Hyperbolic distance versus quasihyperbolic distance in plane domains

By David A. Herron and Jeff Lindquist

Dedicated to David Minda, for decades of interesting discussions.

Abstract

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

1. Introduction

Throughout this section denotes a hyperbolic plane domain: is open and connected and contains at least two points. Each such carries a unique maximal constant curvature -1 conformal metric usually referred to as the Poincaré hyperbolic metric on . The length distance induced by is called hyperbolic distance in . There is also a quasihyperbolic metric on , whose length distance is called quasihyperbolic distance in ; here is the Euclidean distance from to the boundary of . See §2.C for more details.

This work continues that begun in Reference BH20, Reference Her21a, Reference Her21b where we elucidate the geometric similarities and metric differences between the metric spaces and . Our first result, Theorem A below, characterizes when the metric spaces and are quasisymmetrically equivalent. Theorem B reveals that, when these spaces are Gromov hyperbolic, their Gromov boundaries are always quasisymmetrically equivalent.

To set the stage, we begin with some preliminary observations. A straightforward, albeit non-trivial, argument reveals that the metric spaces and are isometric if and only if is an open half-plane and the isometry is the restriction of a Möbius transformation. Furthermore, these metric spaces are bi-Lipschitz equivalent if and only if the identity map is bi-Lipschitz; see §2.C.3.

It is well-known that the identity map enjoys the following properties:

The map is a 2-Lipschitz -quasiconformal homeomorphism.

For any simply connected , is 2-bi-Lipschitz.⁠Footnote1

1

That is 2-Lipschitz is a consequence of Koebe’s One Quarter Theorem.

In general, is bi-Lipschitz if and only if is uniformly perfect.⁠Footnote2

2

The bi-Lipschitz and uniformly perfectness constants depend only on each other.

The last item above is due to Beardon and Pommerenke; see Reference BP78 and §2.C.3.

Our interest is in general (non-simply connected) hyperbolic plane domains where there may be no simple metric control on . For example, given any sequences and of positive numbers with say and , there are sequences of points in the punctured unit disk with hyperbolic and quasihyperbolic distances and . See Reference BH20, Ex. 2.7.

The following striking rigidity theorem contains our first main result. (See §2.A for mapping definitions.) This says that the metric spaces and are either “quite similar” (i.e., bi-Lipschitz equivalent) or “quite different” (i.e., not quasisymmetrically equivalent).

Theorem A.

For any hyperbolic plane domain , the following are quantitatively equivalent.

(A.1)

The metric spaces and are quasisymmetrically equivalent.

(A.2)

The metric spaces and are bi-Lipschitz equivalent.

(A.3)

The identity map is bi-Lipschitz.

(A.4)

is uniformly perfect.

Again, Beardon and Pommerenke Reference BP78 established the equivalence of (A.3) and (A.4).

Recently, the first author and Buckley Reference BH20, Theorem B demonstrated that and are simultaneously Gromov hyperbolic or not, and we call Gromov hyperbolic in the former case. Our next result stands in stark contrast to Theorem A. Even when and are not quasisymmetrically equivalent, the large scale geometry is the same in both spaces—at least when they are Gromov hyperbolic. This further enhances Reference BH20, Theorem A where we proved that these metric spaces have the same quasi-geodesic curves.

Theorem B.

For any Gromov hyperbolic plane domain , the canonical conformal gauges on the Gromov boundaries and are naturally quasisymmetrically equivalent.

If and are quasiisometrically⁠Footnote3 equivalent, the above Gromov boundary equivalence is known and given via a power quasisymmetry; see Reference BS00, Theorem 6.5(2). Whereas our proof of Theorem A is surprisingly simple, the proof of Theorem B employs significant machinery as explained in the first paragraph of §3.B.

3

Our quasiisometries are sometimes called rough bi-Lipschitz maps.

Again, when is uniformly perfect, and are bi-Lipschitz equivalent. A natural conjecture is that this uniform perfectness might be a necessary condition for and to be quasiisometrically equivalent. However, our next result reveals that this is not the case.

Theorem C.

For any finitely connected hyperbolic plane domain , the metric spaces and are quasiisometrically equivalent.

The above is just an easy to state special case of our more general Theorem 3.9 which provides a large class of plane domains whose hyperbolizations and quasihyperbolizations are quasiisometrically equivalent. This raises the natural question of whether the conclusion of Theorem C could be true in general, and we answer this below. Note that a quasiisometry can have an arbitrarily large additive rough constant and this obstacle must be overcome.

Theorem D.

There are uniform (hence Gromov) hyperbolic plane domains with the property that any quasisymmetric equivalence between and , e.g., that given by Theorem B, is not via a power quasisymmetry. In particular, and are not quasiisometrically equivalent.

Domains that satisfy Theorem D include any , where, , and is a strictly decreasing sequence in with as

Section 2 contains the usual definitions and terminology; especially, see §2.C.1 and §2.C.2 for details about the hyperbolic and quasihyperbolic metrics. We prove Theorems A, B, C, D in §§3.A, 3.B, 3.C, 3.D respectively.

2. Preliminaries

We work in the Euclidean plane, and on the Riemann sphere, which we identify, respectively, with the complex number field and its one-point extension . Everywhere is a domain⁠Footnote4 (i.e., an open connected set) and and denote the boundary of with respect to the plane and sphere (respectively). Always, is a hyperbolic domain, i.e., contains at least three points.

4

Our interest is in non-simply connected domains primarily with non-uniformly perfect.

We write and for the closure and boundary of a set in whereas and are the Euclidean closure and boundry of .

We write to indicate a constant that depends only on the data , …. In some cases we write to indicate that for some computable constant that depends only on the relevant data, and means .

The Euclidean line segment joining two points is , and . The open and closed Euclidean disks, and the circle, centered at the point and of radius , are denoted by and and respectively, and is the unit disk. We also define

the definition of is for distinct points , in .

The quantity is the Euclidean distance from to the boundary of , and is the scaling factor (aka, metric-density) for the so-called quasihyperbolic metric on ; see §2.C.1. We use the notation

for the maximal Euclidean disk in centered at a point , and then

is the set of all nearest boundary points for .

The chordal and spherical distances on are and , respectively. Thus

and is the length distance associated with ,⁠Footnote5 , and . Calculations are easier with , but is a geodesic distance whereas is not geodesic.

5

Identifying with the unit sphere in we see that is the angle between .

Each of the metric spaces has an associated length distance, although the latter two are equal. We write for the intrinsic (aka, inner) Euclidean length distance and for the intrinsic chordal length distance (which equals the intrinsic spherical length distance), and then and are the corresponding length spaces. See for example Reference Her10.

It is convenient to let and denote the chordal and spherical distances from to . Again, , and we note that

where is the metric boundary of .

2.A. Maps, paths, and geodesics

An embedding between two metric spaces is a quasisymmetry if there is a homeomorphism (called a distortion function) such that for all triples ,

when this holds, we say that is -QS. These mappings were studied by Tukia and Väisälä in Reference TV80; see also Reference Hei01.

The bi-Lipschitz maps form an important subclass of the quasisymmetric maps; is bi-Lipschitz if and only if there is a constant such that for all ,

and when this holds we say that is -bi-Lipschitz; such an is -QS with .

More generally, a map is an -quasiisometry if , and for all ,

These are often called rough bi-Lipschitz maps, and there seems to be no universal agreement regarding this terminology; some authors use the adjective quasiisometry to mean what we have called bi-Lipschitz, and then a rough quasiisometry satisfies our definition of quasiisometry. A -quasiisometry is simply an isometry (onto its range), and a -quasiisometry is called a -rough isometry.

Two metric spaces are isometrically equivalent (or BL, QS, QC equivalent, respectively) if and only if there is a bijection that is an isometry (or BL, QS, or QC).

Also, are quasiisometrically equivalent if and only if there is a quasiisometry with the property that is cobounded in (i.e., the Hausdorff distance between and is finite). More precisely: are -QI equivalent if there is an -quasiisometry and for each there is an with . An alternative way to describe this is to say that there are quasiisometries in both directions that are rough inverses of each other.

Example 2.1.

Suppose and for each there is an with . The maps

are both rough isometric equivalences: the “identity” inclusion is a -QI equivalence and is a -QI equivalence.

Our metric spaces will always be the domain , either in or in , with either Euclidean distance, chordal distance, spherical distance, an associated length distance, an associated quasihyperbolic distance, or an associated hyperbolic distance.

A path in is a continuous map where is an interval (called the parameter interval for ) that may be closed or open or neither and finite or infinite. The trajectory of such a path is which we call a curve. When is closed and , denotes the set of endpoints of which consists of one or two points depending on whether or not is compact. For example, if , then .

We call a compact path if its parameter interval is compact (which we often assume to be ). We call a rectifiable path if its length is finite, and then we may assume that is parameterized with respect to arclength in which case the parameter interval for is . We note that arclength parameterizations are a priori -Lipschitz continuous.

When , we write (in ) to indicate that is a path (in ) with initial point and terminal point ; this notation implies an orientation: precedes on .

When and are paths that join to and to respectively, denotes the concatenation of and ; so . Of course, . Also, the reverse of is the path defined by (when ) and going from to . Of course, .

An arc is an injective compact path. Every arc is taken to be ordered from its initial point to its terminal point. Given points , there are unique with , and we write . Every compact path contains an arc with the same endpoints; see Reference Väi94.

A path into a metric space is a geodesic if is an isometry (for all , ) and a -quasi-geodesic if is -bi-Lipschitz,⁠Footnote6

6

One can also consider rough-quasi-geodesics where is a quasiisometry; we do not do so.

A characteristic property of geodesics is that the length of each subpath equals the distance between its endpoints. There is a corresponding description for quasi-geodesics: is an -chordarc path if it is rectifiable and

If we ignore parameterizations, then the class of all quasi-geodesics (in some metric space) is exactly the same as the class of all chordarc paths. More precisely, a -quasi-geodesic is a -chordarc path, and if we parameterize an -chordarc path with respect to arclength, then we get an -quasi-geodesic.

In this paper we study the metric spaces or where is a hyperbolic plane domain and and are the hyperbolic and quasihyperbolic distances in . The geodesics and quasi-geodesics in are called hyperbolic geodesics and hyperbolic quasi-geodesics, and similarly in we attach the adjective quasihyperbolic.

2.B. Annuli and uniformly perfect sets

Given and , is an Euclidean annulus with center and conformal modulus ; if or , is a degenerate annulus and . We call the conformal center circle of ; is symmetric about this circle.⁠Footnote7 The inner and outer boundary circles of are, respectively,

7

Reflection across maps to itself interchanging its boundary circles.

A point is inside (outside) if and only if (); that is, is inside (or outside) if and only if is inside (or outside ).

It is convenient to introduce the notation

Then , and ; here , , .

An annulus is a subannulus of , denoted by , provided

Two annuli are concentric if they have a common center, and is a concentric subannulus of , denoted by , provided and is a subannulus of .

An annulus separates if and both components of contains points of ; thus when separates , one of or lies inside and the other lies outside , and if does not meet nor separate , then and are on the same side of . Evidently, if is a subannulus of , then separates the boundary circles of .

We define