The Intrinsic Geometry of Simply and Rectifiably Connected Plane Sets

Abstract

We prove that the metric completion of the intrinsic length space associated with a simply and rectifiably connected plane set is a Hadamard space. We also characterize when such a space is Gromov hyperbolic.

1 Introduction

Throughout this section X is a simply and rectifiably connected plane set; we make no assumption about X being open or closed. Then $\overline{X_l}$ is the metric completion of $X_l:=(X,l)$, the metric space where l is the intrinsic (Euclidean) length distance on X.

Theorem

Suppose X is a simply and rectifiably connected plane set. Then $\overline{X_l}$ is a Hadamard space. Moreover, $\overline{X_l}$ is Gromov hyperbolic if and only if X does not contain Euclidean disks of arbitrarily large radii, i.e., if and only if

$$\begin{aligned} R:=\sup \left\{ r>0 \;\big |\; \exists \, \textsf{D}(x;r)\subset X\right\} <+\infty ; \end{aligned}$$

when this holds, X is 2R-hyperbolic and this is best possible.

Recall that a Hadamard metric space is a complete CAT(0) space¹; see Sect. 2.2.2. Our result provides a bountiful supply of easily constructed Hadamard spaces.

The special case where X is a closed bounded Jordan plane region was established in [1]; there the work of [3] is an essential ingredient. We present an elementary argument for the case where X is a simply connected plane domain; see Sect. 3.1.²

It is easy to construct compact rectifiably connected plane sets X for which $X_l$ is geodesic but fails to have non-positive curvature. It would be useful to have a characterization of the plane sets X for which $\overline{X_l}$ has non-positive curvature. As Bishop mentions, in [5, Ex. 9.1.6, p. 310] the authors assert that for “any locally simply connected plane set X”, $X_l$ has non-positive curvature. However, as Bishop comments, their discussion fails to mention certain essential details including, in particular, the existence of geodesics.

2 Preliminaries

For real numbers r and s,

$$\begin{aligned} r\wedge s:= \min \{r,s\} \quad \text {and}\quad r\vee s:= \max \{r,s\}. \end{aligned}$$

2.1 Metric Space Notation and Terminology

Throughout this section X is an arbitrary metric space with distance denoted $|x-y|$; this is not meant to imply that X possesses any sort of linear or group structure. In this setting, all topological notions refer to the metric topology; here ${{\,\mathrm{\textsf{cl}}\,}}(A), {{\,\mathrm{\textsf{bd}}\,}}(A), {{\,\mathrm{\textsf{int}}\,}}(A)$ are the topological closure, boundary, interior (respectively) of $A\subset X$.

Every metric space can be isometrically embedded into a complete metric space. We let $\bar{X}$ denote the metric completion of the metric space X; thus $\bar{X}$ is the closure of the image of X under such an isometric embedding. We call $\partial X:=\bar{X}{\setminus }X$ the metric boundary of X.

When $A\subset X$, there is a natural embedding ${{\bar{A}}}\hookrightarrow \bar{X}$ and ${{\,\mathrm{\textsf{bd}}\,}}(A)\subset {\partial A}$. Here if $A\subset X$ is open and X complete, then ${\partial A}={{\,\mathrm{\textsf{bd}}\,}}(A)$, but in general ${{\bar{A}}}={\overline{{{\,\mathrm{\textsf{cl}}\,}}}}(A)$ and ${\partial A}={\overline{{{\,\mathrm{\textsf{bd}}\,}}}}(A){\setminus }A$ where ${\overline{{{\,\mathrm{\textsf{cl}}\,}}}}$ and ${\overline{{{\,\mathrm{\textsf{bd}}\,}}}}$ denote topological closure and boundary in $\bar{X}$

2.1.1 Paths, Arcs, Geodesics, and Length

A path in X is a continuous map ${\mathbb R}\supset I\xrightarrow {\gamma }X$ where $I=I_\gamma$ is the parameter interval for $\gamma$ and may be closed or open or neither and finite or infinite. The trajectory of such a path $\gamma$ is $|\gamma |:=\gamma (I)$ which we call a curve and often–when easily understood in context–we abuse notation and just write $\gamma$ in place of $|\gamma |$.

A path $I\xrightarrow {\gamma }X$ is a geodesic if it is an isometry:

$$\begin{aligned} \forall \; s,t\in I, \quad |\gamma (s)-\gamma (t)|=|s-t| \end{aligned}$$

and X is a geodesic metric space if each pair of points can be joined by a geodesic.

When I is closed and $I\ne {\mathbb R}$, $\partial \gamma :=\gamma (\partial I)$ denotes the set of endpoints of $\gamma$ and consists of one or two points depending on whether or not I is compact. For example, if $I_\gamma =[u,v]\subset {\mathbb R}$, then $\partial \gamma =\{\gamma (u),\gamma (v)\}$. When $\partial \gamma =\{a,b\}$, we write $\gamma :a\curvearrowright b$ (in X) to indicate that $\gamma$ is a path (in X) with initial point a and terminal point b; this implies an orientation—a precedes b on $\gamma$.

We call $\gamma$ a compact path if its parameter interval is compact. A compact path $\gamma$ is a loop if $\partial \gamma$ is a single point, and then $|\gamma |$ is often dubbed a closed curve. A loop $\gamma :[u,v]\rightarrow X$ is a Jordan loop (aka, a simple closed curve) if $\gamma \vert _{[u,v)}$ is injective.

An arc $\alpha$ is an injective compact path; here $|\alpha |$ is often called a simple curve; again, we sometimes abuse notation and call $|\alpha |$ an arc. The interior of $\alpha$ is .

Given points $a,b\in |\alpha |$, there is a unique subarc $\alpha [a,b]$ of $\alpha$ with endpoints a, b; precisely, there are unique $u,v\in I$ with $\alpha (u)=a$, $\alpha (v)=b$ and $\alpha [a,b]:=\alpha \vert _{[u,v]}$. (Again, sometimes $\alpha [a,b]$ is this map and sometimes it denotes its trajectory.) We also use this notation for a general path $\gamma$, but here $\gamma [a,b]$ denotes the unique subpath of $\gamma$ that joins a, b obtained by using the last time $\gamma$ is at a up to the first time $\gamma$ is at b.

When $\alpha :a\curvearrowright b$ and $\beta :b\curvearrowright c$ are paths that join a to b and b to c respectively, $\alpha \star \beta$ denotes the concatenation³ of $\alpha$ and $\beta$; so $\alpha \star \beta :a \curvearrowright c$. The reverse of $\gamma$ is the path $\tilde{\gamma }$ defined by $\tilde{\gamma }(t):=\gamma (1-t)$ (when $I_\gamma =[0,1]$) and going from $\gamma (1)$ to $\gamma (0)$. Of course, $|\alpha \star \beta |=|\alpha |\cup |\beta |$ and $|\tilde{\gamma }|=|\gamma |$.

Every compact path contains an arc with the same endpoints; see [12].

The length of a compact path $[0,1]\overset{\gamma }{\rightarrow }X$ is defined in the usual way by

$$\begin{aligned} \ell (\gamma ):=\sup \left\{ \sum _{i=1}^n |\gamma (t_i)-\gamma (t_{i-1})| \,\Big \vert \, 0=t_0<t_1<\cdots <t_n=1 \right\} , \end{aligned}$$

$\gamma$ is rectifiable when $\ell (\gamma )<\infty$, and X is rectifiably connected provided each pair of points in X can be joined by a rectifiable path. An arbitrary path $\gamma$ is locally rectifiable if each compact subpath of $\gamma$ is rectifiable, and such a $\gamma$ is rectifiable if

$$\begin{aligned} \ell (\gamma ):=\sup \bigl \{\ell (\alpha ) \,\big \vert \, \alpha \text { a compact subpath of }\gamma \big \}<+\infty . \end{aligned}$$

Rectifiable paths always have endpoints, and so have unique extensions to compact paths with the same length. Here is a precise statement; cf. [11, Thm. 3.2, p.7].

Fact 2.1

Let ${\mathbb R}\supset I\xrightarrow {\gamma }X$ be a rectifiable path with I a finite interval. Then there is a unique extension $\bar{I}\xrightarrow {{\bar{\gamma }}}\bar{X}$ of $\gamma$ to a compact rectifiable path ${\bar{\gamma }}$ and $\ell ({\bar{\gamma }})=\ell (\gamma )$.

Every rectifiable path can be parametrized with respect to its arclength [11, p. 5]. When $\gamma$ is a rectifiable path, we tacitly assume its parameter interval is $I_\gamma =[0,\ell (\gamma )]$ unless specifically stated otherwise.

2.1.2 Intrinsic Length Distance

Every rectifiably connected metric space X admits a natural intrinsic distance, its so-called (inner) length distance given by

$$\begin{aligned} l(a,b):= \inf \bigl \{\ell (\gamma )\,\big \vert \,\hbox {} \gamma :a\curvearrowright b \hbox { a rectifiable path in}\, X\bigr \} . \end{aligned}$$

A metric space $(X,{{\,\mathrm{|\cdot |}\,}})$ is a length space provided for all points $a,b\in X$, $|a-b|=l(a,b)$, and we call such a ${{\,\mathrm{|\cdot |}\,}}$ a length (or intrinsic) distance function. An l-geodesic $[a,b]_l$ is a shortest path joining a and b, and any shortest path can be parametrized to be an l-geodesic.

The notation $X_l:=(X,l)$ is convenient, and then $\partial _l X:=\overline{X_l}{\setminus }X_l$. We note that $(\overline{X_l})_l=\overline{X_l}$, which is a consequence of the facts that the length distance $l=l_d$ associated with a length distance d is just d, and the completion of a length distance is also a length distance.

More generally, a continuous function $X\xrightarrow {\rho }(0,\infty )$ on a rectifiably connected metric space X induces a length distance $d_\rho$ on X defined by

$$\begin{aligned} d_\rho (a,b):=\inf _{\gamma :a\curvearrowright b} \ell _\rho (\gamma ) \quad \text {where}\quad \ell _\rho (\gamma ):= \int _\gamma \rho \, ds \end{aligned}$$

and where the infimum is taken over all rectifiable paths $\gamma :a\curvearrowright b$ in X. We describe this by calling $\rho \,ds=\rho (x)|dx|$ a conformal metric on X.

There are two useful properties of length spaces that we use repeatedly. First, for any open set U in a length space X, we always have ${{\,\mathrm{\textsf{dist}}\,}}(x,{{\,\mathrm{\textsf{bd}}\,}}U)={{\,\mathrm{\textsf{dist}}\,}}(x,X{\setminus }U)$ for all points $x\in U$. Second, $\bar{X}$ is also a length space. In fact, for all $x\in X, \xi \in \partial X, \varepsilon >0$ there is a path $\gamma :x\curvearrowright \xi$ in $X\cup \{\xi \}$ with $\ell (\gamma )<|x-\xi |+\varepsilon$.

We utilize the fact that rectifiable arcs in $\overline{X_l}$ can be approximated by arcs in X. Here is a precise statement.

Lemma 2.2

Let X be rectifiably connected. Suppose $\tilde{\gamma }:{\tilde{p}}\curvearrowright {\tilde{q}}$ is a rectifiable arc in $\overline{X_l}$. Then for each $\varepsilon >0$, there is a rectifiable path $\gamma :p\curvearrowright q$ in X with $l\bigl (\gamma (t),\tilde{\gamma }(t)\bigr )<\varepsilon$ for all $t\in I:=[0,\ell (\tilde{\gamma })]$ (where I is also the parameter interval for $\gamma$); thus there are rectifiable arcs $\alpha :p\curvearrowright q$ in X and $\tilde{\alpha }:{\tilde{p}}\curvearrowright {\tilde{q}}$ in $X\cup \left\{ {\tilde{p}},{\tilde{q}}\right\}$ with $\alpha ,\tilde{\alpha }\subset \textsf{N}_l(\tilde{\gamma };\varepsilon )$.

Proof Sketch

Given $\varepsilon \in (0,\ell (\tilde{\gamma }))$, let n be the smallest positive integer with $\ell (\tilde{\gamma })/n\le \varepsilon /10$. Put $t_i:=(i/n)\ell (\tilde{\gamma })$ for $0\le i\le n$. Define $x_i:=\tilde{\gamma }(t_i)$ if $\tilde{\gamma }(t_i)\in X$; otherwise, if $\tilde{\gamma }(t_i)\in \partial _l X$, choose any $x_i\in X$ with $l\bigl (x_i,\tilde{\gamma }(t_i)\bigr )<\varepsilon /10$. Then $l(x_{i-1},x_i)<3\varepsilon /10$ so there are rectifiable arcs $\gamma _i:x_{i-1}\curvearrowright x_i$ in X with $\ell (\gamma _i)<3\varepsilon /10$. Then $\gamma :=\gamma _1\star \cdots \star \gamma _n$ has the asserted properties, where $\gamma _i:[t_{i-1},t_i]\rightarrow X$ is parametrized proportional to arc length. $\square$

Here is information that we employ to construct Jordan loops inside X.

Lemma 2.3

Let X be rectifiably connected. Suppose $\gamma _i:{\tilde{p}}\curvearrowright q_i$ ($i=1,2$) are rectifiable arcs in $X\cup \left\{ {\tilde{p}}\right\}$ with $\gamma _1\cap \gamma _2=\left\{ {\tilde{p}}\right\} \subset \overline{X_l}$. Then for each $\varepsilon >0$, there are points $p_i\in \gamma _i$ and a rectifiable arc $\alpha :p_1\curvearrowright p_2$ in X with $l(p_1,{\tilde{p}})<\varepsilon ,l(p_2,{\tilde{p}})<\varepsilon ,\ell (\alpha )<\varepsilon$ and such that $\gamma _1^{-1}[q_1,p_1]\star \alpha \star \gamma _2[p_2,q_2]$ is a rectifiable arc $q_1\curvearrowright q_2$ in X.

Proof

Let $\varepsilon >0$ be given. Choose points $a_i\in \gamma _i$ and a rectifiable arc $\beta :a_1\curvearrowright a_2$ in X with each of $l(a_1,{\tilde{p}}),l(a_2,{\tilde{p}}),\ell (\beta )$ less than $\varepsilon /10$. Let $p_1$ be the last point of $\beta$ in $\gamma _1$ and let $p_2$ be the first point of $\beta [p_1,a_2]$ in $\gamma _2$. Then $\alpha :=\beta [p_1,p_2]$ has the asserted properties. $\square$

Let $[0,1)\xrightarrow {\gamma }X$ be a path in X. If there is a point $\xi \in \partial X$ such that $\lim _{t\rightarrow 1^-}|\gamma (t)-\xi |=0$, then $\xi$ is called a path accessible (metric) boundary point of X. In this situation, we define $\gamma (1):=\xi$ and obtain a path $\gamma :[0,1]\rightarrow X\cup \{\xi \}\subset \bar{X}$. We describe this by saying that $\gamma$ is a path in X with terminal endpoint $\xi \in \partial X$.

We write $\partial ^\textrm{pa}\!X$ for the set of all path accessible boundary points of X. Restricting attention to rectifiable paths $\gamma$ yields rectifiably accessible (metric) boundary points of X, denoted by ${\partial ^\textrm{ra}} X$. Clearly, ${\partial ^\textrm{ra}} X\subset \partial ^\textrm{pa}\!X\subset \partial X$ and each containment may be strict. We define $X^\mathrm{{ra}}:=X\cup {\partial ^\textrm{ra}} X$.

A path in X need not be a path in $X_l$; see [6, Ex. 3.6]. However, a rectifiable path in X is also continuous as a map into $X_l$ and therefore a path in $X_l$. Two rectifiable arcs in X with a common endpoint in $\partial _l X$, say

$$\begin{aligned}{}[0,\ell (\alpha )]\xrightarrow \alpha X\cup \left\{ \xi \right\} , [0,\ell (\beta )]\xrightarrow \beta X\cup \left\{ \xi \right\} \quad \text {with }\; \alpha (0)=\xi =\beta (0)\in \partial _l X, \end{aligned}$$

are l-equivalent if and only if

$$\begin{aligned} \lim _{s\rightarrow 0^+} l\bigl (\alpha (s),\beta (s)\bigr )=0. \end{aligned}$$

There is a natural one-to-one correspondence between $\partial _l X$ and the l-equivalence classes of such rectifiable arcs; see [6, Prop. 3.29].

The identity map $X_l\xrightarrow {{{\,\mathrm{{\textsf{id}}}\,}}}X$ is 1-Lipschitz and so has a 1-Lipschitz extension $\overline{X_l}\xrightarrow {\iota }\bar{X}$. In general, $\iota =\iota _X$ need not be surjective nor injective. However, we always have $\iota (\partial _l X)={\partial ^\textrm{ra}} X$.

We make repeated appeals to the following elementary fact; see [6, Lem. 3.17] and also Fact 2.1.

Lemma 2.4

Let $X=(X,{{\,\mathrm{|\cdot |}\,}})$ be a rectifiably connected metric space with associated length distance space $X_l=(X,l)$. Suppose $[0,1)\xrightarrow {\gamma } X$ is a rectifiable path in X. Then

$$\begin{aligned} \lim _{s,t\rightarrow 1^-}\ell \bigl (\gamma \vert _{[s,t]}\bigr )=0=\lim _{s,t\rightarrow 1^-} l\bigl (\gamma (s),\gamma (t)\bigr ) \end{aligned}$$

so there exist points $z\in \partial X$ and $\zeta \in \partial _l X$ such that

$$\begin{aligned} \lim _{t\rightarrow 1^-}|\gamma (t)-z|=0=\lim _{t\rightarrow 1^-} l\bigl (\gamma (t),\zeta \bigr ); \end{aligned}$$

therefore there are rectifiable paths

that are obtained by defining

$$\begin{aligned} {\bar{\gamma }}(t):= {\left\{ \begin{array}{ll} \gamma (t) &{} \text { for }t\in [0,1), \\ z &{} \text { for }t=1; \end{array}\right. } \quad \text {and}\quad \tilde{\gamma }(t):= {\left\{ \begin{array}{ll} \gamma (t) &{} \text { for } t\in [0,1), \\ \zeta &{} \text { for } t=1; \end{array}\right. } \end{aligned}$$

Moreover, $z\in \partial X$ if and only if   $\zeta \in \partial _l X$. Also, $\ell (\tilde{\gamma })=\ell ({\bar{\gamma }})=\ell (\gamma )$.

Corollary 2.5

Suppose X is rectifiably connected and $a,b\in X^\mathrm{{ra}}:=X\cup {\partial ^\textrm{ra}} X$. Then a, b can be joined by a rectifiable path in $X\cup \left\{ a,b\right\}$. Moreover, if $\gamma :a\curvearrowright b$ in $X\cup \left\{ a,b\right\}$, then there are unique points ${\tilde{a}},{\tilde{b}}\in \overline{X_l}$ and a rectifiable $\tilde{\gamma }:{\tilde{a}}\curvearrowright {\tilde{b}}$ in $X_l\cup \left\{ {\tilde{a}},{\tilde{b}}\right\}$ with $\gamma =\iota {\circ }\tilde{\gamma }$ and $\ell (\gamma )=\ell (\tilde{\gamma })$.

We define $X^\mathrm{{ra}}\times X^\mathrm{{ra}}\xrightarrow {l^\textrm{ra}}[0,+\infty )$ by

$$\begin{aligned} l^\textrm{ra}(a,b):= \inf \bigl \{\ell (\gamma )\,\big \vert \,\hbox {} \gamma :a\curvearrowright b\, \hbox {a rectifiable path in} \; X\cup \left\{ a,b\right\} \bigr \} . \end{aligned}$$

In general, $l^\textrm{ra}$ need not be a distance on $X^\mathrm{{ra}}$ because the triangle inequality may fail. However, its restriction $l^\textrm{ra}_1$ to $X^\mathrm{{ra}}_1\times X^\mathrm{{ra}}_1$, where

$$\begin{aligned} X^\mathrm{{ra}}_1:=X\cup {\partial ^\textrm{ra}_1}X \quad \text {with}\quad {\partial ^\textrm{ra}_1}X:=\left\{ z\in {\partial ^\textrm{ra}} X \mid {{\,\mathrm{\textsf{card}}\,}}\iota ^{-1}(z)=1\right\} , \end{aligned}$$

is a distance on $X^\mathrm{{ra}}_1$. The triangle inequality is easy to check if the intermediate point lies in X and not difficult to verify when this point lies in ${\partial ^\textrm{ra}_1}X$.

Setting

$$\begin{aligned} \partial _l^1X:=\left\{ \xi \in \partial _l X\mid \iota ^{-1}\bigl (\iota (\xi )\bigr )=\{\xi \}\right\} ,\quad X_l^1:=X_l\cup \partial _l^1 X,\quad X^\mathrm{{ra}}_l:=(X^\mathrm{{ra}}_1,l^\textrm{ra}_1) \end{aligned}$$

and $\iota ^1:=\iota \vert _{X_l^1}$, we easily obtain the following.

Lemma 2.6

When X is rectifiably connected, $X_l^1\xrightarrow {\iota ^1}X^\mathrm{{ra}}_l$ is an isometry.

Proof

Let ${\tilde{a}},{\tilde{b}}\in X_l^1$. Then $a:=\iota ({\tilde{a}}),b:=\iota ({\tilde{b}})\in X^\mathrm{{ra}}_1$. Let $\gamma :a\curvearrowright b$ be an arc in $X\cup \left\{ a,b\right\}$. The ends of $\gamma$ determine ${\tilde{a}},{\tilde{b}}$, so by Corollary 2.5 there is a rectifiable $\tilde{\gamma }:{\tilde{a}}\curvearrowright {\tilde{b}}$ in $X_l\cup \left\{ {\tilde{a}},{\tilde{b}}\right\}$ with $\gamma =\iota {\circ }\tilde{\gamma }$ and $\ell (\tilde{\gamma })=\ell (\gamma )$. Thus

$$\begin{aligned} l({\tilde{a}},{\tilde{b}})\le \ell (\tilde{\gamma })=\ell (\gamma ). \end{aligned}$$

Taking an infimum over all such $\gamma$ gives

$$\begin{aligned} l({\tilde{a}},{\tilde{b}})\le l(a,b) \end{aligned}$$

and the opposite inequality holds because $\iota$ is 1-Lipschitz. $\square$

2.2 CAT(0) Metric Spaces

Here our terminology and notation conforms with that in [4]; also, see [5]. We recall a few fundamental concepts, mostly copied directly from [4].

2.2.1 Geodesic and Comparison Triangles

A geodesic triangle $\Delta$ in X consists of three points in X, say $a,b,c\in X$, called the vertices of $\Delta$ and three geodesics, say $\alpha :a\curvearrowright b, \beta :b\curvearrowright c, \gamma :c\curvearrowright a$ (that we may write as [a, b], [b, c], [c, a]) called the sides of $\Delta$. We use the notation

$$\begin{aligned} \Delta =\Delta (\alpha ,\beta ,\gamma ) \quad \text {or}\quad \Delta =[a,b,c]:=[a,b]\star [b,c]\star [c,a] \quad \text {or}\quad \Delta =\Delta (a,b,c) \end{aligned}$$

depending on the context and the need for accuracy.

A Euclidean triangle ${\bar{\Delta }}=\Delta (\bar{a},\bar{b},\bar{c})$ in ${\mathbb C}$ is a comparison triangle for $\Delta =\Delta (a,b,c)$ provided $|a-b|=|\bar{a}-\bar{b}|,|b-c|=|\bar{b}-\bar{c}|,|c-a|=|\bar{c}-\bar{a}|$. We also write ${\bar{\Delta }}={\bar{\Delta }}(a,b,c)$ when a specific choice of $\bar{a},\bar{b},\bar{c}$ is not required. A point $\bar{x}\in [\bar{a},\bar{b}]$ is a comparison point for $x\in [a,b]$ when $|x-a|=|\bar{x}-\bar{a}|$. Assuming that $b\ne a\ne c$ (so $\bar{b}\ne \bar{a}\ne \bar{c}$), the comparison angle of  $\Delta$ at a is defined to be the interior Euclidean angle of ${\bar{\Delta }}$ at $\bar{a}$ and denoted by

$$\begin{aligned} \overline{\measuredangle }_{a}(b,c):=\measuredangle ^\textsf{euc}_{\bar{a}}(\bar{b},\bar{c}). \end{aligned}$$

Assume $a\ne p\ne b$ and let $\alpha :p\curvearrowright a, \beta :p\curvearrowright b$ be rectifiable arcs in X parameterized by arc length. The (upper) Alexandrov angle between $\alpha$ and $\beta$ is defined by

$$\begin{aligned} \measuredangle _p(\alpha ,\beta ):=\limsup _{s,t\rightarrow 0^+} \overline{\measuredangle }_{p}(\alpha (s),\beta (t)); \end{aligned}$$

see [4, 1.12, p.9]. When [p, a], [p, b] are geodesics, $\measuredangle _p(a,b):=\measuredangle _p([p,a],[p,b])$.

2.2.2 CAT(0) Definition

A geodesic triangle $\Delta$ in X satisfies the CAT(0) distance inequality if and only if the distance between any two points of $\Delta$ is not larger than the Euclidean distance between the corresponding comparison points; that is,

$$\begin{aligned} \forall \; x,y\in \Delta \;\text {and corresponding comparison points}\;\bar{x},\bar{y}\in {\bar{\Delta }}, \quad |x-y| \le |\bar{x}-\bar{y}|. \end{aligned}$$

We also say that $\Delta$ is CAT(0)-thin when it satisfies the CAT(0) distance inequality.

A geodesic metric space is CAT(0) if and only if each of its geodesic triangles is CAT(0)-thin. A complete CAT(0) metric space is called a Hadamard space. A geodesic metric space X has non-positive curvature if and only if it is locally CAT(0), meaning that for each point $a\in X$ there is an $r>0$ (that can depend on a) such that the metric ball $\textsf{B}(a;r)$ (endowed with the distance inherited from X) is CAT(0).

Of the many conditions which guarantee that a space is CAT(0), for instance, see [4, Prop. 1.7, p. 161] or [5, Thm. 4.3.5, p.116], we mention only that a geodesic metric space X is CAT(0) if and only if each of its geodesic triangles satisfies the CAT(0) vertex angle criterion. Here $\Delta$ satisfies the CAT(0) vertex angle criterion if and only if $\Delta$ has distinct vertices and the Alexandrov angle between any two sides of $\Delta$ is not greater than the interior Euclidean angle between the corresponding sides of a comparison triangle for $\Delta$; equivalently, if and only if the (Alexandrov) vertex angles of $\Delta$ are not greater than the corresponding (Euclidean) vertex angles of a comparison triangle for $\Delta$.

2.2.3 Triangle Tails

Let $\Delta =[a,b,c]=[a,b]\star [b,c]\star [c,a]$ be a geodesic triangle. Suppose there are points $b_o\in [a,b]$ and $c_o\in [a,c]$ such that the subgeodesics $[a,b_o]\subset [a,b]$ and $[a,c_o]\subset [a,c]$ coincide: i.e., $[a,b_o]=[a,c_o]$. This common geodesic segment is a tail of $\Delta$, and $\Delta$ is tail-less if there are no such tails.⁴

It is not difficult to verify the following. (If the lengths of two sides of an Euclidean triangle are increased by the same amount, then certain angles also increase.)

Fact 2.7

Let X be a geodesic metric space. Suppose every tail-less geodesic triangle in X satisfies the CAT(0) vertex angle criterion. Then X is CAT(0).

2.2.4 Gromov Hyperbolicity Definition

A geodesic metric space X is $\delta$-hyperbolic if and only if for all geodesic triangles $\Delta$ in X, each edge of $\Delta$ lies in the $\delta$-neighborhood of the union of the other two edges, and X is Gromov hyperbolic if and only if it is $\delta$-hyperbolic for some $\delta \in [0,+\infty )$.

2.3 General Plane Information

We view the Euclidean plane as the complex number field ${\mathbb C}$. Everywhere $\Omega$ is a plane domain (i.e., an open connected set), $\Omega ^c:={\mathbb C}{\setminus }\Omega$ and ${\partial \Omega }$ denote the complement and boundary (respectively) of $\Omega$.

The open disk of radius r centered at the point $a\in {\mathbb C}$ is

$$\begin{aligned} \textsf{D}(a;r):=\left\{ z:|z-a|<r\right\} , \end{aligned}$$

${\mathbb D}:=\textsf{D}(0;1)$ is the open unit disk, and the open r-neighborhood of a set $A\subset {\mathbb C}$ is

$$\begin{aligned} \textsf{N}(A;r):= \bigcup _{a\in A} \textsf{D}(a;r) = \left\{ z:{{\,\mathrm{\textsf{dist}}\,}}(z,A)<r\right\} . \end{aligned}$$

2.3.1 Complex Analysis

The well known Riemann and Carathéodory mapping theorems assert that when $\Omega$ is a simply connected plane domain, there is a conformal map (i.e., a holomorphic homeomorphism) $f:{\mathbb D}\rightarrow \Omega$, and if $\Omega$ is a Jordan domain, f extends to a homeomorphism $\bar{\mathbb D}\rightarrow {{\bar{\Omega }}}$. So, each boundary point of a Jordan domain is path accessible from the domain.

We repeatedly use the less known fact that when $\Omega$ is a simply connected plane domain with rectifiable boundary (e.g., if ${\partial \Omega }$ is a rectifiable Jordan loop), then each point of ${\partial \Omega }$ is rectifiably accessible from $\Omega$; that is, ${\partial ^\textrm{ra}}\Omega ={\partial \Omega }$ and $\Omega ^\mathrm{{ra}}={{\bar{\Omega }}}$. I am indebted to Distinguished Professor Chris Bishop for explaining this to me. It is a consequence of the fact that any Riemann map onto such a domain belongs to the Hardy class $\textsf{H}^1$; see the “easy half” of Chris’ result in [2].

A Riemann map ${\mathbb D}\xrightarrow {f}\Omega$ provides a conformal model for the length space $\overline{\Omega _l}$. Indeed, the conformal metric $|f'(z)|\,|dz|$ on ${\mathbb D}$ induces the length distance

$$\begin{aligned} d_f(a,b):= & {} \inf _{\gamma :a\curvearrowright b}\int _\gamma |f'(z)|\,|dz| \end{aligned}$$

where the infimum is over all rectifiable arcs $\gamma :a\curvearrowright b \hbox { in } {\mathbb D}$ and ${\mathbb D}_f:=({\mathbb D},d_f)\xrightarrow {f}(\Omega ,l)=:\Omega _l$ is an isometry. One can demonstrate that $\overline{{\mathbb D}_f}={\mathbb D}\cup \partial _f{\mathbb D}$, where $\partial _f{\mathbb D}:=\left\{ \zeta \in \partial {\mathbb D}\;\big \vert \; f\bigl ([0,\zeta )\bigr ) \text { is rectifiable}\right\}$; evidently, $\partial _f{\mathbb D}\subset \partial {\mathbb D}_f$, and the opposite containment can be established with the help of [10, Prop. 2.14-p.29, Cor. 2.17-p.35, Thm. 4.20-p.88]. Thus $\overline{\Omega _l}$ is isometrically equivalent to $\overline{{\mathbb D}_f}$. With this model, the map $\iota :\overline{\Omega _l}\rightarrow \Omega ^\mathrm{{ra}}$ can be realized as the radial limit extension $f:\overline{{\mathbb D}_f}\rightarrow \Omega ^\mathrm{{ra}}\subset {{\bar{\Omega }}}$.

For example, we now see that a Jordan loop $\Lambda$ in $\overline{\Omega _l}$ corresponds to a Jordan loop in $\overline{{\mathbb D}_f}\subset \bar{\mathbb D}$ whose interior is a simply connected domain in ${\mathbb D}$ with f image a simply connected $D\subset \Omega$ satisfying ${\partial D}=\iota (\Lambda )$, which is a rectifiably connected loop (perhaps not Jordan) in $\Omega ^\mathrm{{ra}}$.

3 Proofs

Here we establish the Theorem stated in the Introduction. Now X is a given simply and rectifiably connected plane set with $\overline{X_l}$ the metric completion of the intrinsic (Euclidean) length space $X_l$ associated with X. Also, $\overline{X_l}\xrightarrow {\iota }\bar{X}$ is the 1-Lip extension of the identity map $X_l\rightarrow X$.

First, we consider simply connected plane domains, then arbitrary simply and rectifiably connected plane sets.

3.1 CAT(0) Proof for X a Simply Connected Plane Domain

Assume $X=\Omega$ is a simply connected plane domain. Evidently, $\overline{\Omega _l}$ is a complete length metric space. We demonstrate that it is a 4-point limit of CAT(0) spaces, so by [4, Thm. 3.9, p.196] it is also CAT(0) and hence a Hadamard space.

The Riemann Mapping Theorem provides a conformal map ${\mathbb D}\xrightarrow {f}\Omega$ (i.e., a holomorphic homeomorphism). Let $(r_\nu )$ be a strictly increasing sequence in (0, 1) with $r_\nu \nearrow 1$. For each $\nu \in {\mathbb N}$, define

$$\begin{aligned} f_\nu (\zeta ):=f(r_\nu \zeta ), \quad \Omega _\nu :=f_\nu ({\mathbb D})=f(r_\nu {\mathbb D}), \end{aligned}$$

and let $U_\nu :=f_\nu (r_\nu {\mathbb D})=f(r_\nu ^2{\mathbb D}),\quad \lambda _\nu :=\lambda _{\Omega _\nu }, \quad M_\nu :=\max _{{{\bar{U}}}_\nu }\lambda _\nu , \text {and} \varepsilon _\nu :=\bigl (10{{\,\mathrm{\textsf{diam}}\,}}_l(\Omega _\nu )M_\nu ^2\bigr )^{-1}$.

Note that $U_\nu$ is compactly contained in $\Omega _\nu$ which in turn is compactly contained in $\Omega$, and

$$\begin{aligned} {{\,\mathrm{\textsf{diam}}\,}}_l(\Omega _\nu )={{\,\mathrm{\textsf{diam}}\,}}(\Omega _\nu ,l) \quad \text { with } l=l_\Omega . \end{aligned}$$

Also, for any plane domain D, $\lambda _D\,ds$ denotes the Poincaré hyperbolic metric in D, i.e., $\lambda _D\,ds$ is the maximal complete metric in D with constant Gaussian curvature $-1$; see [8]. Note that $\left\{ U_\nu \mid \nu \in {\mathbb N}\right\}$ is an increasing open cover of $\Omega$ and $(M_\nu ),(\varepsilon _\nu )$ are increasing, decreasing positive sequences with $M_\nu \nearrow +\infty ,\;\varepsilon _\nu \searrow 0$ respectively.

It is easy to check that $\overline{\Omega _l}$ is a 4-point limit of the spaces $({{\bar{\Omega }}}_\nu ,l_\nu )$ where $l_\nu$ is Euclidean length distance in $\Omega _\nu$. Since $\Omega _\nu$ is a Jordan domain, we could appeal to Bishop’s result now, but it is easy to provide a simple alternative argument.

Consider the conformal metric $\rho _\nu \,ds$ in $\Omega _\nu$ where

$$\begin{aligned} \rho _\nu :=1+\varepsilon _\nu \lambda _\nu \quad (\text {and note that}\rho _\nu \le 1+\bigl (10{{\,\mathrm{\textsf{diam}}\,}}_l(\Omega _\nu )M_\nu \bigr )^{-1} \text {in } {{\bar{U}}}_\nu ). \end{aligned}$$

Let $d_\nu$ be the length distance obtained from the metric $\rho _\nu \,ds$ in $\Omega _\nu$. Since $\log \lambda _\nu$ is subharmonic and ${{\mathscr {C}}}^\infty$ smooth in $\Omega _\nu$, so is $\log \rho _\nu$ (see [7, 2.1, 2.2]) and therefore by classical results (e.g., see [4, Thm.1A.6, Thm. 4.1, pp. 173,193]) each space $(\Omega _\nu ,d_\nu )$ is CAT(0).

We check that $\overline{\Omega _l}$ is a 4-point limit of the CAT(0) spaces $(\Omega _\nu ,d_\nu )$.

Let $x_1,x_2,x_3,x_4\in \overline{\Omega _l}$ and $\varepsilon \in (0,{{\,\mathrm{\textsf{diam}}\,}}_l\Omega _1)$ be given. Define $z_1,z_2,z_3,z_4\in \Omega$ as follows: if $x_i\in \Omega$, let $z_i:=x_i$; otherwise, $x_i\in \partial _l\Omega$, and we pick any $z_i\in \Omega$ with $l(z_i,x_i)<\varepsilon /10$.

Next, for all $1\le i<j\le 4$, choose arcs $\sigma _n^{ij}:z_i\curvearrowright z_j$ in $\Omega$ with $\ell (\sigma _n^{ij})$ decreasing to $l(z_i,z_j)$. Fix N so that for all $1\le i<j\le 4$, $n\ge N \implies \ell (\sigma _n^{ij})<l(z_i,z_j)+\varepsilon /10$; so $\ell (\sigma _n^{ij})<\frac{11}{10}{{\,\mathrm{\textsf{diam}}\,}}_l(\Omega _\nu )$.

For each n, $K_n:=\bigcup _{1\le i<j\le 4}\sigma _n^{ij}$ is a compact subset of $\Omega$, so there is an increasing sequence $(\nu _n)_{n\ge N}$ such that for each $n\ge N$, $M_{\nu _n}>2\varepsilon ^{-1}$ and $K_n\subset U_{\nu _n}\subset {{\bar{U}}}_{\nu _n}\subset \Omega _{\nu _n}$. Then for each $n\ge N$ and all $1\le i<j\le 4$,

$$\begin{aligned} l(z_i,z_j)&\le l_{\nu _n}(z_i,z_j) \le d_{\nu _n}(z_i,z_j) \le \ell _{\rho _{\nu _n}}(\sigma _n^{ij})=\int _{\sigma _n^{ij}}\rho _{\nu _n}\,ds \\&\le \Big (1+\bigl (10{{\,\mathrm{\textsf{diam}}\,}}_l(\Omega _\nu )M_{\nu _n}\bigr )^{-1}\Bigr )\ell (\sigma _n^{ij}) \quad (\text {because}\, \sigma _{ij}^n\subset K_n\subset U_{\nu _n}) \\&\le \ell (\sigma _n^{ij}) + \frac{11}{100M_{\nu _n}}< l(z_i,z_j)+\frac{\varepsilon }{10}+\frac{11\varepsilon }{200} < l(z_i,z_j)+\frac{\varepsilon }{5} \end{aligned}$$

and so

$$\begin{aligned} l(x_i,x_j)&\le l(x_i,z_i) + l(z_i,z_j) + l(z_j,x_j) \le d_{\nu _n}(z_i,z_j)+\frac{\varepsilon }{5} \le l(z_i,z_j)+\frac{2\varepsilon }{5} \\&\le l(z_i,x_i) + l(x_i,x_j) + l(x_j,z_j) + \frac{2\varepsilon }{5} \le l(x_i,x_j) + \frac{3\varepsilon }{5}. \end{aligned}$$

Thus, for all $n\ge N$ and $1\le i<j\le 4$: $z_i,z_j\in \Omega _{\nu _n}$ and $|l(x_i,x_j)-d_{\nu _n}(z_i,z_j)|<\varepsilon$. $\square$

3.2 CAT(0) Proof for General Case

Let X be a simply and rectifiably connected plane set. Our primary goal here is to demonstrate that $\overline{X_l}$ is uniquely geodesic; the CAT(0) property follows.

Since X is simply connected, whenever $\Lambda$ is a Jordan loop in X, ${{\mathscr {D}}}(\Lambda ):=\Lambda \cup {{\,\mathrm{\textsf{Int}}\,}}(\Lambda )\subset X$. As we employ this observation again and again, it is worthwhile to review methods for constructing Jordan loops.

Given distinct points p, q in X,

$$\begin{aligned} \Gamma (p,q):=\left\{ \text {all rectifiable arcs}\;\gamma :p\curvearrowright q \;\text {in} X\right\} \ne \varnothing . \end{aligned}$$

Suppose $\beta ,\gamma \in \Gamma (p,q)$ and there is a point $c\in \gamma {\setminus }\beta$. There are several ways to construct a Jordan loop $\Lambda$ in X that contains an open subarc of $\gamma$ which in turn contains c. Most simply, we move backwards, forwards along $\gamma$ from c (towards p, q respectively) and let a, b be (respectively) the first points of $\beta \cap \gamma$. Here $\Lambda :=\gamma [a,b]\cup \beta [a,b]$ has the asserted properties with $c\in \gamma (a,b)$.

A minor possible problem is that we do not know the order of a, b along $\beta$. To remedy this, set $b_1:=b$ and then, move backwards along $\gamma$ (from c to p), and let $a_1$ be the first point of $\beta [p,b_1]\cap \gamma$. Now $\Lambda _1:=\gamma [a_1,b_1]\star \beta ^{-1}[b_1,a_1]$ has the asserted properties and $p\le a<b\le q$ along both $\beta$ and $\gamma$. Yet another alternative is to set $a_2:=a$, move forwards along $\gamma$ (from c to q), let $b_2$ be the first point of $\beta [a_2,q]\cap \gamma$, and use $\Lambda _2:=\gamma [a_2,b_2]\star \beta ^{-1}[b_2,a_2]$. Note that the three Jordan loops $\Lambda ,\Lambda _1,\Lambda _2$ could all be different.

For definitiveness, we always use the first alternative construction.

We assume ${{\,\mathrm{\textsf{int}}\,}}(X)\ne \varnothing$, so

$$\begin{aligned} {{\mathscr {O}}}:=\left\{ \text {all components}\; \Omega \;\text {of} {{\,\mathrm{\textsf{int}}\,}}(X)\right\} \ne \varnothing . \end{aligned}$$

Note that even if some $\Omega \in {{\mathscr {O}}}$ has non-rectifiably accessible boundary points, ${{\bar{\Omega }}}\subset X$ is still possible. For each $\Omega \in {{\mathscr {O}}}$, $\overline{\Omega _l}\xrightarrow {\iota _\Omega }{{\bar{\Omega }}}$ is the 1-Lip extension of the identity map $\Omega _l\rightarrow \Omega$.

The following facts are useful.

• (3.1a) Rectifiable Jordan loops. Suppose $\Lambda$ is a rectifiable Jordan loop in X. Then there is a unique $\Omega \in {{\mathscr {O}}}$ with ${{\mathscr {D}}}(\Lambda )\subset \Omega ^\mathrm{{ra}}\cap X$, and if $\Lambda \cap \Omega =\varnothing$, then $\Omega ={{\,\mathrm{\textsf{Int}}\,}}(\Lambda )$.

• (3.1b) Components of  ${{\,\mathrm{\textsf{int}}\,}}(X)$. For distinct $\Omega _1,\Omega _2\in {{\mathscr {O}}}$, ${{\,\mathrm{\textsf{card}}\,}}(\Omega ^\mathrm{{ra}}_1\cap \Omega ^\mathrm{{ra}}_2\cap X)\le 1$.

• (3.1c) Unique length boundary points. For each $\Omega \in {{\mathscr {O}}}$, $z\in {\partial ^\textrm{ra}}\Omega \cap X \implies {{\,\mathrm{\textsf{card}}\,}}\iota _\Omega ^{-1}(z)=1$.⁵

Proof of ()

Since X is simply connected, ${{\mathscr {D}}}(\Lambda )\subset X$, so $D:={{\,\mathrm{\textsf{Int}}\,}}(\Lambda )\subset X$ and there is an $\Omega \in {{\mathscr {O}}}$ with $D\subset \Omega$. Evidently, ${{\mathscr {D}}}(\Lambda )={{\bar{D}}}=D^\textrm{ra}\subset \Omega ^\mathrm{{ra}}\cap X$; see Sect. 2.3.1. Fix a point $o\in \textsf{D}$. Given $p\in \Omega$, let $\alpha :o\curvearrowright p$ in $\Omega$. If $\Lambda \cap \Omega =\varnothing$, then $\alpha \cap \Lambda =\varnothing$, so $\alpha \subset D$ whence $p\in D$ and $\Omega =D$. $\square$

Proof of ()

Let a, b be distinct points in $\Omega ^\mathrm{{ra}}_1\cap \Omega ^\mathrm{{ra}}_2\cap X$ for some $\Omega _1,\Omega _2\in {{\mathscr {O}}}$. For $j\in \left\{ 1,2\right\}$, pick rectifiable arcs $\alpha _j:a\curvearrowright b$ in $\Omega _j\cup \left\{ a,b\right\}$. Since , , so we may assume that $\alpha _1\cap \alpha _2=\left\{ a,b\right\}$. Then $\Lambda :=\alpha _1\star \alpha _2^{-1}$ is a rectifiable Jordan loop in X, so by (3.1a) there is a unique $\Omega \in {{\mathscr {O}}}$ with ${{\mathscr {D}}}(\Lambda )\subset \Omega ^\mathrm{{ra}}\cap X$.

Fix a point $o\in D:={{\,\mathrm{\textsf{Int}}\,}}(\Lambda )$ and points . Then for each $j\in \left\{ 1,2\right\}$, any $z\in \Omega _j$ can be joined to $z_j$ (by a rectifiable path in $\Omega _j$) and then to o (by a rectifiable path in $\Omega \cup \left\{ z_j\right\}$), so there is a rectifiable path $z\curvearrowright o$ in ${{\,\mathrm{\textsf{int}}\,}}(X)$. It follows that $\Omega _1=\Omega =\Omega _2$. $\square$

Proof of ()

Suppose $z\in {\partial ^\textrm{ra}}\Omega \cap X$ for some $\Omega \in {{\mathscr {O}}}$. Let $\alpha ,\beta$ be rectifiable arcs in $\Omega \cup \left\{ z\right\}$ both having z an endpoint. We show that $\alpha$ and $\beta$ determine the same point in $\partial _l\Omega$.

Assume $[0,\ell (\alpha )]\xrightarrow {\alpha }\Omega \cup \left\{ z\right\}$ and $[0,\ell (\beta )]\xrightarrow {\beta }\Omega \cup \left\{ z\right\}$ with $\alpha (0)=z=\beta (0)$. We verify that $\lim _{s\rightarrow 0+} l\bigl (\alpha (s),\beta (s)\bigr )=0$.

First, suppose that for all $\upsilon \in \bigl (0,\ell (\alpha )\wedge \ell (\beta )\bigr )$, $\alpha \bigl ((0,\upsilon )\bigr )\cap \beta \bigl ((0,\upsilon )\bigr )\ne \varnothing$. Given such an $\upsilon$, pick $\sigma ,\tau \in (0,\upsilon )$ with $\alpha (\sigma )=\beta (\tau )$. Then for any $s\in (0,\upsilon )$,

$$\begin{aligned} l\bigl (\alpha (s),\beta (s)\bigr ) \le l\bigl (\alpha (s),\alpha (\sigma )\bigr ) + l\bigl (\alpha (\sigma ),\beta (\tau )\bigr ) + l\bigl (\beta (\tau ),\beta (s)\bigr ) \le 2\upsilon < \varepsilon \end{aligned}$$

provided $\upsilon <2\varepsilon$.

Otherwise, we may assume $\alpha \cap \beta =\left\{ z\right\}$. Let $\gamma$ be a rectifiable arc in $\Omega$ from the terminal point a of $\alpha$ to the terminal point b of $\beta$. Then $\alpha \star \beta ^{-1}$ and $\gamma$ are arcs in $\Gamma (a,b)$ and $z\in (\alpha \star \beta ^{-1}){\setminus }\gamma$, so there is a rectifiable Jordan loop $\Lambda$ in X that contains an open subarc of $\alpha \star \beta ^{-1}$ which in turn contains z. Evidently, ${{\,\mathrm{\textsf{Int}}\,}}(\Lambda )\subset \Omega$ and ${{\mathscr {D}}}(\Lambda )\subset \Omega ^\mathrm{{ra}}\cap X$, and thus $\lim _{s\rightarrow 0+} l\bigl (\alpha (s),\beta (s)\bigr )=0$. $\square$

3.2.1 Entry and Exit Points

Let p, q be distinct points in X and $\Omega \in {{\mathscr {O}}}$. We say that $\gamma \in \Gamma (p,q)$ enters $\Omega$ if ${{\,\mathrm{\textsf{card}}\,}}(\gamma \cap \Omega ^\mathrm{{ra}})\ge 2$. We employ the following crucial facts.

• (3.2a) Both points in $\Omega ^\mathrm{{ra}}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \Omega ^\mathrm{{ra}}\bigr )=2 \implies \forall \;\gamma \in \Gamma (p,q), \;\gamma \subset \Omega ^\mathrm{{ra}}.$

• (3.2b) One point in $\Omega ^\mathrm{{ra}}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \Omega ^\mathrm{{ra}}\bigr )=1 \implies \exists \; e:=e_\Omega \in {\partial ^\textrm{ra}}\Omega \cap X$ such that $\forall \;\gamma \in \Gamma (p,q)$

  $$\begin{aligned}&\left\{ p,q\right\} \cap \Omega ^\mathrm{{ra}}=\left\{ p\right\} \implies e\in \gamma ,\;\gamma [p,e]\subset \Omega ^\mathrm{{ra}}, \;\text {and}\; \gamma (e,q]\cap \Omega ^\mathrm{{ra}}=\varnothing , \\&\left\{ p,q\right\} \cap \Omega ^\mathrm{{ra}}=\left\{ q\right\} \implies e\in \gamma ,\;\gamma [e,q]\subset \Omega ^\mathrm{{ra}}, \;\text {and}\; \gamma [p,e)\cap \Omega ^\mathrm{{ra}}=\varnothing . \end{aligned}$$

• (3.2c) Neither point in $\Omega ^\mathrm{{ra}}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \Omega ^\mathrm{{ra}}\bigr )=0 \implies$ if some arc in $\Gamma (p,q)$ enters $\Omega$, then $\exists \;a:=a_\Omega ,b:=b_\Omega \in {\partial ^\textrm{ra}}\Omega \cap X$ such that $\forall \;\gamma \in \Gamma (p,q)$

  $$\begin{aligned} a,b\in \gamma ,\; \gamma [a,b]\subset \Omega ^\mathrm{{ra}},\; \text { and } \bigl (\gamma [p,a)\cup \gamma (b,q]\bigr )\cap \Omega ^\mathrm{{ra}}=\varnothing . \end{aligned}$$

The points a, b in (3.2c) (and e in (3.2b)) are called entry, exit points (respectively) for $\Omega$ relative to p, q. These entry, exit points depend only on p, q, and $\Omega$.

Proof of

Assume $p,q\in \Omega ^\mathrm{{ra}}\cap X$ and let $\gamma \in \Gamma (p,q)$. We show that $\gamma \subset \Omega ^\mathrm{{ra}}$.

Let $\alpha :p\curvearrowright q$ be a rectifiable arc in $\Omega \cup \left\{ p,q\right\} \subset \Omega ^\mathrm{{ra}}\cap X$. Suppose there is a point $o\in \gamma {\setminus }\alpha$. As discussed in the third paragraph at the beginning of this subsection, there are points $a,b\in \alpha \cap \gamma$ such that $p\le a<b\le q$ along both $\alpha$ and $\gamma$ with $\Lambda :=\gamma [a,b]\star \alpha ^{-1}[b,a]$ a rectifiable Jordan loop in X and with $o\in \gamma (a,b)$.

By (3.1a) there is a unique $\Omega _o\in {{\mathscr {O}}}$ with ${{\mathscr {D}}}(\Lambda )\subset \Omega ^\mathrm{{ra}}_o\cap X$. We claim that $D:={{\,\mathrm{\textsf{Int}}\,}}(\Lambda )\subset \Omega$, so $\Omega =\Omega _o$ and $o\in \Omega ^\mathrm{{ra}}$, and as o is an arbitrary point of $\gamma {\setminus }\alpha$, $\gamma \subset \Omega ^\mathrm{{ra}}$ as asserted.

Let $z\in D$ and fix any point $c\in \alpha (a,b)$. Since $\Lambda ={\partial D}$ is rectifiable, there is a rectifiable arc $\beta :z\curvearrowright c$ in $D\cup \left\{ c\right\}$. As $c\in \Omega \subset {{\,\mathrm{\textsf{int}}\,}}(X)$ and $D={{\,\mathrm{\textsf{Int}}\,}}(\Lambda )\subset \Omega _o\subset {{\,\mathrm{\textsf{int}}\,}}(X)$, $\beta \subset {{\,\mathrm{\textsf{int}}\,}}(X)$ and so $z\in \beta \subset \Omega$. $\square$

The proof of (3.2b) is similar to, but easier than, the proof of (3.2c) and so left to the reader.

Proof of

Assume $\gamma \in \Gamma (p,q)$ enters $\Omega$. There are distinct points $p_o,q_o\in \gamma \cap \Omega ^\mathrm{{ra}}$ and we label these so that $p<p_o<q_o<q$ along $\gamma$. According to (3.2a), $\gamma [p_o,q_o]\subset \Omega ^\mathrm{{ra}}$. Roughly speaking, a, b are the endpoints of the maximal subarc of $\gamma$ that contains $\gamma [p_o,q_o]$ and lies in $\Omega ^\mathrm{{ra}}$. Some care is required because $\Omega ^\mathrm{{ra}}$ need not be closed in ${\mathbb {C}}$ nor in X.

The set $A{:=}\left\{ z{\in }\gamma [p,p_o]\;\big |\;z{\in }\Omega ^\mathrm{{ra}}\right\}$ is non-empty and bounded below, so it has a greatest lower bound a. Similarly, there is a least upper bound b for $B:=\left\{ z\in \gamma [q_o,q]\;\big |\;z\in \Omega ^\mathrm{{ra}}\right\}$. Clearly $a,b\in \gamma \cap {\partial \Omega }$, $\bigl (\gamma [p,a)\cup \gamma (b,q]\bigr )\cap \Omega ^\mathrm{{ra}}=\varnothing$, and it is not difficult to check that $\gamma (a,b)\subset \Omega ^\mathrm{{ra}}$.

To corroborate that $a,b\in {\partial ^\textrm{ra}}\Omega$, we employ (3.1c) in conjunction with Lemma 2.6 as follows. Since $\gamma (a,b)\subset \Omega ^\mathrm{{ra}}\cap X$, it lies in the image of the isometry $\iota ^1_\Omega :\Omega _l^1\rightarrow \Omega ^\mathrm{{ra}}_l$. Thus $\tilde{\gamma }:=(\iota ^1_\Omega )^{-1}{\circ }\gamma (a,b)$ is a rectifiable arc in $\Omega _l^1$ and so has endpoints ${\tilde{a}},{\tilde{b}}$ that we label to have $\iota _\Omega$ images a, b. Thus $a=\iota _\Omega ({\tilde{a}}),b=\iota _\Omega ({\tilde{b}})\in \iota _\Omega (\partial _l\Omega )={\partial ^\textrm{ra}}\Omega$.

Let $\beta \in \Gamma (p,q)$. The path $\gamma ^{-1}[a,p]\star \beta \star \gamma ^{-1}[q,b]$ contains a rectifiable arc $a\curvearrowright b$ that must lie in $\Omega ^\mathrm{{ra}}$. Since $\bigl (\gamma [p,a)\cup \gamma (b,q]\bigr )\cap \Omega ^\mathrm{{ra}}=\varnothing$, it must be that $a,b\in \beta$, so $\beta [a,b]\subset \Omega ^\mathrm{{ra}}$. If, e.g., there were a point $c\in \beta (b,q]\cap \Omega ^\mathrm{{ra}}$, then letting d be the first point of $\beta [c,q]$ in $\gamma [b,q]$ would give an arc $\gamma [b,d]\star \beta ^{-1}[d,c]$, but $b,c\in \Omega ^\mathrm{{ra}}$ would imply $\gamma [b,d]\subset \Omega ^\mathrm{{ra}}$ violating our choice of b. Similarly $\beta [p,a)\cap \Omega ^\mathrm{{ra}}=\varnothing$. $\square$

Here is a noteworthy consequence of (3.1c) and (3.2a):

$$\begin{aligned} \forall \;\Omega \in {{\mathscr {O}}}, \quad \text {there is an isometric embedding}\;\; \overline{\Omega _l}\xrightarrow {h_\Omega }\overline{X_l}\text { with } \iota _X{\circ } h_\Omega =\iota _\Omega . \end{aligned}$$

(3.3)

Proof of (3.3)

The identity map induces a 1-Lipschitz embedding which then has a 1-Lipschitz extension $h_\Omega :\overline{\Omega _l}\rightarrow \overline{X_l}$. We explain why ${{\,\mathrm{{\textsf{id}}}\,}}_{\Omega _l}$ is an isometric embedding.

Fix $a,b\in \Omega$ and let $\gamma \in \Gamma (a,b)$. By (3.2a), $\gamma \subset \Omega ^\mathrm{{ra}}\cap X$, so by (3.1c) $\gamma$ lies in the image of the isometry $\iota ^1_\Omega :\Omega _l^1\rightarrow \Omega ^\mathrm{{ra}}_l$; see Lemma 2.6. Thus $\tilde{\gamma }:=(\iota ^1_\Omega )^{-1}{\circ }\gamma$ is a rectifiable arc in $\Omega _l^1$ with $\ell (\gamma )=\ell (\tilde{\gamma })\ge l_\Omega (a,b)$. Taking an infimum over all such arcs $\gamma$, and using the fact that ${{\,\mathrm{{\textsf{id}}}\,}}_{\Omega _l}$ is 1-Lipschitz, we now obtain $l_X(a,b)=l_\Omega (a,b)$. It now follows that $h_\Omega$ is an isometric embedding.

Evidently, $h_\Omega (z)=z$ for $z\in \Omega$. Suppose $\zeta \in \partial _l\Omega$. Let $[0,1)\xrightarrow {\alpha }\Omega$ be a rectifiable path that represents $\zeta$. According to Lemma 2.4, $\alpha$ extends to rectifiable arcs $\tilde{\alpha }$ in $\overline{\Omega _l}$ and ${\bar{\alpha }}=\iota _\Omega {\circ }\tilde{\alpha }$ in $\Omega ^\mathrm{{ra}}\subset {{\bar{\Omega }}}$ with $\zeta =\tilde{\alpha }(1)$ and $z={\bar{\alpha }}(1)=\iota _\Omega (\zeta )\in {\partial ^\textrm{ra}}\Omega$.

But, $\alpha ={{\,\mathrm{{\textsf{id}}}\,}}_{\Omega _l}{\circ }\alpha$ is also a rectifiable arc in X and so has extensions ${\bar{\alpha }}$ in $X^\mathrm{{ra}}$ (i.e., $z\in X^\mathrm{{ra}}$) and $\alpha _X$ in $\overline{X_l}$. If $z\in X$, then $z\in \Omega ^\mathrm{{ra}}\cap X$, so $\left\{ \zeta \right\} =\iota _\Omega ^{-1}(z)$, $h_\Omega (\zeta )=z$, and $\iota _X\bigl (h_\Omega (\zeta )\bigr )=\iota _X(z)=z=\iota _\Omega (\zeta )$. If $z\in {\partial ^\textrm{ra}} X$, then $\xi :=\alpha _X(1)\in \partial _l X$, $z=\iota _X(\xi )$, and $h_\Omega (\zeta )=\xi$, so $\iota _X\bigl (h_\Omega (\zeta )\bigr )=\iota _X(\xi )=z=\iota _\Omega (\zeta )$. $\square$

To simplify notation, often we identify $\overline{\Omega _l}$ with its image $h_\Omega (\overline{\Omega _l})\subset \overline{X_l}$, but we must remember that some points⁶ in $\partial _l\Omega$ may lie in X (and so not in $\partial _l X$).

We require similar information to deal with points in $\partial _l X$. Given distinct points p, q in $\overline{X_l}$, define

$$\begin{aligned} \Gamma _l(p,q):=\left\{ \text {all rectifiable arcs}\;\gamma :p\curvearrowright q\;\text {in} X\cup \left\{ p,q\right\} \right\} \ne \varnothing \end{aligned}$$

and

$$\begin{aligned} {\bar{\Gamma }}_l(p,q):=\left\{ \text {all rectifiable arcs}\;\gamma :p\curvearrowright q \;\text {in }\overline{X_l}\right\} \ne \varnothing . \end{aligned}$$

The arcs in $\Gamma _l(p,q)$ are easier to work with (and all we need to compute l(p, q)), but facts about ${\bar{\Gamma }}_l(p,q)$ will help us establish uniqueness of l-geodesics.

Let p, q be distinct points in $\overline{X_l}$ and $\Omega \in {{\mathscr {O}}}$. The arcs in ${\bar{\Gamma }}(p,q)$ also have unique entry, exit points as in (3.2c); here $\gamma$ enters $\overline{\Omega _l}$ if ${{\,\mathrm{\textsf{card}}\,}}(\gamma \cap \overline{\Omega _l})\ge 2$. We identify $\overline{\Omega _l}$ with its image $h_\Omega (\overline{\Omega _l})\subset \overline{X_l}$.

• (3.4a) Both points in $\overline{\Omega _l}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \overline{\Omega _l}\bigr )=2 \implies \forall \;\gamma \in {\bar{\Gamma }}_l(p,q), \;\gamma \subset \overline{\Omega _l}$.

• (3.4b) One point in $\overline{\Omega _l}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \overline{\Omega _l}\bigr )=1 \implies \exists \; e:=e_\Omega \in {\partial ^\textrm{ra}}\Omega \cap X \text { such that } \forall \;\gamma \in {\bar{\Gamma }}_l(p,q)$

  $$\begin{aligned}&\left\{ p,q\right\} \cap \overline{\Omega _l}=\left\{ p\right\} \implies e\in \gamma ,\; \gamma [p,e]\subset \overline{\Omega _l}, \;\text {and}\; \gamma (e,q]\cap \overline{\Omega _l}=\varnothing , \\&\left\{ p,q\right\} \cap \overline{\Omega _l}=\left\{ q\right\} \implies e\in \gamma ,\; \gamma [e,q]\subset \overline{\Omega _l}, \;\text {and}\; \gamma [p,e)\cap \overline{\Omega _l}=\varnothing . \end{aligned}$$

• (3.4c) Neither point in $\overline{\Omega _l}$.    ${{\,\mathrm{\textsf{card}}\,}}\bigl (\left\{ p,q\right\} \cap \overline{\Omega _l}\bigr )=0 \implies$ if some arc in ${\bar{\Gamma }}_l(p,q)$ enters $\overline{\Omega _l}$, then $\exists \;a:=a_\Omega ,b:=b_\Omega \in {\partial ^\textrm{ra}}\Omega \cap X$ such that $\forall \;\gamma \in {\bar{\Gamma }}_l(p,q)$

  $$\begin{aligned} a,b\in \gamma ,\; \gamma [a,b]\subset \overline{\Omega _l}, \text { and } \bigl (\gamma [p,a)\cup \gamma (b,q]\bigr )\cap \overline{\Omega _l}=\varnothing . \end{aligned}$$

Again, we call the points a, b in (3.4c) (and e in (3.4b)) entry, exit points (respectively) for $\overline{\Omega _l}$ relative to p, q. These entry, exit points depend only on p, q, and $\Omega$.

Proof of

Suppose $p,q\in \overline{\Omega _l}$ and $\gamma \in {\bar{\Gamma }}_l(p,q)$, but $\gamma \not \subset \overline{\Omega _l}$. By replacing $\gamma$ with an appropriate subarc, we may assume $\gamma \cap \overline{\Omega _l}=\left\{ p,q\right\}$. Let $p_1,q_1\in \gamma$ be the first points⁷ at distance $d:=\frac{1}{10}l(p,q)$ from p, q respectively. Put $\varepsilon :=d\wedge {{\,\mathrm{\textsf{dist}}\,}}_l(\gamma [p_1,q_1],\overline{\Omega _l})$.

Pick any points $p_0,q_0\in \Omega$ with $l(p_0,p)<\varepsilon , l(q_0,q)<\varepsilon$. Mimicking the proof of Lemma 2.2 gives us a rectifiable arc $\alpha :p_0\curvearrowright q_0$ in X with $\alpha \subset \textsf{N}_l(\gamma ;\varepsilon )$. According to (3.2a), $\alpha \subset \Omega ^\mathrm{{ra}}\cap X\subset \overline{\Omega _l}$. Since $l(p,q)<l(p_0,q_0)+2\varepsilon$,

$$\begin{aligned} l(p_0,q_0)\ge 10d-2\varepsilon . \end{aligned}$$

Fix a point $a\in \alpha$ with $l(a,p_0)=l(a,q_0)\ge \frac{1}{2}l(p_0,q_0)\ge 5d-\varepsilon$. Evidently, for all $b\in \gamma [p_1,q_1]$,

$$\begin{aligned} a\in \alpha \subset \overline{\Omega _l}\implies l(b,a)\ge {{\,\mathrm{\textsf{dist}}\,}}_l(b,\overline{\Omega _l})\ge {{\,\mathrm{\textsf{dist}}\,}}_l(\gamma [p_1,q_1],\overline{\Omega _l})\ge \varepsilon . \end{aligned}$$

Also, if $b\in \gamma [p,p_1]$, then

$$\begin{aligned} l(b,a)\ge l(p,a)-l(p,b)\ge l(p_0,a)-l(p_0,p)-l(p,b)\ge 4d-2\varepsilon \ge 2\varepsilon . \end{aligned}$$

Similarly, $b\in \gamma [q_1,q]\implies l(b,a)>\varepsilon$. This contradicts $\alpha \subset \textsf{N}_l(\gamma ;\varepsilon )$. $\square$

Items (3.4b) and (3.4c) now readily follow. To see that the entry and exit points lie in X, note that we can use arcs in $X\cup \left\{ p,q\right\}$ to determine these points.

3.2.2 Stable Points

Given distinct $p,q\in \overline{X_l}$, we call $x\in \overline{X_l}$ a (p,q)-stable point if x lies in every $\gamma \in {\bar{\Gamma }}_l(p,q)$. Let $\Sigma (p,q)$ be the set of all (p, q)-stable points. Evidently, p, q, and all entry and exit points associated with p, q belong to $\Sigma (p,q)$. It is not difficult to see that $\Sigma (p,q)$ is closed in $X\cup \left\{ p,q\right\}$, ordered via any arc in ${\bar{\Gamma }}(p,q)$, and $l(p,q)=l(p,x)+l(x,q)$ for any $x\in \Sigma (p,q)$. To see that all arcs induce the same ordering on any $x,y\in \Sigma (p,q)$: note that if $\beta ,\gamma \in {\bar{\Gamma }}_l(p,q)$ with $x<y, y<x$ along $\gamma ,\beta$, respectively, then $\gamma [p,x]\star \beta [x,q]$ is a path (which contains an arc) $p\curvearrowright q$ but avoids y contradicting $y\in \Sigma (p,q)$.

By Lemma 2.2, $x\in \Sigma (p,q)$ provided $x\in \gamma$ for all $\gamma \in \Gamma _l(p,q)$. Also

$$\begin{aligned} \Sigma (p,q)=\left\{ p,q\right\} \cup \bigcup _{x<y}\Sigma (x,y) \end{aligned}$$

where the union is over all $x,y\in X\cap \Sigma (p,q)$, and,

$$\begin{aligned} x,y\in \Sigma (p,q) \text { with } p\le x<y\le q \implies x\in \Sigma (p,y). \end{aligned}$$

Indeed, $x,y\in \gamma \in \Gamma _l(p,q)\text { and }\exists \; x\not \in \beta \in \Gamma _l(p,y)\implies x\not \in \beta \star \gamma [y,q]\in \Gamma _l(p,q)$.

Here are two especially useful facts.

(3.5a)

Consequently,

$$\begin{aligned} \Sigma (p,q)=\left\{ p,q\right\} \implies \exists \;\Omega \in {{\mathscr {O}}}\text { such that } p,q\in \overline{\Omega _l}\subset \overline{X_l}. \end{aligned}$$

(3.5b)

Proof Sketch for (3.5)

Suppose $\gamma \in \Gamma _l(p,q)$ and $z\in \gamma {\setminus }\Sigma (p,q)\subset X$. Pick an arc $\beta \in \Gamma _l(p,q)$ with $z\not \in \beta$. Again, we can construct a Jordan loop $\Lambda$ in X that contains an open subarc of $\gamma$ which in turn contains z, but a wee bit of care is required. Since the ends of $\beta ,\gamma$ both determine the same points $p,q\in \overline{X_l}$, there are points $p_1,p_2$ and $q_1,q_2$ on $\beta ,\gamma$ respectively, close in $X_l$, and as close to p, q as desired. Then $\gamma [p_1,q_1]\cup \beta [p_2,q_2]$, together with short arcs $p_1\curvearrowright p_2, q_1\curvearrowright q_2$ in X, forms a loop in X which contains the asserted Jordan loop $\Lambda$; see Lemma 2.3 for details. Now (3.5b) follows from (3.1a).

To corroborate (3.5b), start with any $\gamma \in \Gamma _l(p,q)$. By (3.5b), for each , there is an $\Omega _z\in {{\mathscr {O}}}$ and an arc $\alpha _z\subset \gamma \cap \Omega ^\mathrm{{ra}}_z$ with . If $w\in \alpha _z$, then , so by (3.1b) $\Omega _w=\Omega _z$. It now follows that there is a single $\Omega \in {{\mathscr {O}}}$ with , but then $\gamma \subset \overline{\Omega _l}$. $\square$

3.2.3 Contructing Geodesics

Let $p,q\in \overline{X_l}$. We exhibit an l-geodesic $p\curvearrowright q$ in $\overline{X_l}$.

Assume $p,q\in X$. Suppose there exists an $\Omega \in {{\mathscr {O}}}$ with $p,q\in \Omega ^\mathrm{{ra}}$. By (3.1c), there are unique points ${\tilde{p}},{\tilde{q}}\in \overline{\Omega _l}$ with $p=\iota _\Omega ({\tilde{p}}),q=\iota _\Omega ({\tilde{q}})$. By Sect. 3.1, there is a unique l-geodesic $\sigma _\Omega :{\tilde{p}}\curvearrowright {\tilde{q}}$ in $\overline{\Omega _l}$. Then by (3.3) and its proof, $\sigma _X:=h_\Omega {\circ }\sigma _\Omega$ is an l-geodesic in $\overline{X_l}$ with endpoints $h_\Omega ({\tilde{p}})=p$ and $h_\Omega ({\tilde{q}})=q$.

Now suppose that for all $\Omega \in {{\mathscr {O}}}$, $\left\{ p,q\right\} \not \subset \Omega ^\mathrm{{ra}}$. We construct a path $\sigma :p\curvearrowright q$ in $\overline{X_l}$ that has $\ell (\sigma )\le \ell (\gamma )$ for all $\gamma \in \Gamma (p,q)$. Thus $\sigma$ is a shortest path and hence an l-geodesic

If $p\in \Omega ^\mathrm{{ra}}_p$ for some $\Omega _p\in {{\mathscr {O}}}$, let $e_p\in {\partial ^\textrm{ra}}\Omega _p\cap X$ be the exit point associated with $q,\Omega _p$ as given in (3.2b) and let $\sigma _p$ be the $h_{\Omega _p}$ image of the l-geodesic ${\tilde{p}}\curvearrowright {\tilde{e}}_p$ in $\overline{(\Omega _p)_l}$ where ${\tilde{p}},{\tilde{e}}_p$ are the unique points in $\overline{(\Omega _p)_l}$ with $p=\iota _{\Omega _p}({\tilde{p}}),e_p=\iota _{\Omega _p}({\tilde{e}}_p)$. If no such $\Omega _p$ exists, put $\Omega _p:=\varnothing , e_p:=p, \sigma _p:=\left\{ p\right\}$. Define $\Omega _q,e_q,\sigma _q$ in a similar manner.

Let $\gamma \in \Gamma (p,q)$. Note that $\left\{ e_p,e_q\right\} \subset \gamma$. Suppose $z\in \gamma [e_p,e_q]{\setminus }\Sigma (p,q)$. By (3.5b), there is an $\Omega \in {{\mathscr {O}}}$ and an arc $\alpha \subset \gamma [e_p,e_q]\cap \Omega ^\mathrm{{ra}}$ with . Thus $\gamma$ enters $\Omega$ and so $z\in \gamma (a_\Omega ,b_\Omega )$ where $a_\Omega ,b_\Omega \in {\partial ^\textrm{ra}}\Omega \cap X$ are the entry, exit points associated with $\Omega$ as given in (3.2c).

It now follows that $\gamma [e_p,e_q]{\setminus }\Sigma (p,q)$ is a union of countably many $\gamma _n:=\gamma (a_n,b_n)$ where $a_n:=a_{\Omega _n},b_n:=b_{\Omega _n}$ are the entry, exit points (given by (3.2c)) associated with the countably many $\Omega _n$ that satisfy ${{\,\mathrm{\textsf{card}}\,}}(\gamma \cap \Omega ^\mathrm{{ra}}_n)\ge 2$ with $\Omega _p\ne \Omega _n\ne \Omega _q$. Note that $a_n,b_n\in {\partial ^\textrm{ra}}\Omega _n\cap X$ and these entry, exit points correspond to unique points ${\tilde{a}}_n,{\tilde{b}}_n\in \partial _l\Omega _n\subset \overline{(\Omega _n)_l}$.

For each n, let $\sigma _n:a_n\curvearrowright b_n$ in $\overline{X_l}$ be the $h_{\Omega _n}$ image of the l-geodesic ${\tilde{a}}_n\curvearrowright {\tilde{b}}_n$ in $\overline{(\Omega _n)_l}$. Replacing each of the subarcs $\gamma [p,e_p],\gamma [e_q,q],\gamma _n$ of $\gamma$ with $\sigma _p,\sigma _q,\sigma _n$, respectively, we obtain an arc

$$\begin{aligned} \sigma :=\sigma _p\cup \Sigma (p,q)\cup \sigma _q\cup \bigcup _n \sigma _n:p\curvearrowright q \quad \text {in}\, \overline{X_l}. \end{aligned}$$

Since the new subarcs have lengths no larger than the replaced subarcs, $\ell (\sigma )\le \ell (\gamma )$. Since the entry, exit points relative to p, q do not depend on $\gamma$, the construction of $\sigma$ is independent of $\gamma$ and $\sigma$ is indeed an arc $p\curvearrowright q$ in $\overline{X_l}$ with shortest length.

Assume $p\in X$ and $q\in \partial _l X$. Suppose $q\in \overline{\Omega _l}$ for some $\Omega \in {{\mathscr {O}}}$. Assume $p\not \in \Omega ^\mathrm{{ra}}$. By (3.4b), there is a unique exit point $e\in {\partial ^\textrm{ra}}\Omega \cap X$ (that depends only on p), and by earlier work, there are unique l-geodesics $\sigma _q:e\curvearrowright q$ in $\overline{\Omega _l}\subset \overline{X_l}$ and $\sigma _p:p\curvearrowright e$ in $\overline{X_l}$, and we see that $\sigma :=\sigma _p\star \sigma _q$ is a shortest arc $p\curvearrowright q$ in $\overline{X_l}$.

Suppose that for all $\Omega \in {{\mathscr {O}}}$, $q\not \in \overline{\Omega _l}$. Start with any $\gamma \in \Gamma _l(p,q)$ and let $(z_n)$ be an increasing sequence along $\gamma$ with $\ell (\gamma [z_n,q])\rightarrow 0$. If $z_n\in \Sigma (p,q)$, set $x_n:=z_n$. If $z_n\not \in \Sigma (p,q)$, then (3.5b) and (3.4c) provide entry, exit points $a_n,b_n\in \gamma \cap {\partial ^\textrm{ra}}\Omega _n$ with $z_n\in \gamma (a_n,b_n)$; here we set $x_n:=b_n$. Thus $(x_n)$ is an increasing sequence in $\Sigma (p,q)$ with $l(x_n,q)\rightarrow 0$ as $n\rightarrow +\infty$.

As $p,x_n\in X$, there are l-geodesics $\sigma _n:p\curvearrowright x_n$ in $\overline{X_l}$. However, $x_n\in \Sigma (p,x_{n+1})$, so $\sigma _n\subset \sigma _{n+1}$.⁸ Therefore, it follows that $\sigma :=\bigcup _{n\ge 1}\sigma _n$ is a rectifiable arc in $\overline{X_l}$ with terminal endpoint q and with $\ell (\sigma )=l(p,q)$. Thus $\sigma$ is a shortest arc, hence an l-geodesic in $\overline{X_l}$.

Assume $p,q\in \partial _l X$. If $\Sigma (p,q)=\left\{ p,q\right\}$, then by (3.5b) $p,q\in \overline{\Omega _l}\subset \overline{X_l}$ for some $\Omega \in {{\mathscr {O}}}$ and hence there is an l-geodesic joining these points. Suppose there exists an $x\in X\cap \Sigma (p,q)$. Then by a previous case there are l-geodesics $p\curvearrowright x$ and $x\curvearrowright q$ which paste together to give a path $\sigma :p\curvearrowright q$ in $\overline{X_l}$ with $\ell (\sigma )=l(p,x)+l(x,q)=l(p,q)$.

3.2.4 Uniqueness and CAT(0)

Our penultimate task is to verify uniqueness of l-geodesics in $\overline{X_l}$. Let p, q be distinct points in $\overline{X_l}$, let $\sigma :p\curvearrowright q$ be the l-geodesic in $\overline{X_l}$ constructed above, and suppose $\psi :p\curvearrowright q$ is also an l-geodesic in $\overline{X_l}$. Then

$$\begin{aligned} \sigma \cap \Sigma (p,q)=\Sigma (p,q)=\psi \cap \Sigma (p,q). \end{aligned}$$

Let $z\in \sigma {\setminus }\Sigma (p,q)$.

Appealing to (3.5b) we obtain an $\Omega \in {{\mathscr {O}}}$ and an arc $\alpha \subset \sigma \cap \Omega ^\mathrm{{ra}}$ with . This means that ${{\,\mathrm{\textsf{card}}\,}}(\sigma \cap \overline{\Omega _l})\ge 2$, so by (3.4c) $z\in \sigma [a,b]\subset \overline{\Omega _l}$ where $a,b\in {\partial ^\textrm{ra}}\Omega \subset \overline{X_l}$ are the entry, exit points associated with $p,q,\Omega$. Also by (3.4c), $a,b\in \psi$. Since $\psi [a,b]\subset \overline{\Omega _l}$ is a shortest arc and $\overline{\Omega _l}$ is CAT(0) (by Sect. 3.1), it must be that $\psi [a,b]=\sigma [a,b]$.

By symmetry it now follows that $\sigma =\psi$.

Finally, we confirm the CAT(0) property for $\overline{X_l}$. Let $\Delta :=[a,b,c]_l=[a,b]_l\cup [b,c]_l\cup [c,a]_l$ be a geodesic triangle in $\overline{X_l}$. By Fact 2.7 we may assume that $\Delta$ is tail-less. Since l-geodesics in $\overline{X_l}$ are unique, this means that $\Delta$ is a rectifiable Jordan loop in $\overline{X_l}$.

Since $[a,b]_l\cap \bigl ([b,c]_l\star [c,a]_l\bigr )=\left\{ a,b\right\}$, $\Sigma (a,b)=\left\{ a,b\right\}$. Now (3.5b) produces an $\Omega \in {{\mathscr {O}}}$ with $a,b\in \overline{\Omega _l}\subset \overline{X_l}$ and so $[a,b]_l\subset \overline{\Omega _l}$. Similarly, $\Sigma (b,c)=\left\{ b,c\right\}$ and $\Sigma (c,a)=\left\{ c,a\right\}$, so $[b,c]_l\cup [c,a]_l\subset \overline{\Omega _l}$. Thus $\Delta \subset \overline{\Omega _l}$ and therefore $\Delta$ satisfies the CAT(0) vertex angle criterion.

3.3 Gromov Hyperbolicity Proof

Clearly, if X contains Euclidean disks of arbitrarily large radius, then $\overline{X_l}$ is not Gromov hyperbolic. Suppose

$$\begin{aligned} R:=\sup \left\{ r>0 \;\big \vert \; \exists \, \textsf{D}(x;r)\subset X\right\} <+\infty . \end{aligned}$$

We show that X is 2R-hyperbolic, and that this is best possible.

Let $\Delta =[a,b,c]_l$ be a geodesic triangle in $\overline{X_l}$. We may assume that $\Delta$ is tail-less; therefore, as explained immediately above, $\Delta \subset \overline{\Omega _l}$ for some $\Omega \in {{\mathscr {O}}}$.

There is a simply connected $\Omega _o\subset \Omega$ with ${\partial \Omega }_o=\iota (\Delta )$; see the last paragraph of Sect. 2.3.1. Let $D_o:=\textsf{D}(o;r)$ be a maximal open disk in $\Omega _o$. Then ${\partial D}_o\cap {\partial \Omega }_o$ either consists of two antipodal points or has cardinality at least three. Below we verify that ${{\,\mathrm{\textsf{card}}\,}}({\partial D}_o\cap {\partial \Omega }_o)=3$.

Notice that ${{\bar{D}}}_o=D_o\cup {\partial D}_o$ isometrically embeds into $\overline{\Omega _l}\hookrightarrow \overline{X_l}$ so $S_l:=\partial _l D_o$ is a Euclidean circle in $\overline{X_l}$ that bounds a Euclidean disk $D_l$ in $\overline{X_l}$. Note too that $S_l\cap \Delta \subset \overline{\Omega _l}$ is isometrically equivalent to $\partial D_o\cap {\partial \Omega }_o$; one way to see this is to use the conformal model for $\overline{\Omega _l}$ (see the last paragraph of Sect. 2.3.1) in conjunction with [10, Prop. 2.14, Cor.2.17, pp. 29,35].

So, ${{\,\mathrm{\textsf{card}}\,}}(S_l\cap \Delta )\ge 2$; we show that $S_l\cap \Delta =\left\{ a_o,b_o,c_o\right\}$ with $a_o\in (b,c)_l,b_o\in (c,a)_l,c_o\in (a,b)_l$.

First, let $E\in \left\{ [a,b]_l,[b,c]_l,[c,a]_l\right\}$ be an edge of $\Delta$. If $E\cap S_l$ contained two distinct points x, y, then the Euclidean segment $[x,y]\ne E[x,y]$ would be an l-geodesic $x\curvearrowright y$ in $\overline{\Omega _l}$ which would violate unique geodesicity for $\overline{\Omega _l}$; thus

$$\begin{aligned} {{\,\mathrm{\textsf{card}}\,}}(E\cap S_l)\le 1. \end{aligned}$$

(3.6a)

Also,

$$\begin{aligned} \exists \; p\in E\cap S_l \quad \implies \quad \forall \; q\in E{\setminus }\left\{ p\right\} , \;\; \measuredangle _p(q,o)\ge \frac{\pi }{2}. \end{aligned}$$

(3.6b)

Indeed, E is a complete convex subspace of $\overline{\Omega _l}$, so any $p\in E\cap S_l$ is the unique point of E nearest to o, and so (3.6b) follows from [4, Prop. II.2.4(3), p. 177]. Here $\measuredangle _p(q,o)=\measuredangle _p(E[p,q],[p,o])$.

Thus $2\le {{\,\mathrm{\textsf{card}}\,}}(S_l\cap \Delta )\le 3$. Suppose $S_l\cap \Delta =\left\{ p,q\right\}$. Then $\iota (p),\iota (q)$ are antipodal points of ${\partial D}_o$, so the Euclidean segment [p, q] is the l-geodesic $p\curvearrowright q$ in $\overline{\Omega _l}$. Now p, q are not both vertices of $\Delta$ (otherwise (3.6a) would be violated) so we can select a vertex, say a, of $\Delta$ so that

$$\begin{aligned} p\in [a,b]_l, \; q\in [a,c]_l, \; p\ne a\ne q \ne c \text { but maybe } p=b. \end{aligned}$$

According to (3.6b), $\measuredangle _p(a,o)\ge \pi /2$ and $\measuredangle _q(a,o)\ge \pi /2$. However, the comparison triangle ${\bar{\Delta }}(a,p,q)$ (in $\mathbb {C}$) cannot have two vertex angles that are both of size $\pi /2$ or larger.

Thus ${{\,\mathrm{\textsf{card}}\,}}(S_l\cap \Delta )=3$. Employing (3.6a) again we see that $S_l\cap \Delta \cap \left\{ a,b,c\right\} =\varnothing$. It now follows that there are points $a_o\in (b,c)_l,b_o\in (c,a)_l,c_o\in (a,b)_l$ with $S_l\cap \Delta =\left\{ a_o,b_o,c_o\right\}$.

Take any point $z\in \Delta$, say $z\in [a,b]_l$, or even $z\in [a,c_o]_l$. Look at a comparison triangle $\Delta (\bar{a},\bar{c}_o,\bar{b}_0)$ for $[a,c_o,b_o]_l$ and pick a point $w\in [a,b_o]_l$ so that the Euclidean segment $[\bar{z},\bar{w}]$ is parallel to $[\bar{c}_o,\bar{b}_o]$. Now we see that

$$\begin{aligned} l(z,w) \le |\bar{z}-\bar{w}| \le |\bar{c}_o-\bar{b}_o| = l(c_o,b_o) = |\iota (c_o)-\iota (b_o)| \le 2r\le 2R \end{aligned}$$

and therefore $\overline{X_l}$ is 2R-hyperbolic.

To see that this is best possible, fix $R>0$ and consider the set

$$\begin{aligned} X:=\left\{ z\in {\mathbb C}\;\big \vert \; |{{\,\mathrm{{\text {Im}}}\,}}(z)|\le R\right\} ; \end{aligned}$$

the points $a:=t, b:=Ri, c:=-Ri$ where $t>2R$; and $\Delta :=[a,b,c]=[a,b]\cup [b,c]\cup [c,a]$. Let $2\varphi$ be the vertex angle for $\Delta$ at a; i.e., the angle between the edges [a, b] and [a, c]. Pick $y\in (0,R)$ so that $z:=2R+iy\in [a,b]$. Then

$$\begin{aligned} \frac{t}{H} = \cos \varphi = \frac{2R}{h} \quad \text {and}\quad \frac{R}{H} = \sin \varphi = \frac{R-y}{h} \end{aligned}$$

where

$$\begin{aligned} H:=|a-b|=\sqrt{t^2+R^2} \quad \text {and}\quad h:=|z-b|=\frac{2RH}{t}. \end{aligned}$$

Now $\dfrac{{{\,\mathrm{\textsf{dist}}\,}}(z,[a,c])}{H-h}=\sin 2\varphi =2\dfrac{tR}{H^2}$, so

$$\begin{aligned} {{\,\mathrm{\textsf{dist}}\,}}(z,[a,c])=2R\frac{t}{H}\frac{H-h}{H}= 2R\frac{t}{\sqrt{t^2+R^2}}\Bigl (1-\frac{2R}{t}\Bigr ) \rightarrow 2R \text { as } t\rightarrow +\infty . \end{aligned}$$

Since ${{\,\mathrm{\textsf{dist}}\,}}(z,[b,c])=2R$, we see that X is $\delta$-hyperbolic for $\delta :=2R$ but no smaller $\delta$ works. $\square$