H}_{i+1}^{\prime}$ (which is in parallel with $\tilde{H}_{i+1}$) is at least $6^{-\ell}\theta_{i}/8$ and so $6^{-k+1}\geq 6^{-\ell}\theta_{i}/8$ by the assumption of the lemma. This tells us that $k\leq\ell+3+\log\frac{1}{\theta_{i}}$ and so $|S|\leq 3+\log\frac{1}{\theta_{i}}\leq\frac{1}{100}\sqrt{i+M}$ for large enough $M$. Thus,

[TABLE]

which is a contradiction. ∎

We also choose $M$ sufficiently large so that $100\theta_{1}<\frac{1}{100}$.

Fix some $x\in X$ and choose some $k\in\mathbb{N}$. Let $y\in G_{\infty}$ be any point so that $f(y)=x$. For each $i\in\mathbb{N}$, there exists some $H_{i}$ that contains $y$. We call this subset $Y_{i}$. If $y$ is a terminal point of some $H_{i}$, then the choice of $Y_{i}$ may not be unique. In this case, choose $Y_{i}$ arbitrarily while ensuring that, at each step, $Y_{i+1}\subset Y_{i}$. We let $s_{i}$ and $t_{i}$ denote the source and sink of $Y_{i}$.

Let $(u,v)\in\mathbb{S}^{1}$ be a $90^{\circ}$ clockwise rotation of the direction from $\tilde{s}_{k}$ to $\tilde{t}_{k}$. Let

[TABLE]

and define the ball $B=B(x^{\prime},6^{-k})$. Our goal is to prove that $B\cap X=\emptyset$. It suffices to show that $B\cap(f(Y_{i})\backslash f(Y_{i+1}))=\emptyset$ for every $i\in\mathbb{N}$. If $k\leq i+5$, then it is obvious that this intersection is empty by a simple triangle inequality argument and the fact that $f(Y_{i})\subseteq B(f(s_{k}),6^{-k+5})$.

We may now suppose $i<k-5$. Let $S\subseteq\{i+2,...,k\}$ be the indices $j$ so that $Y_{j}$ is forked in $Y_{j-1}$. Let $P:\mathbb{R}^{2}\to\mathbb{R}$ denote the orthogonal projection onto the linear subspace spanned by $\tilde{s}_{i}$ and $\tilde{t}_{i}$ with the orientation that $P(\tilde{s}_{i})<P(\tilde{t}_{i})$.

Lemma 6.7**.**

Suppose at least 100 of $\{Y_{j}\}_{j=i+2}^{k}$ are forked. Then

[TABLE]

Proof.

Let $\ell\in\{i+2,...,k\}$ be the smallest integer so that $Y_{\ell}$ is forked. Thus, in $Y_{\ell}$, there exists a non-forked $H_{\ell+1}$ separating $Y_{k}$ from $s_{\ell}$ and $t_{\ell}$ (since at least one $Y_{j}$ is forked for $j\in\{\ell+2,\dots,k\}$). As $\ell$ was minimal in $\{i+2,...,k\}$, $P(\tilde{s}_{\ell})\geq P(\tilde{s}_{i+1})$ and we get that $P(\tilde{s}_{k})-P(\tilde{s}_{\ell})>6^{-\ell}/2$. As $k\geq\ell+99$ and since the group multiplication acts in the first two coordinates simply as addition, this proves the lemma (arguing similarly for the right hand side of the interval). ∎

Lemma 6.8**.**

If $B$ intersects $f(Y_{i})\backslash f(Y_{i+1})$, then $Y_{i+1}$ must be forked and $B$ must intersect some other $f(H_{i+1})$ where $H_{i+1}$ is in parallel with $Y_{i+1}$.

Proof.

Since $B\cap f(Y_{i})\neq\emptyset$, the ball $B$ intersects some $f(H_{i+1})$ in $f(Y_{i})$. We denote this $H_{i+1}$ as $Y_{i+1}^{\prime}$. Let $\theta$ denote the angle between the line segments connecting the terminals of $\tilde{Y}_{i}$ and $\tilde{Y}_{k}$. First suppose that $\theta\leq 1/100$. Then an elementary geometric argument in $\mathbb{R}^{2}$ using Lemma 6.3 shows that $\pi(B)$ cannot intersect any $\tilde{H}_{i+1}$ where $H_{i+1}$ is in series with $Y_{i+1}$. Thus $Y_{i+1}^{\prime}$ must be in parallel with $Y_{i+1}$, and this proves the lemma in this case.

Now suppose $\theta>1/100$. Then as $100\theta_{1}\leq 1/100$, we must have that at least 100 of $\{Y_{j}\}_{j=i+2}^{k}$ are forked. The lemma then follows from the previous lemma. Indeed, $P(\pi(B))\cap P(\tilde{Y}_{i+1}^{\prime})\neq\emptyset$, and so $Y_{i+1}^{\prime}$ cannot be in series with $Y_{i+1}$. ∎

Recall that $C_{1}$ is the 1-cone defined in (6.1).

Lemma 6.9**.**

$6B\subseteq f(s_{k})\cdot C_{1}$.

Proof.

We may suppose $f(s_{k})=(0,0,0)$. Let $p\in B(0,6^{-k+1})$. Then $(u6^{-k+10},v6^{-k+10},0)\cdot p$ has a $z$-coordinate of absolute value at most $6^{-2k+2}+6^{-k+10}\cdot 6^{-k+1}\leq 6^{-2k+12}$. On the other hand, the $x$ and $y$ coordinates have absolute value at least $6^{-k+9}$. This easily proves the lemma (with drastic overestimation). ∎

We now show Proposition 6.1.

Proof of Proposition 6.1.

Suppose $B$ intersects $f(Y_{i})\backslash f(Y_{i+1})$. Let $Y_{i+1}^{\prime}$ denote the $H_{i+1}$ that is in parallel with $Y_{i+1}$ such that $B\cap f(Y_{i+1}^{\prime})$ is nonempty (from Lemma 6.8). Let $p\in B\cap f(Y_{i}^{\prime})$. In a similar way, for $j\in\{i+2,...,k\}$, let $Y_{j}^{\prime}$ denote the $H_{j}$ that contain $p$ with $Y_{j}^{\prime}\subseteq Y_{j-1}^{\prime}$ and $s_{k}^{\prime},t_{k}^{\prime},\tilde{s}_{k}^{\prime},\tilde{t}_{k}^{\prime}$ be defined similarly. Note that the inflated ball $6B$ must intersect $\tilde{s}_{k}^{\prime}$. Let $S^{\prime}$ be the indices $j$ of $\{i+2,...,k\}$ where $Y_{j}^{\prime}$ is forked in $Y_{j-1}^{\prime}$. By Proposition 6.6, we have that both $\sum_{j\in S}\theta_{j}$ and $\sum_{j\in S^{\prime}}\theta_{j}$ are less than 1/100.

Take a geodesic path from $s_{k}$ to $s_{k}^{\prime}$. This is pushed into $\mathbb{H}$ via $f$ to a piecewise affine horizontal path $p_{1}p_{2}\cdots p_{n}$ so that $p_{1}=f(s_{k})$ and $p_{n}=f(s_{k}^{\prime})$. As $\max\{\sum_{j\in S}\theta_{j},\sum_{j\in S^{\prime}}\theta_{j}\}\leq 1/100$ and the line from $\pi(p_{1})$ to $\pi(p_{n})$ is almost orthogonal to the line between the terminals of $\tilde{Y}_{i}$, the closed piecewise affine path $\pi(p_{1})\cdots\pi(p_{n})\pi(p_{1})$ in $\mathbb{R}^{2}$ does not self intersect. It follows from Remark 6.4 that this path encloses an isosceles triangle of angle $\theta_{i}-\frac{\theta_{i+1}}{2}$ and base at least $\frac{|\pi(p_{n})-\pi(p_{1})|}{2}$ which has area

[TABLE]

Thus, we see that $p_{n}\notin p_{1}\cdot C_{1/16\theta_{i-1}}$. As $\theta_{i}$ is decreasing, this means $p_{n}\notin p_{1}\cdot C_{1/16\theta_{1}}$. However, from Lemma 6.9 we have that $p_{n}\in p_{1}\cdot C_{1}$. This is a contradiction, which means $B$ does not intersect $f(Y_{i})\backslash f(Y_{i+1})$. This proves that $X=f(G_{\infty})$ is porous in $\mathbb{H}$. ∎