Gromov hyperbolicity and quasihyperbolic geodesics

We characterize Gromov hyperbolicity of the quasihyperbolic metric space (\Omega,k) by geometric properties of the Ahlfors regular length metric measure space (\Omega,d,\mu). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition.


Introduction
Given a proper subdomain Ω of the Euclidean space Ê n , n ≥ 2, equipped with the usual Euclidean distance, one defines the quasihyperbolic metric k in Ω as the path metric generated by the density where d(z) = dist(z, ∂Ω). Precisely, one sets k(x, y) = inf γxy γxy ρ(z) ds, where the infimum is taken over all rectifiable curves γ xy that join x and y in Ω and the integral is the usual line integral. Then Ω equipped with k is a geodesic: there is a curve γ xy whose length in the above sense equals k(x, y). Let us denote by [x, y] any such geodesic; these geodesics are not necessarily unique as can be easily seen, for example for Ω = Ê n \{0}. The quasihyperbolic metric k was introduced in [GP] and [GO] where the basic properties of it were established. If for all triples of geodesics [x, y], [y, z], [z, x] in Ω every point in [x, y] is within k-distance δ from [y, z] ∪ [z, x], the space (Ω, k) is called δ-hyperbolic. Roughly speaking, this means that geodesic triangles in Ω are δ-thin. Moreover, we say that (Ω, k) is Gromov hyperbolic if it is δ-hyperbolic for some δ. The following theorem from [BB] that extends results from [BHK] gives a complete characterization of Gromov hyperbolicity of (Ω, k). Above, the Gehring-Hayman condition means that there is a constant C gh ≥ 1 such that for each pair of points x, y in Ω and for each quasihyperbolic geodesic [x, y] it holds that length ([x, y]) ≤ C gh length(γ xy ), where γ xy is any other curve joining x to y in Ω. In other words, it says that quasihyperbolic geodesics are essentially the shortest curves in Ω. The other condition, a ball separation condition, requires the existence of a constant C bs ≥ 1 such that for each pair of points x and y, for each quasihyperbolic geodesic [x, y], for every z ∈ [x, y], and for every curve γ xy joining x to y it holds that B(z, C bs d(z)) ∩ γ xy = ∅.
Notice that the three conditions in Theorem 1.1, Gromov hyperbolicity and the Gehring-Hayman and the ball separation conditions, are only based on metric concepts. It is then natural to ask for an extension of this characterization to an abstract metric setting. Such an extension was given in [BB], relying on an analytic assumption that essentially requires that the space in question supports a suitable Poincaré inequality. This very same condition, expressed in terms of moduli of curve families [HeiK], is already in force in [BHK].
The purpose of this paper is to show that Poincaré inequalities are not critical for geometric characterizations of Gromov hyperbolicity of a non-complete metric space, equipped with the quasihyperbolic metric. Our main result reads as follows.
Theorem 1.2. Let Q > 1 and let (X, d, µ) be a Q-regular metric measure space with (X, d) a locally compact and annularly quasiconvex length space. Let Ω be a bounded and proper subdomain of X, and let d Ω be the inner metric on Ω associated to d. Then (Ω, k) is Gromov hyperbolic if and only if (Ω, d Ω ) satisfies both a Gehring-Hayman condition and a ball separation condition.
Above, annularly quasiconvexity means that there is a constant λ ≥ 1 so that, for any x ∈ X and all 0 < r ′ < r, each pair of points y, z in B(x, r) \ B(x, r ′ ) can be joined with a path γ yz in B(x, λr) \ B(x, r ′ /λ) such that length(γ yz ) ≤ λd(y, z), Q-regularity requires the existence of a constant C q so that for all r > 0 and all x ∈ X, and the other concepts are defined analogously to the Euclidean setting described in the beginning of our introduction. See Section 2 for the precise definitions. In fact, the assumptions of Theorem 1.2 can be somewhat relaxed, see Section 5.
The main point in Theorem 1.2 is the necessity of the Gehring-Hayman and ball separation conditions; their sufficiency is already given in [BB].
This paper is organized as follows. Section 2 contains necessary definitions. In Section 3 we give preliminaries related the quasihyperbolic metric and Whitney balls. Section 4 is devoted to the proof of our main technical estimate, and Section 5 contains the proof of our main result and some generalizations.

Definitions
Let (X, d) be a metric space. A curve means a continuous map where the supremum is taken over all partitions a = t 0 < t 1 < · · · < t m = b of the interval [a, b]. If ℓ d (γ) < ∞, then γ is said to be a rectifiable curve. When the parameter interval is open or half-open, we set where the supremum is taken over all compact subintervals [c, d].
When every pair of points in (X, d) can be joined with a rectifiable curve, the space (X, d) is called rectifiably connected. If ℓ d (γ xy ) = d(x, y) for some curve γ xy joining points x, y ∈ X, then γ xy is said to be a geodesic. If every pair of points in (X, d) can be joined with a geodesic, then (X, d) is called a geodesic space. Moreover, a geodesic ray in X is an isometric image in (X, d) of the interval [0, ∞). Furthermore, for a rectifiable curve γ we define the arc length s : Let (X, d) be a geodesic metric space and let δ ≥ 0. Denote by [x, y] any geodesic joining two points x and y in X. If for all triples of In other words, geodesic triangles in X are δ-thin. Moreover, we say that a space is Gromov hyperbolic if it is δ-hyperbolic for some δ. All Gromov hyperbolic spaces in this paper are assumed to be unbounded.
Next, let (X, d) be a locally compact, rectifiably connected and noncomplete metric space, and denote by X d its metric completion. Then the boundary ∂ d X := X d \ X is nonempty. We write If γ also satisfies the cigar condition for every t ∈ [a, b], the curve is called a D-uniform curve. A metric space (X, d) is called a D-quasiconvex space or D-uniform space if every pair of points in it can be joined with a D-quasiconvex curve or a D-uniform curve respectively.
Let ρ : X → (0, ∞) be a continuous function. For each rectifiable Because (X, d) is rectifiably connected, the density ρ determines a metric d ρ , called a ρ-metric, where the infimum is taken over all rectifiable curves γ xy joining x, y ∈ X. If ρ ≡ 1, then ℓ ρ (γ) = ℓ d (γ) is the length of the curve γ with respect to the metric d, and the metric d ρ = ℓ d is the inner metric associated with d. Generally, if the distance between every pair of points in the metric space is the infimum of the lengths of all curves joining the points, then the metric space is called a length space.
If we choose we obtain the quasihyperbolic metric in X. In this special case, we denote the metric d ρ by k d and the quasihyperbolic length of the curve γ by ℓ k d (γ). Moreover, [x, y] k d refers to a quasihyperbolic geodesic joining points x and y in X. Because we are dealing with many different metrics, the usual metric notations will have an additional subscript that refers to the metric in use. For ease of notation, terms which refer to the metric d ρ will have an additional subscript ρ instead of d ρ .
We say that (X, d) satisfies a ball separation condition if there is a constant C bs ≥ 1 such that for each pair of points x, y ∈ X, for every quasihyperbolic geodesic [x, y] k d ⊂ X, for every z ∈ [x, y] k d , and for every curve γ xy joining points x and y, it holds that Thus the condition says that the ball B d (z, C bs d(z)) either includes at least one of the endpoints of the quasihyperbolic geodesic or it separates the endpoints. This condition was introduced in [BHK,§7]. We also say that (X, d) satisfies the Gehring-Hayman condition if there is a constant C gh ≥ 1 such that for every [x, y] k d it holds that where γ xy is any other curve joining x to y in X.
is a locally compact, rectifiable connected and non-complete metric space, and the identity map from (X, d) to (X, ℓ d ) is continuous. If (X, d) is minimally nice, then the identity map from (X, d) to (X, k d ) is a homeomorphism, and (X, k d ) is complete, proper (i.e. closed balls are compact) and geodesic (cf. [BHK,Theorem 2.8]). Furthermore, we define a proper, geodesic space (X, k d ) to be K-roughly starlike, K > 0, with respect to a base point w ∈ X, if for every point x ∈ X there exists some geodesic ray emanating from w whose distance to x is at most K.
Let µ be a Borel regular measure on (X, d) with dense support. We call the density ρ a conformal density provided it satisfies both a Harnack inequality HI(A) for some constant A ≥ 1 : ) and all z ∈ X, HI(A) and a volume growth condition VG(B) for some constant B > 0 : for all z ∈ X and r > 0.

VG(B)
Here µ ρ is the Borel measure on X defined by and Q is a positive real number. Generally Q will be the Hausdorff dimension of our space (X, d). There is nothing special about the constant 1 2 in condition HI(A): we may replace it by any constant 0 < c ≤ 1 2 , actually by any constant c ∈ (0, 1). Suppose that we have fixed 0 < c ≤ 1 2 . Then each ball B d (z, cd(z)) is called a Whitney type ball.
In general, we say that for each z ∈ X and every r > 0. We also say that (X, d, µ) is Q-regular on Whitney type balls for some Q > 0 if there are constants C w ≥ 1 and 0 < ǫ ≤ 1 such that for each z ∈ X and every r ≤ ǫd(z)/2.

The metric measure space
Let (X, d, µ) be a minimally nice metric measure space so that the measure µ is Borel regular and (X, k d ) is Gromov hyperbolic. Let w ∈ X be a base point. We define two deformations by setting where the infimum is taken over all curves γ xy joining points x and y in X, and d k d s = ds/d(z) denotes the quasihyperbolic distance element.
For ease of notation we write d ǫ instead of d σǫ . We also refer to the metric d ǫ via an additional subscript ǫ, for instance X ǫ denotes the metric completion of (X, d ǫ ). If Q > 0 is fixed, we also attach the Borel Thus (X, d ǫ ) is always bounded. Moreover, by the triangle inequality the density ρ ǫ satisfies a Harnack type inequality: for all x, y ∈ X and all ǫ > 0. We also obtain that the density σ ǫ satisfies the Harnack inequality HI(A) with the constant A = 3 exp{2ǫ} : for all x, y ∈ B d (z, 1 2 d(z)) and for every z ∈ X. Bonk, Heinonen and Koskela proved in [BHK,§4 and Theorem 5.1] that there is ǫ 0 > 0 depending on δ such that the metric space (X, d ǫ ) is D ǫ -uniform for every 0 < ǫ ≤ ǫ 0 , where k d -quasihyperbolic geodesics serve as D ǫ -uniform curves with D ǫ = D(δ, ǫ, ǫ 0 ) ≥ 1. Especially, we have a version of the Gehring-Hayman condition: there is a constant D ǫ ≥ 1 such that when ǫ ≤ ǫ 0 , for each quasihyperbolic geodesic [x, y] k d in X and for each curve γ xy joining x to y in X. Furthermore, if (X, k d ) is K-roughly starlike with respect to the base point w, then by [BHK,Lemma 4.17] we have that for all ǫ > 0 and every x ∈ X. Thus there exists c = c(δ, K) ∈ (0, 1) such that for all x, y ∈ X and every 0 < ǫ ≤ ǫ 0 , where k ǫ is the quasihyperbolic metric derived from d ǫ . Moreover, we obtain from [BHK,Theorem 6.39] that Whitney type balls in (X, d ǫ ) are also Whitney type balls in (X, d).

and by HI(A) it follows that
(3.7) The lower bound follows similarly: When (X, d) is D-quasiconvex, by HI(A) we have that B d (z, 1 ADσǫ(z) r) ⊂ B ǫ (z, r). Thus HI(A) together with (2.2) yields (3.8) Let Q > 1 and let (X, d, µ) be a minimally nice D-quasiconvex and Q-upper regular space so that the measure µ is Q-regular on Whitney type balls, and (X, k d ) is a K-roughly starlike Gromov hyperbolic space. Let the constants ǫ 0 and C be as in the paragraph containing (3.2) and (3.5). We may define a Whitney covering of the space  [KL,§3].
Let r(z) = ǫd ǫ (z)/50. From the family {B ǫ (z, r(z))} z∈X of balls we select a maximal (countable) subfamily {B ǫ (z i , r(z i )/5)} i∈I of pairwise disjoint balls. We write B = {B i } i∈I , where B i = B ǫ (z i , r i ) and r i = r(z i ). We call the family B the Whitney covering of (X, d ǫ ). We list the basic properties of the Whitney covering in Lemma 3.1. As in [KL,Lemma 3.2], the property (iv) is a consequence of the Q-regularity condition of µ ǫ on Whitney type balls, and the property (v) follows from the proof of Lemma 3.2 in [KL] via uniformity and Q-regularity condition of µ ǫ on Whitney type balls.
Lemma 3.1. There is N ∈ AE such that (i) the balls B ǫ (z i , r i /5) are pairwise disjoint, Furthermore, suppose d ǫ (x, y) ≥ d ǫ (x)/2 and let γ xy be a D ǫ -uniform curve joining x to y. There exists a constant C ǫ > 0, that depends quantitatively on ǫ and the hypotheses, such that ( Fix a ball B 0 from the Whitney covering B and let z 0 be its center. For each B i ∈ B we fix a geodesic [z 0 , z i ] kǫ . Furthermore, for each Similarly as in [KL,Lemma 3.3 and Lemma 3.4] we may prove, when ǫ ≤ min{ǫ 0 , 1 8D , 1 2C 2 }, that there is a constant C o > 0 that depends quantitatively on ǫ and the hypotheses, such that

The main lemma
Next we prove a lemma which is the central tool for proving our main theorem. From now on we assume that Q > 1, (X, d, µ) is a minimally nice Q-upper regular D-quasiconvex metric measure space such that the measure µ is Q-regular on Whitney type balls, and (X, k d ) is a K-roughly starlike Gromov hyperbolic space. We also assume that (X, d ǫ , µ ǫ ) is a deformation of (X, d, µ) as described above, where w ∈ X is a base point, ǫ 0 > 0 is as in the paragraph containing (3.2), C > 1 as in (3.5) and 0 < ǫ ≤ min{ǫ 0 , 1 8D , 1 2C 2 }. Lemma 4.1. Let u ∈ X be a point and γ ⊂ X be a curve such that dist ǫ (u, γ) ≤ min{C 1 d ǫ (u), C 2 diam ǫ (γ)} for some C 1 , C 2 > 0. Then there exists a constant M ≥ 1 that depends quantitatively on ǫ and the constants in our hypotheses, so that Proof. Suppose that for a fixed M > 2 there is u ∈ X and a curve γ such that dist ǫ (u, γ) ≤ min{C 1 d ǫ (u), C 2 diam ǫ (γ)} but dist d (u, γ) > Md(u). We will show that such an M has an upper bound in terms of our data. Towards this end, let us fix u and γ as above. By replacing γ with a suitable subcurve of γ we may assume without loss of generality that Let B be a Whitney covering of (X, d ǫ , µ ǫ ) as in Section 3. We choose B ǫ (u, r(u)) as the fixed ball B 0 ∈ B . Letû ∈ ∂ d X be such that d(u) = d(u,û). Let y ∈ γ and let [u, y] k d be a k d -quasihyperbolic geodesic joining u to y. Moreover, let I y , J y ⊂ AE be index sets so that We define g : X → (0, ∞) by setting Because k d -quasihyperbolic geodesics are D ǫ -uniform curves in (X, d ǫ ), we obtain that Here d ǫ s = σ ǫ (z) ds is the length element in the metric d ǫ .
Moreover, by (WB2) and Lemma 3.1 (iv) we obtain that [u,ỹ] . (4.2) Let j ∈ J y . Because diam kǫ (B j ) ≤ 2ǫ 50−ǫ , using (WB1) and (3.4), we obtain the estimate [u,y]  Combining this with (4.2) and (4.1) we have that Now, from (4.4) we obtain the estimate where P ≥ 1 is a constant that depends on ǫ and the constants of the hypotheses. Let ν be a Radon measure on γ ⊂ (X, d ǫ ) given by Frostman's lemma (cf. [Ma] and [KL,Theorem 4.1]) so that For every y ∈ γ we set We may choose a finite number of points y n ∈ γ such that γ ⊂ n S yn . Hence, using (4.5), Fubini's theorem and Hölder's inequality we obtain that (4.7) Let us first estimate the first double sum in (4.7). Let Pick an integer m with 2 m−1 < M + 1 ≤ 2 m . WriteB = B d (z, ǫC 2 50 d(z)) for B = B ǫ (z, r(z)) ∈ B . Given B ∈ B, (WB1) ensures that B ⊂B. Moreover, if B ∩ A n = ∅ for some n ∈ , thenB ⊂ A n−1 ∪ A n ∪ A n+1 . Thus, by the Q-upper regularity condition, the Q-regularity condition on Whitney type balls in (X, d) and Lemma 3.1 (iv) we deduce that Above, in moving from the second line to the third, we used the pairwise disjointness of the balls 1 5 B, (WB2) and the Q-regularity of µ on Whitney type balls.
Combining (4.8) and (4.14) with (4.7) we obtain Inserting (4.6) we conclude that This gives the desired upper bound on M and the claim follows.

Proof of Theorem 1.2
We begin by proving the following theorem.
Theorem 5.1. Let Q > 1 and let (X, d, µ) be a minimally nice Q-upper regular D-quasiconvex space such that the measure µ is Q-regular on Whitney type balls. Suppose that (X, k d ) is a K-roughly starlike Gromov hyperbolic space. Then (X, d) satisfies both the Gehring-Hayman condition and the ball separation condition.
Proof. Let us first prove that (X, d) satisfies the Gehring-Hayman condition. Because (X, k d ) is Gromov hyperbolic and K-roughly starlike, (X, d ǫ , µ ǫ ) is uniform for a deformation as in Section 3 with respect to a base point w ∈ X, where we choose ǫ ≤ min{ǫ 0 , 1 8D , 1 2C 2 }, where ǫ 0 > 0 is as in the paragraph containing (3.2), C > 1 as in (3.5). We know from (3.7) and (3.8) that the measure µ ǫ is Q-regular on Whitney type balls. We will consider (X, d, µ) as a conformal deformation of (X, d ǫ , µ ǫ ).
Hence,ρ is a conformal density. The corresponding deformation of the metric space (X, d ǫ ) withρ results in an inner metric space (X, ℓ d ) which is bi-Lipschitz equivalent to the original metric space (X, d).