Quasiconformal Jordan domains

We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $\Phi$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $\Phi$ to $\mathbb{S}^{1}$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane.


Introduction
Let (X, d X ) be a metric space with locally finite Hausdorff 2-measure. If X is also homeomorphic to a 2-manifold, we say that (X, d X ) is a metric surface. A homeomorphism φ : (X, d X ) → (Y, d Y ) between metric surfaces is quasiconformal if there exists K ≥ 1 such that for all path families Γ, where mod Γ is the conformal modulus of Γ, see Section 2.3. We say that a metric surface (Y, d Y ) is a metric Jordan domain if the metric completion Y is homeomorphic to the closed unit disk D, the boundary ∂Y = Y \ Y is homeomorphic to the unit circle S 1 , and the Hausdorff 2-measure of Y is finite.
A metric Jordan domain is a quasiconformal Jordan domain if there exists a quasiconformal homeomorphism φ : D → (Y, d Y ). A metric Jordan domain is a quasiconformal one if and only if (Y, d Y ) is reciprocal as introduced in [Raj17, Theorem 1.4]; see Definition 2.5. This uses the facts that H 2 Y (Y) < ∞ and that ∂Y is a non-trivial continuum.
In general, it is not true that the completion Y of a quasiconformal Jordan domain is a quasiconformal image of the closed unit disk D. We illustrate this with an example after Theorem 1.1. Contrast this with the classical case when Y is a Jordan domain in the plane R 2 . Then any 1-quasiconformal homeomorphism φ : D → Y, i.e., any Riemann map from the unit disk onto Y extends to a homeomorphism Φ : D → Y by a result known as the Carathéodory extension theorem [GM05, Chapter I, Theorem 3.1]. In fact, the extension still satisfies (1) with K = 1. Additionally, if φ : D → Y is K-quasiconformal for some K ≥ 1, it still has a homeomorphic extension to the boundary.
1.1. Carathéodory's theorem. We prove the following generalization of the classical Carathéodory extension theorem of quasiconformal maps. Theorem 1.1. Let φ : D → Y be a quasiconformal map onto a quasiconformal Jordan domain. Then there exists an extension Φ : D → Y of φ that is surjective, monotone and Φ(S 1 ) = ∂Y.
Here we say that a map is monotone if it is continuous and the preimage of every point is a continuum, i.e., a compact and connected set.
Proposition 1.2. The extension Φ in Theorem 1.1 is quasiconformal if and only if for every y ∈ ∂Y and R > 0 for which Y \ B Y (y, R) = ∅, Moreover, if (2) holds at each y ∈ ∂Y and φ is K-quasiconformal, then Φ is Kquasiconformal.
A well-known fact is that if there exists C U > 0 such that for all y ∈ ∂Y and 0 < r < diam ∂Y, H 2 Y (B Y (y, r)) ≤ C U r 2 , then (2) holds; see Lemma 2.8. The condition (2) has a close link to the reciprocality condition introduced in [Raj17]; see Definition 2.5. The aforementioned example of the collapsed disk [0, 1] 2 fails (2) at exactly one point.
1.2. Quasicircles. Consider a quasiconformal Jordan domain Y whose boundary points satisfy the area growth inequality (3). We know from Proposition 1.2 that the extension Φ : D → Y of any quasiconformal homeomorphism φ : D → Y is a quasiconformal homeomorphism. In particular, the boundary map g φ = Φ| S 1 : S 1 → ∂Y is a homeomorphism.
We are especially interested when we can deduce that ∂Y is a quasicircle, i.e., a quasisymmetric image of S 1 . We refer the reader to Section 2 for definitions. Theorem 1.3 (Beurling-Ahlfors extension). Suppose that Y is a quasiconformal Jordan domain whose boundary points satisfy the area growth (3).
If φ : D → Y is a quasiconformal homeomorphism, then the boundary map g φ is a quasisymmetry if and only if ∂Y has bounded turning. If ∂Y has bounded turning, then any quasisymmetry g : S 1 → ∂Y is the boundary map of some quasiconformal map φ : D → Y. Theorem 1.3 has a parallel in the classical literature. Ahlfors and Beurling proved in [BA56] that every quasisymmetry g : S 1 → S 1 is the boundary homeomorphism of some quasiconformal map φ : D → D. In fact, we apply their result in proving that quasisymmetries g : S 1 → ∂Y extend like claimed. It is also known that the boundary homeomorphisms of quasiconformal maps φ : D → D are quasisymmetries. So we also recover this result with the assumptions of Theorem 1.3.
We now know from Theorem 1.3 that ∂Y is a quasicircle in some situations. We are interested whether or not ∂Y can be bi-Lipschitz embedded into the plane. We say that a quasicircle Z is planar, or a planar quasicircle, if there exists a bi-Lipschitz embedding h : Z → R 2 .
One of the main results obtained in [HM12] states that a quasicircle is planar if and only if its Assouad dimension is strictly less than two, see Definition 2.10. There are quasicircles for every Assouad dimension between 1 and ∞ since Z = (S 1 , · α 2 ) for 0 < α ≤ 1 has Assouad dimension α −1 . Proposition 1.4. Let Y be a quasiconformal Jordan domain. If ∂Y is a quasicircle and the boundary points satisfy the area growth (3), then the Assouad dimension of ∂Y is at most two.
It is not clear if ∂Y in the above statement must be planar. However, if Y is annularly linearly locally connected (ALLC) and Ahlfors 2-regular, then ∂Y is a planar quasicircle [MW13, Theorems 8.1 and 8.2]; see [MW13] for the proofs and terminology. Quasiconformal Jordan domains satisfying these stronger assumptions appear in [Wil08] and [BHR01].
We localize these assumptions in the following statement and obtain the same conclusion.
Theorem 1.5. Let Y be a quasiconformal Jordan domain such that ∂Y is a quasicircle and its boundary points satisfy the area growth (3). Then the Assouad dimension of ∂Y is strictly less than two if the following two conditions are satisfied for some r 0 > 0, C > 0 and λ > 1: (a) For every y ∈ ∂Y and 0 < 2r < R < r 0 and any pair If (a) holds we say that ∂Y is relatively ALLC and if (b) holds, we say that Y satisfies the Ahlfors lower bound near ∂Y. The main point of Theorem 1.5 is to only restrict the geometry of Y near the boundary ∂Y.
The relative ALLC guarantees that ∂Y is porous in Y below the given scale r 0 (Lemma 5.2). The porosity allows us to pack many balls in Y, well-disjoint from ∂Y, near all points of ∂Y at all scales below r 0 . Now the Ahlfors lower bound, valid for such balls, combined with the upper bound (3) allow us to control quantitatively the total amount of such non-overlapping balls in a given interval of scales. This quantification allows us to prove planarity for ∂Y. This idea appears in [BHR01, Lemma 3.12], where the authors prove that a compact set in an Ahlfors regular space is porous if and only if its Assouad dimension is strictly smaller than the homogeneous dimension of the space. A similar argument also works in the setting of Theorem 1.5.
1.3. Outline. In Section 2, we introduce the notations we use and some preliminary results. In Section 3, we prove Theorem 1.1. Theorem 1.3 and Proposition 1.4 are proved in Section 4. Theorem 1.5 is proved in Section 5. Section 6 contains some concluding remarks.
2. Preliminaries 2.1. Notation. Let (Y, d Y ) be a metric space. The open ball centered at a point y ∈ Y of radius r > 0 with respect to the metric d is denoted by B Y (y, r). The closed ball is denoted by B Y (y, r). We sometimes omit the subscript from d Y , from B Y , and from B Y , respectively.
We recall the definition of Hausdorff measure. Let (Y, d) be a metric space. For all Q ≥ 0, the Q-dimensional Hausdorff measure (Hausdorff Q-measure) is defined by is chosen in such a way that H n R n coincides with the Lebesgue measure L n for all positive integers.
The length of a path γ : the supremum taken over all finite partitions whenever this limit exists. If γ is rectifiable, its metric speed exists at L 1 -almost with v γ ∈ L 1 ([a, b]) and L 1 the Lebesgue measure on the real line. Equivalently, γ is absolutely continuous if it maps sets of L 1 -measure zero to sets of H 1 Y -measure zero in its image [Dud07, Section 3].
Let γ : [a, b] → X be an absolutely continuous path. Then the path integral of a Borel function ρ : If γ is rectifiable, then the path integral of ρ over γ is defined to be the path integral of ρ over the arc length parametrization γ s of γ; see for example Chapter 5 of [HKST15].
2.2. Quasiconformal Jordan domains. We assume that Y is a quasiconformal Jordan domain. In particular, its completion Y is homeomorphic to [0, 1] 2 and has finite Hausdorff 2-measure.
Given a Borel set A ⊂ Y, the length of a path γ : The equality (5)   Via a standard approximation argument using simple functions, we obtain the following.
Theorem 2.1. Let f : Y → R be 1-Lipschitz, α ∈ {0, 1} and C α as in (6). Then, for every Borel function g : When g is H α+1 Y -integrable, the upper integral can be replaced with the usual one.
In the following, we say that C ⊂ Y is a continuum if C is compact and connected. A compact set F ⊂ Y separates x, y ∈ Y if x, y ∈ Y \ F and the points are in different connected components of Y \ F. Lemma 2.2. Let F ⊂ Y be compact and x, y ∈ Y separated by F. Then there exists a continuum C ⊂ F that separates x and y.
Proof. Let x, y and F be as in the claim. We consider a homeomorphism h : Y → Z ⊂ S 2 , where Z is the union of the equator and the southern hemisphere of S 2 . Then there exists a nested sequence of quadrilaterals Z n ⊃ Z n+1 ⊃ Z such that h(x), h(y) ∈ Z n is an interior point of Z n for each n ∈ N and Z = ∞ n=1 Z n . Since F n = ∂Z n ∪ h(F) ⊂ Z n separates h(x) and h(y) in S 2 , there exists a continuum C n ⊂ F n separating h(x) and h(y) in S 2 [Wil79, Chapter 2, Lemma 5.20]. In particular, for every path γ : [0, 1] → Z joining h(x) to h(y), there exists z n ∈ C n ∩ |γ| for every n ∈ N. Up to passing to a subsequence and relabeling, 2.3. Metric Sobolev spaces. In this section we give an overview of Sobolev theory in the metric surface setting, and refer to [HKST15] for a comprehensive introduction. Let where the infimum is taken over all admissible functions ρ. Observe that if Γ 1 and Γ 2 are path families and every path γ 1 ∈ Γ 1 contains a subpath γ 2 ∈ Γ 2 , then mod Γ 1 ≤ mod Γ 2 . In particular, this holds if Γ 1 ⊂ Γ 2 . A property holds for almost every path if the family of paths for which the property fails has zero modulus.
The function ρ is a weak upper gradient of ψ if the same holds for almost every rectifiable path.
A weak upper gradient ρ ∈ L 2 loc (Y) of ψ is minimal if it satisfies ρ ≤ ρ almost everywhere for all weak upper gradients ρ ∈ L 2 loc (Y) of ψ. If ψ has a weak upper gradient ρ ∈ L 2 loc (Y), then ψ has a minimal weak upper gradient, which we denote by ρ ψ . We refer to Section 6 of [HKST15] and Section 3 of [Wil12] for details.
Fix a point z ∈ Z, and let Definition 2.3 is sometimes called the geometric definition of quasiconformality. A special case of [Wil12, Theorem 1.1] yields the following.
Theorem 2.4. Let Y and Z be metric spaces with locally finite Hausdorff 2measure and ψ : Y → Z a homeomorphism. The following are equivalent for the same constant K > 0: For a set G ⊂ Y and disjoint sets F 1 , F 2 ⊂ G, let Γ(F 1 , F 2 ; G) denote the family of paths that start from F 1 , end in F 2 and whose images are contained in G. A quadrilateral is a set Q homeomorphic to [0, 1] 2 with boundary ∂Q consisting of four boundary arcs, overlapping only at the end points, labelled ξ 1 , ξ 2 , ξ 3 , ξ 4 in cyclic order.
We note that the product in (8) is always bounded from below by a universal constant κ 0 > 0 [RR19]. We also have the following.
Proposition 2.6 (Corollary 12.3 of [Raj17]). Let Y be a metric surface, U ⊂ Y a domain, and ψ : Recall the definition of quasiconformal Jordan domain from the introduction.
for every x ∈ D. The constants K and C depend on each other quantitatively.
Proof. Since the Lebesgue 2-measure on D is doubling, Theorem 1.2 of [Wil12] states that an upper bound K O (Ψ) ≤ K is quantitatively equivalent to the following statement: there exists C ′ ≥ 1 such that for every x ∈ D, (10) holds for some C, with K and C depending on one another quantitatively. It remains to prove that . This can be seen by extending Φ to a neighbourhood of D via reflection over S 1 and by applying [KS93, Theorem 1.12.3]. The minimal weak upper gradient of Φ has a representative that vanishes in S 1 since L 2 (S 1 ) = 0. Therefore ρ 2 Φ ≤ C 0 K J Φ holds L 2 -almost everywhere in D. This implies that Φ satisfies the second condition in Theorem 2.4 with the constant C 0 K.
We recall a sufficient condition for (2) for later use.
Lemma 2.8. Suppose that there exists C U > 0 such that for all y ∈ ∂Y and 0 < r < diam ∂Y, Then for C U = 8C U / log 2 and for every y ∈ ∂Y and 0 < 2r < R < 2 −1 diam ∂Y, In particular, (2) in Proposition 1.2 holds under the assumption (11).
Proof. The inequality (12) follows from (11) by considering the admissible func- The L 2 -norm of ρ is estimated from above by applying the area growth (11) on The inequality (12) follows. We claim now that (2) in Proposition 1.2 holds. Let y ∈ ∂Y and R ′ > R > 0 such that Y \ B Y (y, R ′ ) = ∅ and 2 −1 diam ∂Y > R. Then for every 0 < 2r < R, every The right-hand side converges to zero as r → 0 + , given (12). This establishes (2).
2.5. Quasicircles. In this section we recall some basic properties of quasisymmetries and quasicircles. If g : between metric spaces is η-quasisymmetric if for every y ∈ Y and 0 < r 1 , L g (y, r 1 ) ≤ η r 1 r 2 ℓ g (y, r 2 ).
A homeomorphism g is quasisymmetric if it is η-quasisymmetric for some homeo- and an r-net if for every y ∈ Y, there exists x ∈ S for which d Y (x, y) < r. An r-separated set is maximal if it is also an r-net.
Definition 2.10. A metric space (Y, d Y ) has its Assouad dimension bounded from above by Q > 0 if for every 0 < ǫ < 1 and every (y, r) where C is a constant independent of ǫ, y, r and S. Here #S refers to the counting measure of S. The Assouad dimension of Y is the infimum of such Q.
has λ-bounded turning if for every y, z ∈ Y there exists a compact and connected set E ⊂ Y containing y and z such that diam E ≤ λd Y (y, z).
We recall that a metric space C homeomorphic to S 1 is a quasisymmetric image of S 1 if and only if C has bounded turning and is doubling [TV80]. We refer to any quasisymmetric image of S 1 as a quasicircle.
3. Carathéodory's theorem 3.1. Proof of Theorem 1.1. We fix a quasiconformal homeomorphism φ : D → Y and claim that it has a monotone and surjective extension Φ : D → Y. We are assuming that Y is homeomorphic to [0, 1] 2 and has finite Hausdorff 2-measure. Fix x 0 ∈ S 1 . For each 0 < r < 2 −1 , denote E r = S 1 (x 0 , r) ∩ D. Let V r be the component of D \ E r whose closure contains {x 0 }.
Let U r = φ(V r ). Since V r is connected, so is U r . Moreover, as V r ′ ⊂ V r whenever r ′ < r, we have U r ′ ⊂ U r . Therefore Notice that C ⊂ ∂Y.
Lemma 3.1 implies that C is a singleton. Let y 0 denote the unique element. We define Φ(x 0 ) := y 0 . We repeat the argument for every x 0 ∈ S 1 . By setting Φ(x) = φ(x) for every x ∈ D, we obtain a mapping We prove in Lemma 3.4 that Φ is continuous and surjective. Lemma 3.5 shows the monotonicity of Φ. Hence Theorem 1.1 follows after we verify these lemmas.
Lemma 3.1. Let d r denote the diameter of U r . Then d r → diam C = 0 for every x 0 ∈ S 1 . Before proving Lemma 3.1, we prove a couple of technical lemmas. In the following, an arc refers to a set homeomorphic to [0, 1].
Lemma 3.2. Let C ′ ⊂ Y be an arc and C ⊂ ∂Y a compact and connected set. Then Proof of Lemma 3.2. Since C and C ′ are disjoint, there are Borel functions ρ ∈ L 2 (Y) admissible for Γ(C, C ′ ; Y ∪ C). We fix such a function ρ and find a lower bound for the L 2 -norm of ρ, depending only on C and C ′ . The claim (16) follows from this. We argue as in the proof of [Raj17, Proposition 3.5]. First, we join C and C ′ with an arc γ : [0, 1] → Y ∪ C for which r 0 = d(|γ| , ∂Y \ C) > 0, and consider the Lipschitz function f (z) = d(|γ| , z). Since C and C ′ are arcs, we can choose γ in such a way that γ(0) separates C into two arcs J 1 and J 2 , γ(1) separates C ′ into two arcs J 3 and J 4 , and γ(t) ∈ Y \ (C ∪ C ′ ) for every 0 < t < 1.
Fix 0 < r 1 < r 0 such that every J i intersects f −1 (r) for every 0 < r < r 1 . For every such r > 0, the level set f −1 (r) separates |γ| from the arc ∂Y \ C. Then Lemma 2.2 provides us with a continuum Γ r ⊂ f −1 (r) that separates |γ| from ∂Y \ C. The continuum Γ r must intersect every J i , since otherwise we find a path joining |γ| to ∂Y \ C that does not intersect Γ r .
By applying Theorem 2.1 to the function g(y) = χ Y (y), we conclude that the level set f −1 (r) has finite Hausdorff 1-measure for L 1 -almost every 0 < r < r 1 . In particular, the continuum Γ r has finite Hausdorff 1-measure. Then every pair of points from Γ r can be joined with a rectifiable path within Γ r [Sem96, Proposition 15.1]. Consequently, there exists a rectifiable arc θ : Then Theorem 2.1 and Hölder's inequality imply Rearranging this inequality establishes the claim. Proof of Lemma 3.3. Let 0 < s < t < 1. Then By the absolute continuity of the integral, given ǫ > 0, there exists δ > 0 for which This fact and (17) imply that for any given (s j ) ∞ j=1 ⊂ (0, 1) converging to zero, the sequence (θ(s j )) ∞ j=1 is Cauchy. Since Y is complete, the sequence converges to some y 0 ∈ G. We define θ(0) := y 0 . The inequalities (17) and (18) imply that y 0 is independent of the sequence (s j ) ∞ j=1 , and setting θ(s) = θ(s) for 0 < s defines a continuous extension of θ to [0, 1).
By arguing similarly for t = 1, we find a continuous extension θ : [0, 1] → G of θ. The inequality (17) extends to every 0 ≤ s < t ≤ 1, implying the absolute continuity of θ. Notice that for every y ∈ G, there exists a sequence (t j ) ∞ j=1 ⊂ (0, 1) such that θ(t j ) → y. By passing to a subsequence and relabeling, we may assume that (t j ) ∞ j=1 has a limit in [0, 1]. This implies that θ is surjective.
Proof of Lemma 3.1. Since C is the intersection of the U r and U r are nested, we have d r → diam C. Hence the difficulty lies in proving diam C = 0.
Fix an arc C ′ ⊂ Y for which φ −1 (C ′ ) ⊂ D \ (E r ∪ V r ) for every 0 < r < 2 −1 . We assume diam C > 0 and derive a contradiction. Since diam C > 0, there exist a subarc C ⊂ C such that r 0 = d(C, C ′ ∪ (∂Y \ C)) > 0. We claim that If (19) holds for C, we obtain a contradiction with Lemma 3.2. So it suffices to prove (19). We claim that there exists a sequence r n → 0 + such that every θ ∈ Γ(C, C ′ ; Y ∪ C) has a subpath in Γ(φ(E r n ∩ D), C ′ ; Y). If this can be proved, then the K-quasiconformality of φ yields Given Lemma 2.8, the right-hand side converges to zero as n → ∞, and we conclude (19). The rest of the proof is spent on finding the sequence of radii (r n ) ∞ n=1 . The K-quasiconformality of φ yields that the minimal weak upper gradient ρ φ satisfies ρ 2 φ ≤ K J φ ∈ L 1 (D). The integrability of J φ follows from the fact that Y has finite Hausdorff 2-measure. This implies that φ has an L 2 (D)-integrable upper gradient g [HKST15, Lemma 6.2.2].
We consider g 0 = gχ D ∈ L 2 (D). Polar coordinates centered at x 0 yield (20) In particular, g 0 has a finite path integral over E r for almost every 0 < r < 2 −1 . Let I denote those 0 < r < 2 −1 for which this holds. Let Γ 0 be the family of non-constant rectifiable paths in D along which g 0 is not integrable. Consider an absolutely continuous non-constant path γ : [a, b] → Y with image in Y and which is not an element of Γ 0 . Since g is an upper gradient of φ, we have that Consider a surjective Lipschitz γ r : [0, 1] → E r for r ∈ I. Then, for every 0 < s < t < 1, (21) implies where the last term on the right is finite. Therefore satisfies the assumptions of Lemma 3.3. Hence there exists a continuous extension Since φ is a homeomorphism, θ r (s) ∈ Y for both s = 0, 1. Hence F r is homeomorphic to a circle or an arc. We note that φ(V r ) is one of the connected components of Y \ F r . In particular, U r = φ(V r ) is homeomorphic to [0, 1] 2 , U r ∩ ∂Y is a point or an arc, and C ⊂ C ⊂ U r ∩ ∂Y. As the ends of U r ∩ ∂Y and F r coincide, d(C, ∂Y \ C) > 0 implies that U r ∩ ∂Y is an arc and C ∩ F r = ∅. This means that every path θ ∈ Γ(C, C ′ ; Y ∪ C) has a subpath in Γ(F r ∩ Y, C ′ ; Y). Then (19) follows by taking a sequence (r n ) ∞ n=1 ⊂ I converging to zero.
Lemma 3.4. The mapping Φ : D → Y is continuous and surjective.
Proof. Let x 0 ∈ S 1 . If D ∋ x n → x 0 , the accumulation points of (Φ(x n )) ∞ n=1 are contained in the intersection of U r , where U r are as in the definition of C in (15). Lemma 3.1 shows that the intersection is a singleton, which we defined to be More generally, if D ∋ x n → x 0 , we find for every x n an element z n ∈ D such that d Y (Φ(z n ), Φ(x n )) ≤ 2 −n and z n − x n ≤ 2 −n . Then D ∋ z n → x 0 and Consider now y 0 ∈ Y. Then there exists a sequence Y ∋ y n → y 0 . Up to passing to a subsequence and relabeling, (Φ −1 (y n )) ∞ n=1 converges to some x 0 ∈ D. The continuity of Φ implies Φ(x 0 ) = y 0 .
Proof. Let y ∈ ∂Y and suppose that there are two distinct points x 1 and x 2 from S 1 such that Φ(x 1 ) = y = Φ(x 2 ). Let I i be radial lines from x i to 0 for i = 1, 2 and define I = I 0 ∪ I 1 . Here Φ(I) is a Jordan loop intersecting ∂Y exactly at y. Let U denote the component of Y \ Φ(I) whose closure intersects ∂Y only at y. Then V = Φ −1 (U) is one of the components of D \ I. By construction, V ∩ S 1 is connected and is mapped to the singleton y. Therefore x 1 and x 2 can be joined with a path in Φ −1 (y). The monotonicity of Φ follows.

Proof of Proposition 1.2.
Recall that we fix a quasiconformal homeomorphism φ : D → Y and study its extension Φ under the assumption (2). This is done in several parts. First, Lemma 3.6 proves that the extension is a homeomorphism. Lemma 3.8 implies that Φ is quasiconformal. After this is verified, Lemma 3.9 proves that K O (Φ) = K O (φ) and K I (Φ) = K I (φ). In particular, the maximal dilatations of Φ and φ coincide.
To finish up the proof of Proposition 1.2, we also need to verify that if Φ is a quasiconformal homeomorphism, then the assumption (2) holds. But this follows from the corresponding Euclidean result. So Proposition 1.2 follows after we prove the lemmas mentioned above.
Lemma 3.6. The mapping Φ is a homeomorphism.
Proof. Let y ∈ ∂Y and suppose that there exists a non-trivial continuum E ⊂ Φ −1 (y). Let F ⊂ Y \ B Y (y, R) be a non-trivial arc for some R > 0. For 0 < r < R, let Γ(y, r, where AC(Y) refers to those absolutely continuous paths in Y whose images lie in Y. Since φ = Φ| D is K-quasiconformal, we have that The right-hand side is a strictly positive lower bound; recall Lemma 3.2. Therefore mod Γ(y, r, R) ≥ C 0 > 0 for a constant independent of r. By passing to the limit r → 0 + , we find a contradiction with (2). The injectivity of Φ follows. Since Φ is continuous, surjective, and monotone, we conclude that Φ is a homeomorphism.
Here mod Γ(x, r, 2r) ≤ C 1 for a constant depending only on D. Then Lemma 3.7 yields M ≤ 2KC 1 . The claim follows by passing to the limit r → 0 + . Lemma 3.9. The outer (resp. inner) dilatation of Φ coincides with the outer (resp. inner) dilatation of φ.
Proof. Lemmas 3.6 and 3.8 prove that Φ is a quasiconformal homeomorphism. In D, the minimal weak upper gradients of Φ and φ coincide. This is also true for their Jacobians. Therefore they satisfy (ii) in Theorem 2.4 with the same constant.

Beurling-Ahlfors extension
For this section we fix a quasiconformal Jordan domain Y satisfying (3) and a quasiconformal homeomorphism φ : D → Y. Let Φ : D → Y denote the quasiconformal homeomorphic extension of φ, obtained from Proposition 1.2. We refer to g φ = Φ| S 1 as the boundary map of φ. The goal of this section is to prove Theorem 1.3. We reduce the proof to Proposition 4.1.
Observe that if g φ is a quasisymmetry, then ∂Y has bounded turning as this property is preserved by quasisymmetries [TV80]. Moreover, if we fix an arbitrary quasisymmetry g : S 1 → ∂Y, then h = g −1 φ • g : S 1 → S 1 is a quasisymmetry. The Beurling-Ahlfors extension theorem [BA56] yields the existence of a quasiconformal map H : D → D whose boundary map equals h. Then G = Φ • H : D → Y is the quasiconformal extension of g whose existence we wanted to establish. So Theorem 1.3 is a consequence of the following result.
Proposition 4.1. If ∂Y has bounded turning and satisfies the mass upper bound (3), then the boundary map g φ = Φ| S 1 is a quasisymmetry.
We start the proof of Proposition 4.1 by first establishing Proposition 1.4. There we claim that ∂Y has Assouad dimension at most two.
Let E ⊂ ∂Y be the (closure of the) component of ∂Y \ C r that contains y and let a and b denote the ends of E. Here a, b ∈ E ∩ C r ⊂ f −1 (r) so d(a, y) = r = d(b, y).
Let E a ⊂ ∂Y be the arc ending at a and y with diam E a ≤ λr, and let E b denote the corresponding arc for b and y. Then Let E ′ ⊂ ∂Y be the arc ending at a and b with smaller diameter.
By integrating over the interval (0, r) and by applying Theorem 2.1, we con- The claim follows for C = (4λ) 3 2 π .
Proof of Proposition 1.4. Let 0 < r < diam ∂Y and 0 < ǫ < 1. We consider a point y ∈ ∂Y and an ǫr-separated set S ⊂ B Y (y, r) ∩ ∂Y. We conclude from Lemma 4.2 and (3) that for some C ≥ 1 Therefore #S ≤ Cǫ −2 for some constant C independent of r, y and ǫ. Hence the Assouad dimension of ∂Y is at most 2.
Proof of Proposition 4.1. Let 0 < r 0 be such that for every x ∈ S 1 and every 0 < r ≤ r 0 , we have that 4L g (x, 2r) < diam ∂Y. Fix x ∈ S 1 and 0 < r < r 0 . Let z, a, b ∈ S 1 ∩ D(x, r) be such that 0 < a − z 2 ≤ b − z 2 . We proved in Proposition 1.4 that ∂Y is doubling, so by a result of Tukia-Väisälä [TV80, Theorems 2.15 and 2.23], the quasisymmetry of g = g φ follows if there exists a constant H > 0 depending only on the constants in (3), Lemma 4.2, the bounded turning constant λ of ∂Y, and the maximal dilatation K of φ such that d(g(a), g(z)) ≤ Hd(g(b), g(z)).
Let ℓ = d(g(a), g(z)) and let M > 0 be such that ℓ > Md(g(b), g(z)). If we find H 0 such that M ≤ H 0 independently of a, b and z, we may set H = H 0 . If M ≤ (2λ) 2 , any choice H 0 ≥ (2λ) 2 suffices. So we may assume M > (2λ) 2 . Fix Therefore 2 −1 ℓ < d(g(z ′ ), g(z)). We conclude that Let A ′ be the subarc of ∂Y joining g(a) to g(z ′ ) that does not contain g(z). Then any arc joining A ′ to g(z) within ∂Y must pass through either g(a) or g(z ′ ). Using this fact, the bounded turning of ∂Y yields Let B ′ be the subarc of ∂Y with smallest diameter which ends at g(b) and g(z).
The bounded turning of ∂Y implies that The inclusions (23) and (24) imply that every path γ ∈ Γ(A ′ , B ′ ; Y) has a sub-

Planar quasicircles
We prove Theorem 1.5 in this section. The main result of this section is the following.
Proposition 5.1. Under the assumptions of Theorem 1.5, the Assouad dimension of ∂Y is strictly less than 2.
Proof of Theorem 1.5 assuming Proposition 5.1. Proposition 5.1 states that ∂Y has Assouad dimension strictly less than 2. Having verified this, [HM12] yields the existence of a bi-Lipschitz embedding h : ∂Y → R 2 , i.e., ∂Y is planar. So Theorem 1.5 follows from Proposition 5.1. We split the proof of Proposition 5.1 into a couple of sublemmas.
Lemma 5.2. Suppose that ∂Y has λ-bounded turning and satisfies (a) in Theorem 1.5. Then there exists a constant C p ≥ 1 depending only on λ such that for every x ∈ ∂Y and 0 < r < min {r 0 , diam ∂Y}, there exists Proof. This is only a small modification of the proof of Theorem 8.2 of [MW13] but we include the details here for the convenience of the reader. Let s = 8λ 2 (2λ + 1) and C p = 8λs. Let x ∈ ∂Y and 0 < r < min {r 0 , diam ∂Y}. We claim that there exists a point v ∈ Y such that Suppose for now that r < (4λ) −1 min {r 0 , diam ∂Y}. Then there exists a point z ∈ ∂Y for which d(x, z) ≥ 2λr. We fix such a z.
Let |γ| denote the (closure of the) subarc of ∂Y \ w ∈ ∂Y : d(x, w) = r 4λ that contains x. Let a and b denote the end points of |γ| labelled in such a way that {x, a, z, b} is cyclically ordered on ∂Y. We have that d(x, a) = d(x, b) = r 4λ and |γ| ⊂ B Y x, r 4λ . The relative ALLC condition of ∂Y implies that there exists a path α joining a to b in B Y x, r 4 \ B Y x, r 8λ 2 . We assume without loss of generality that α is an arc.
The lower bound on d(a, |γ b |) and connectedness of |α| imply the existence of v ∈ |α| such that d (v, In either case, (28) holds.
Let r 1 = min {r 0 , diam ∂Y}, where r 0 is the parameter from the assumptions of Theorem 1.5. There exists a constant C ≥ 1 with the following properties: (i) For every y ∈ ∂Y and every 0 < r, H 2 Y (B Y (y, r)) ≤ Cr 2 . Moreover, for every y ∈ ∂Y and every 0 < r < r 1 , H 2 Y (B Y (y, r)) ≥ C −1 r 2 . (ii) For every y ∈ ∂Y and 0 < r < r 1 , there exists z ∈ Y such that B Y (z, z, r)). Remark 5.3. We have assumed that for all radii 0 < r < diam ∂Y and every y ∈ ∂Y, H 2 Y (B Y (y, r)) ≤ C ′ r 2 for some C ′ > 0. Then the upper bound in (i) holds for every r > 0 if we replace The lower bound for such balls follows from Lemma 4.2.
The property (ii) follows from Lemma 5.2 for some constant. Recall that, under the assumptions of Theorem 1.5, the lower bound in (iii) holds for each 0 < r < d(z, ∂Y) ≤ r 0 for some constant C ′ . Hence (iii) follows.
Our claim is qualitative, so we use a uniform constant C for these various conditions.
In the following, if B is a ball and ξ > 0, ξB refers to the ball with the same center and ξ times the radius. where A 1 ≃ A 2 means that there exists a constant of comparability D > 0 for which D −1 A 1 ≤ A 2 ≤ DA 1 . Here the constants of comparability depend only on the constant C.
Proof of Lemma 5.4. For each x ∈ ∂Y and each integer m such that 0 < 2 m < r 1 , with the constants of comparison depending only on C.
For each m ∈ Z with 2 m < r 1 , let B m denote the collection of the B x,m as x ∈ ∂Y varies. The 5r-covering theorem [Fed69,2.8.4] states that there exists subcollection B ′ m ⊂ B m whose elements are pairwise disjoint and (31) Let 1 ≤ N ∈ N and m 1 ∈ Z with 2 m 1 < r 1 ≤ 2 m 1 +1 . Consider the collection B = ∞ k=1 B ′ m 1 −kN . By choosing a sufficiently large N, the elements of B are pairwise disjoint and each element satisfies max {d(∂Y, B), diam B} < r 1 /2. The choice of N depends only on the constants in (30). The inclusion (31) implies that for each 0 < r < r 1 and x ∈ ∂Y, there exists B ∈ B such that (30) holds with B x,m replaced by B and 2 m by r, with constants of comparison depending only on C. Then (29) follows.
Lemma 5.5. There exist 1 > a > 0, b > 0 and ξ > 1, with the choices depending only on the constants of comparability in (29), such that (32) where χ ξB is the characteristic function of the ball ξB.
Proof of Lemma 5.5. For each (w, s) ∈ B(y, 3r) ∩ ∂Y × (0, r 1 ), let B s (w) ∈ B be the ball obtained from Lemma 5.4. Lemma 5.4 implies the existence of A > 1 for which A −1 s ≤ diam B s (w) < As and for the center c s (w) of B s (w), A −1 s ≤ d(w, c s (w)) < As.
Let k ∈ Z be the largest integer for which be a strictly increasing sequence in the interval Aaǫr, r/A 2k+1 . Denote s i = A 2(i−1) s i for each i = 1, 2, . . . k. Here s k < r/A and As i−1 < A −1 s i for each i. Hence the collection {B s i (w)} k i=1 contains k different balls. This implies (33) T(z) ≥ k for every z ∈ B(w, 2 −1 ǫr).
Suppose that the claim holds whenever 0 < r < r 1 . Consider r 1 ≤ r < diam ∂Y. Let E 0 ⊂ ∂Y be a maximal r 1 /2-separated net. For every f ∈ E 0 , the set E f = There are some ways to construct quasiconformal Jordan domains. For example, if X is a metric surface satisfying local versions of annular linear local connectivity and Ahlfors 2-regularity, Theorem 4.17 of [Wil10] yields the existence of many quasiconformal Jordan domains Y ⊂ X satisfying the assumptions of Theorem 1.5. Some examples of quasiconformal Jordan domains can also be obtained from [Haï09, Theorem 2].