Uniformizing Gromov hyperbolic spaces with Busemann functions

The goal of this paper is to construct an unbounded analogue of the uniformizations of Gromov hyperbolic spaces built by Bonk, Heinonen and Koskela in their extensive study of a number of problems in conformal analysis [4]. The most familiar special case of our procedure is the construction of the upper half-space $\{(x,y):y>0\}$ in $\mathbb{R}^{2}$ from the hyperbolic plane $\mathbb{H}^{2}$, which is discussed in Example 1.5. The guiding example in [4], by comparison, is the relationship between $\mathbb{H}^{2}$ and the Euclidean unit disk $\{(x,y):x^{2}+y^{2}<1\}$. As can be seen from these examples, the input for uniformization is a geodesic Gromov hyperbolic space $X$ and the output is an incomplete metric space $\Omega$, obtained from a conformal deformation of $X$, that is uniform in the sense of Definition 1.1 below. The density used for uniformizing a Gromov hyperbolic space $X$ in [4] is exponential in the distance to a fixed point $z$ of $X$. In contrast we will be using a density that is exponential in a Busemann function associated to a particular point of the Gromov boundary of $X$. This choice of density is natural as Busemann functions are often interpreted as distance functions “from infinity” and can themselves be used to define a boundary of the space $X$ [1, $\S$3]. Unlike in [4], we will not assume any local compactness properties on $X$, so specializing our results back to their setting yields a small generalization of their results as well.

Our principal application of this uniformization construction will be to hyperbolic fillings $X$ of a metric space $Z$, with a particular focus on the case in which $Z$ is unbounded. When $Z$ is bounded a hyperbolic filling $X$ of $Z$ can be thought of as a Gromov hyperbolic graph whose Gromov boundary can be canonically identified with $Z$; in the case that $Z$ is unbounded there are some additional subtleties to this notion owing to the fact that the Gromov boundary of a Gromov hyperbolic space is always bounded. We refer to the discussion prior to Theorem 1.12 for further information on this, as well as the contents of Section 5. Our use of Busemann functions in this setting is inspired by the hyperbolic filling construction of Buyalo and Schroeder [12, Chapter 6] for arbitrary metric spaces $Z$.

Our uniformization construction for hyperbolic fillings is used in a followup work [11] in order to establish a correspondence between Newton-Sobolev classes of functions on the hyperbolic filling of $Z$ and Besov classes of functions on $Z$ in the special case that $Z$ carries a doubling measure. This is heavily inspired by work of A. Björn, J. Björn, and Shanmugalingam [3] that establishes the corresponding result in the case that $Z$ is bounded. In a closely related work [10] we also generalize to our setting their results [2] on how local Poincaré inequalities transform under the uniformization in [4]. This yields some interesting new examples of uniform metric spaces satisfying Poincaré inequalities. There are a number of known variants on the correspondence between function spaces on the hyperbolic filling and function spaces on $Z$, see for instance [6], [7], [9]. Such correspondences were one of the original motivating factors in the use of hyperbolic fillings in analysis on metric spaces. For applications to trace theorems on Ahlfors regular metric spaces that demonstrate the power of these correspondences we refer to [19].

Lastly we remark that the idea of uniformizing Gromov hyperbolic spaces using Busemann functions has been developed independently by Zhou [22] for the purpose of an entirely different set of applications, including a study of Teichmueller’s displacement problem for quasi-isometries of Gromov hyperbolic spaces. The work [22] in particular gives alternative proofs of the main uniformization theorems (Theorems 1.4 and 1.6) restricted to the case of proper Gromov hyperbolic spaces and the original range $0<\varepsilon\leq\varepsilon_{0}$ of exponents considered by Bonk-Heinonen-Koskela (see Theorem 1.9 below). Our applications to CAT$(-1)$ spaces and hyperbolic fillings require us to consider exponents outside this range however; this is a key point of departure from [22].

Stating our main theorems require some preliminary definitions. We opt to give precise definitions in the corresponding sections throughout the paper, while mostly only outlining the necessary definitions here in the introduction. For a metric space $(X,d)$ and a curve $\gamma:I\rightarrow X$, $I\subset\mathbb{R}$ a subinterval, we write $\ell(\gamma)$ for the length of $\gamma$ measured in $X$. We will follow the standard practice of using $\gamma$ to denote both the parametrization of the curve and the image of the curve in $X$. The curve $\gamma$ is a geodesic if it is isometric as a mapping of $I$ into $X$. We say that $X$ is geodesic if any two points can be joined by a geodesic. We will use the following distance notation for distance from a point $x$ to a set $E$ in any metric space $(X,d)$,

and in particular will write $\mathrm{dist}(x,\gamma)$ for the distance of a point $x\in X$ to (the image of) a curve $\gamma$.

We now define uniform metric spaces. We start with an incomplete metric space $(\Omega,d)$. We denote the boundary of $\Omega$ in its completion $\bar{\Omega}$ by $\partial\Omega=\bar{\Omega}\backslash\Omega$. For $x\in\Omega$ we write $d_{\Omega}(x):=\mathrm{dist}(x,\partial\Omega)$ for the distance from $x$ to the boundary $\partial\Omega$. For the definition below we allow $I=[a,b]\subset\mathbb{R}$ to be any closed interval, and for a curve $\gamma:I\rightarrow X$ we denote its endpoints by $\gamma_{-}:=\gamma(a)$ and $\gamma_{+}:=\gamma(b)$. For such an interval $I\subset\mathbb{R}$ we write $I_{\leq t}=\{s\in I:s\leq t\}$ and $I_{\geq t}=\{s\in I:s\geq t\}$.

Many reasonable domains in Euclidean space such as the unit ball or upper half-space provide natural examples of uniform metric spaces when they are equipped with the Euclidean metric. The first requirement (1.1) implies that $A$-uniform curves minimize the distance between their endpoints up to the multiplicative constant $A$. The second requirement (1.2) implies that if we cut $\gamma$ at any point $\gamma(t)$ then at least one of the two subcurves $\gamma|_{I_{\leq t}}$ or $\gamma|_{I_{\geq t}}$ must have length controlled by the distance $d_{\Omega}(\gamma(t))$ of $\gamma(t)$ to $\partial\Omega$. We note that it is easily verified from the definitions that the property of a curve $\gamma$ being $A$-uniform is independent of the choice of parametrization of $\gamma$. For the purpose of formulating our theorems it is convenient to extend the definition of $A$-uniform curves to allow for arbitrary subintervals $I\subset\mathbb{R}$ and to allow the possibility $\ell(\gamma)=\infty$; as this extension is somewhat technical we refer to Definition 4.16 for the exact details.

For a continuous function $\rho:X\rightarrow(0,\infty)$ we write

for the line integral of $\rho$ along $\gamma$. We refer to [4, Appendix] for a detailed discussion of line integrals in our context. We will often refer to such a positive continuous function $\rho$ as a density on $X$. The following definition plays a key role in the statement of our main theorems.

We will refer to the inequality (1.3) as the GH-inequality and will sometimes refer to the constant $M$ as the $GH$-constant. The terminology here is inspired by the work of Gehring-Hayman [16], which shows that in a simply connected hyperbolic domain $\Omega$ in the complex plane the hyperbolic geodesics minimize Euclidean length among all curves in the domain with the same end points, up to a universal multiplicative constant. Here the density $\rho$ is given by the conformal change of metric relating the Euclidean metric on $\Omega$ to the hyperbolic metric. Note that if $X$ is a tree then the GH-inequality holds for any density $\rho$ with $M=1$ since any path joining two points in a tree must contain the geodesic joining those points.

We next discuss the notions we will need regarding Gromov hyperbolic spaces. Most formal definitions regarding Gromov hyperbolicity and the Gromov boundary are postponed to Section 2, as they can be found in any standard reference such as [12], [17]. A geodesic metric space $X$ is Gromov hyperbolic if there is a $\delta\geq 0$ such that all geodesic triangles are $\delta$-thin, meaning that for any geodesic triangle $\Delta$ each edge of $\Delta$ is contained in a $\delta$-neighborhood of the other two edges of $\Delta$. In this case we will also say that $X$ is $\delta$-hyperbolic. We write $\partial_{G}X$ for the Gromov boundary of $X$, to be defined in Section 2; for now we note that a geodesic ray $\gamma:[0,\infty)\rightarrow X$ can always be identified with an equivalence class $[\gamma]\in\partial_{G}X$, but in general not every point in $\partial_{G}X$ can be realized in this way.

We consider a complete geodesic $\delta$-hyperbolic space $X$ and a geodesic ray $\gamma:[0,\infty)\rightarrow X$. The Busemann function $b_{\gamma}:X\rightarrow\mathbb{R}$ associated to $\gamma$ is defined by the limit

Using the triangle inequality and the fact that $d(\gamma(t),\gamma(0))=t$, it’s easy to check that the right side is nonincreasing in $t$ and bounded below by $-d(\gamma(0),x)$, so this limit exists. It’s also easily verified that $b_{\gamma}$ is 1-Lipschitz, thus in particular is continuous. As is customary when considering Busemann functions, we will refer to any translate $b=b_{\gamma}+s$ of $b_{\gamma}$ for a constant $s\in\mathbb{R}$ as a Busemann function as well. We write

for the set of all Busemann functions on $X$. Given a Busemann function $b=b_{\gamma}+s$ we will write $\omega_{b}=[\gamma]\in\partial_{G}X$ for the point in the Gromov boundary determined by the geodesic ray $\gamma$. We will refer to $\omega_{b}$ as the basepoint of $b$ and say that $b$ is based at $\omega_{b}$.

To state our theorems in their appropriate generality it is useful to augment the set $\mathcal{B}(X)$ with the distance functions on $X$: for $z\in X$ we write $b_{z}(x)=d(x,z)$ for the distance function to $z$ and write

for the set of all translates of distance functions on $X$. We then write $\hat{\mathcal{B}}(X)=\mathcal{B}(X)\cup\mathcal{D}(X)$. In the case $b=b_{z}+s$ we write $\omega_{b}=z$ and refer to $\omega_{b}$ as the basepoint of $b$ as well. The defining formula (1.4) for Busemann functions shows that any $b\in\mathcal{B}(X)$ can be realized as a pointwise limit of functions $b_{t}\in\mathcal{D}(X)$ defined by $b_{t}(x)=d(\gamma(t),x)-t$.

Given $b\in\hat{\mathcal{B}}(X)$ and $\varepsilon>0$ we define a density $\rho_{\varepsilon,b}$ on $X$ by

We write $X_{\varepsilon,b}=(X,d_{\varepsilon,b})$ for the conformal deformation of $X$ with conformal factor $\rho_{\varepsilon,b}$. In the theorem below we will be assuming that $X$ is $K$-roughly starlike from the basepoint $\omega_{b}$ of $b$. This is a technical condition on geodesics starting from $\omega_{b}$ that is described in Definition 2.3. The main purpose of this hypothesis is to rule out cases such as trees that have arbitrarily long finite branches. This $K$-rough starlikeness condition will be satisfied with $K=\frac{1}{2}$ in our application of Theorem 1.4 to hyperbolic fillings in Theorem 1.12.

In this statement and all subsequent ones the notation $A=A(\delta,K,\varepsilon,M)$ is used to indicate that a particular constant depends on the indicated parameters. We refer to Definition 4.16 for the extension of the definition of $A$-uniform curves that is necessary to cover the case of an arbitrary geodesic in $X$; Definition 1.1 only covers the case of geodesics defined on closed intervals. The claim that $X_{\varepsilon,b}$ is bounded if and only if $b\in\mathcal{D}(X)$ does not require either the rough starlikeness hypothesis or the assumption that $\rho_{\varepsilon,b}$ is a GH-density, see Proposition 4.4.

We describe the motivating example for Theorem 1.4 in the case $b\in\mathcal{B}(X)$ below.

A metric space $(X,d)$ is proper if its closed balls are compact. In the case $b\in\mathcal{D}(X)$, Theorem 1.4 generalizes [4, Proposition 4.5] as it does not require $X$ to be proper and allows for a potentially larger range of values for the parameter $\varepsilon$; this larger range will be relevant to Theorem 1.10.

For a $\delta$-hyperbolic space $X$ and a point $\omega\in\partial_{G}X$ we write $\partial_{\omega}X=\partial_{G}X\backslash\{\omega\}$ for the complement of $\omega$ in the Gromov boundary of $X$. We will refer to $\partial_{\omega}X$ as the Gromov boundary relative to $\omega$ for reasons that will be explained prior to Proposition 2.8. We formally extend this definition to $\omega\in X$ by defining $\partial_{\omega}X=\partial_{G}X$; in this case we will still refer to $\partial_{\omega}X$ as the Gromov boundary relative to $\omega$, with the understanding that this simply coincides with the standard Gromov boundary for $\omega\in X$. As part of the proof of Theorem 1.4, we will show that there is a canonical identification $\varphi_{\varepsilon,b}:\partial_{\omega_{b}}X\rightarrow\partial X_{\varepsilon,b}$ between the Gromov boundary of $X$ relative to $\omega_{b}$ and the boundary of $X_{\varepsilon,b}$ in its completion. The most important property of this identification is summarized in Theorem 1.6 below.

A function $b\in\hat{\mathcal{B}}(X)$ can be used to define a natural class of metrics on $\partial_{\omega_{b}}X$ known as visual metrics based at $\omega_{b}$ (see [12, Chapter 3] as well as Section 2.3). These visual metrics have an associated parameter $\varepsilon>0$ and a comparison constant $L$ to a specific model quasi-metric $\theta_{\varepsilon,b}$ on $\partial_{\omega_{b}}X$ defined in (2.13). We continue to write $d_{\varepsilon,b}$ for the canonical extension of the metric on the uniformization $X_{\varepsilon,b}$ to its completion $\bar{X}_{\varepsilon,b}$.

For a precise description of the identification $\varphi_{\varepsilon,b}$ we refer to the disucssion after (4.15).

Let’s now discuss when the hypotheses of Theorems 1.4 and 1.6 are satisfied in practice. The two key hypotheses are the rough starlikeness hypothesis from the basepoint $\omega_{b}$ of $b$ and the assumption that $\rho_{\varepsilon,b}$ is a GH-density on $X$. The rough starlikeness hypothesis is always easily verified in applications of interest, so as a consequence it is typically not a concern when trying to apply these theorems. Thus the main hypothesis to verify is that of $\rho_{\varepsilon,b}$ being a GH-density. The most general result available regarding verifying this condition is the following theorem of Bonk-Heinonen-Koskela.

The inequality (1.7) is satisfied for $\rho_{\varepsilon,b}$ for any $b\in\hat{\mathcal{B}}(X)$ since all functions in $\hat{\mathcal{B}}(X)$ are $1$-Lipschitz. Theorem 1.9 builds on a number of previous works that are summarized at the beginning of [4, Chapter 5]. Thus if one is not concerned about obtaining Theorems 1.4 and 1.6 for a specific value of $\varepsilon$ then it is always possible to assume that $\rho_{\varepsilon,b}$ is a GH-density with constant $M=20$ by taking $\varepsilon$ sufficiently small.

In general one wants to establish that $\rho_{\varepsilon,b}$ is a GH-density for as large a value of $\varepsilon$ as possible, as this property is then inherited for smaller values of $\varepsilon$ by Proposition 4.12. This is particularly important for applications in which there is a preferred visual metric on $\partial_{\omega}X$, such as Theorems 1.10 and 1.12 below.

CAT$(-1)$ spaces are geodesic metric spaces in which geodesic triangles are thinner than corresponding comparison geodesic triangles in the hyperbolic plane $\mathbb{H}^{2}$. We refer to [14, Definition 3.2.1] for a precise definition; since the proper definition is somewhat lengthy to state and we will only be using easily stated consequences of the CAT$(-1)$ property, we omit a full description of the definition here. These spaces encompass many natural examples such as trees and simply connected Riemannian manifolds with sectional curvatures $\leq-1$. A CAT$(-1)$ space is $\delta$-hyperbolic with the same hyperbolicity constant $\delta=\delta(\mathbb{H}^{2})$ as the hyperbolic plane, for which the optimal constant can be computed explicitly to be $\delta=\log(1+\sqrt{2})$.

For a CAT$(-1)$ space $X$ and a function $b\in\hat{\mathcal{B}}(X)$ with basepoint $\omega$ the model quasi-metric $\theta_{1,b}$ on $\partial_{\omega}X$ in fact defines a distinguished choice of visual metric on $\partial_{\omega}X$ with parameter $\varepsilon=1$. This metric is known as the Bourdon metric on $\partial_{\omega}X=\partial X$ when $b\in\mathcal{D}(X)$ and the Hamenstädt metric on $\partial_{\omega}X$ when $b\in\mathcal{B}(X)$. For further details we refer to Remark 2.9. Our next theorem applies Theorems 1.4 and 1.6 to the special case of CAT$(-1)$ spaces at the special value $\varepsilon=1$.

The constant $M$ in Theorem 1.10 is universal in the sense that it is the same for any CAT$(-1)$ space $X$ and any $b\in\hat{\mathcal{B}}(X)$. The dependence of $A$ on $K$ can be removed when $b\in\mathcal{D}(X)$ by mimicking the arguments of [4, Proposition 4.5]; this same comment applies to Theorem 1.4 as well. The conclusions of Theorem 1.10 show in particular that the boundary $\partial X_{1,b}$ of the uniformization has a canonical biLipschitz identification with the Gromov boundary $\partial_{\omega}X$ relative to $\omega$ equipped with the distinguished visual metric $\theta_{1,b}$.

We will also apply our uniformization results to hyperbolic fillings of an arbitrary metric space $(Z,d)$. We briefly describe the construction of the hyperbolic filling here, with further details in Section 5, including proofs for the claims made here. Our construction will depend in part on two parameters $\alpha>1$ and $\tau>1$. For an $r>0$ we say that a subset $S\subset Z$ is $r$-separated if for each $x,y\in S$ we have $d(x,y)\geq r$. Given a parameter $\alpha>1$, we choose for each $n\in\mathbb{Z}$ a maximal $\alpha^{-n}$-separated subset $S_{n}$ of $Z$. For $n\in\mathbb{Z}$ we write $V_{n}=\{(z,n):z\in S_{n}\}$ and set $V=\bigcup_{n\in\mathbb{Z}}V_{n}$. The set $V$ will serve as the vertex set for $X$. We define the height function $h:V\rightarrow\mathbb{Z}$ on this vertex set by $h(v)=n$ for $v=(z,n)\in V_{n}$.

We associate to each vertex $v=(z,n)\in V$ the ball $B(v)=B(z,\tau\alpha^{-n})$ of radius $\tau\alpha^{-n}$ centered at $z$. We place an edge between vertices $v,w\in V$ if and only if their heights satisfy $|h(v)-h(w)|\leq 1$ and their associated balls satisfy $B(v)\cap B(w)\neq\emptyset$. We write $X$ for the resulting graph and call this a hyperbolic filling of $Z$. If $\tau$ is sufficiently large (see inequality (5.1)) then $X$ will be a connected graph by Proposition 5.5. We make $X$ into a geodesic metric space by declaring all edges to have unit length. We extend the height function $h$ to a $1$-Lipschitz function $h:X\rightarrow\mathbb{R}$ by linearly interpolating the values of $h$ from the vertices to the edges of $X$.

As a metric space $X$ is $\delta$-hyperbolic with $\delta=\delta(\alpha,\tau)$ depending only on the parameters $\alpha$ and $\tau$. There is a distinguished point $\omega\in\partial_{G}X$ in the Gromov boundary that can be thought of as an ideal point at infinity for $Z$. We have an identification $\partial_{\omega}X\cong\bar{Z}$ of the Gromov boundary relative to $\omega$ with the completion $\bar{Z}$ of $Z$. Under this identification the extension of the metric $d$ to $\bar{Z}$ defines a visual metric on $\partial_{\omega}X$ with parameter $\varepsilon=\log\alpha$. All of the results mentioned here are proved in Section 5.

We define a density $\rho$ on $X$ by $\rho(x)=\alpha^{-h(x)}$ for $x\in X$. We write $X_{\rho}$ for the conformal deformation of $X$ with conformal factor $\rho$. By Lemma 5.12 there is a Busemann function $b$ based at $\omega$ such that $|h(x)-b(x)|\leq 3$ for all $x\in X$ and therefore $h\in\mathcal{B}_{3}(X)$ in the notation of Remark 1.7. For such a Busemann function $b\in\mathcal{B}(X)$ we have that the density $\rho$ is uniformly comparable to the density $\rho_{\varepsilon,b}$ with $\varepsilon=\log\alpha$.

We compare our results to those of [3] in Remark 6.4. Another notable predecessor to Theorem 1.12 in the case that $Z$ is compact is the work of Piaggio [13, Section 2].

We provide here an outline of the contents of the rest of the paper. In Section 2 we review several key notions in the setting of Gromov hyperbolic spaces. Section 3 establishes some basic properties of geodesic triangles in Gromov hyperbolic spaces with vertices on the Gromov boundary and gives a rough formula for evaluating certain distance functions and Busemann functions on their edges. We then use these results in Section 4 to obtain estimates for the uniformized distance and prove Theorems 1.4, 1.6, and 1.10. In Section 5 we construct the hyperbolic fillings of metric spaces that we use in Theorem 1.12 and establish their basic properties. Lastly Theorem 1.12 is proved in Section 6.

We are very grateful to Nageswari Shanmugalingam for providing multiple drafts of the work [3] on which a significant part of this paper is based. We also thank Tushar Das for making us aware of the results of [14] that are used to prove Theorem 1.10.