Quasiconformal parametrization of metric surfaces with small dilatation
We verify a conjecture of Rajala: if is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain , then there exists a quasiconformal mapping satisfying the modulus inequality for all curve families in . This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball.
Key words and phrases:Quasiconformal mapping, conformal modulus, convex body
2010 Mathematics Subject Classification:30L10, 52A10
A growing body of recent literature has studied the following quasiconformal uniformization problem: for a metric space homeomorphic to a domain in or , under what conditions does there exist a quasiconformal (or quasisymmetric) parametrization of by ? This question originated largely in the work of Semmes; see especially [Sem:96b] and [HeiS:97, Qu. 3-7]. A landmark paper of Bonk and Kleiner [BonkKleiner] gives a complete description of those spaces admitting a quasisymmetric parametrization by under the assumption that is Ahlfors 2-regular; a necessary and sufficient condition is that be linearly locally contractible.
A similar theorem for geometrically quasiconformal parametrizations was recently proven by Rajala [Raj16] in the setting of metric surfaces homeomorphic to or with locally finite Hausdorff 2-measure. A new condition called reciprocality is introduced which is necessary and sufficient for the existence of the desired quasiconformal parametrization. The original Bonk–Kleiner theorem can then be obtained as a corollary. We refer the reader to the introduction of Rajala’s paper for additional background and references. See also Lytchak and Wenger [LyWen:17] for other new results on quasiconformal parametrizations in somewhat the same spirit.
We recall now the relevant definitions. Let be a metric measure space. Given a family of curves in , the -modulus of is
the infimum taken over all Borel functions such that for every locally rectifiable curve . A homeomorphism is -geometrically quasiconformal with exponent if
for all curve families in . The smallest value such that for all curve families in is called the outer dilatation of . Similarly, the smallest value such that for all curve families in is the inner dilatation.
If is understood, we say simply that is -quasiconformal or quasiconformal. In this note, we always take and we write in place of . We will assume that a metric space is equipped with the Hausdorff 2-measure.
The same paper of Rajala also examines a related question: if such a quasiconformal parametrization exists, can one find a quasiconformal mapping which improves the dilatation constants and to within some universal constants? If so, what is the best result of this type? Rajala obtains the following theorem [Raj16, Thm. 1.5]:
(Rajala) Let be a simply connected domain and a metric space of locally finite Hausdorff 2-measure. There exists a quasiconformal homeomorphism if and only if there exists a 2-quasiconformal homeomorphism .
This result is proved using the measurable Riemann mapping theorem along with the classical John’s theorem on convex bodies. The latter theorem asserts, in part, that any convex body in contains a unique ellipsoid of maximal volume satisfying , where the constant is the best possible. The constant 2 in Theorem 1.1 is derived from the constant in John’s theorem for dimension two.
In this note, we prove the following improvement to Theorem 1.1, which was conjectured by Rajala in [Raj16].
Let be a simply connected domain and a metric space of locally finite Hausdorff 2-measure. There exists a quasiconformal homeomorphism if and only if there exists a quasiconformal homeomorphism satisfying
Rajala’s techniques, together with standard volume ratio estimates (see for instance [Ball:97, Thm. 6.2]), guarantee the existence of a quasiconformal map with outer dilatation , and a quasiconformal map with inner dilatation . The improvement in Theorem 1.2 is in finding a map which satisfies both modulus inequalities simultaneously.
Inequality (1) cannot be improved, as shown by taking , where is the metric. That is, every quasiconformal map must satisfy and . Moreover, the identity map satisfies (1). These facts are proved in Example 2.2 of [Raj16].
The simple connectedness assumption is essential to Theorem 1.2. For example, any -quasiconformal mapping between the annular regions and , , must satisfy . In particular, for any there exist annuli such that any -quasiconformal map must satisfy . A similar fact holds for wedge domains in , , which is one indication that a result like Theorem 1.2 is only possible in dimension two. See Väisälä [Vais1, Sec. 39-40] for a discussion of quasiconformal mappings between annular and wedge domains.
Finally, Theorem 1.2 remains true when is replaced by , though for simplicity we do not address that case explicitly.
2. An area inequality for planar convex bodies
The key innovation for proving Theorem 1.2 is to replace the application of John’s theorem by the two lemmas in this section, after which Theorem 1.2 follows by a straightforward modification of Rajala’s proof. See the introductory notes by Ball [Ball:97] for an overview of John’s theorem and related results on volume ratios. However, we have not found any result comparable to our lemmas in the literature on convex bodies.
In the following, will denote the area of the set . We also let denote the outer radius of and denote the inner radius of . A convex body is a compact convex set in with nonempty interior; it is symmetric if implies . There is a natural correspondence between the set of norms on and the set of symmetric convex bodies in . Namely, the unit ball for a norm on is a symmetric convex body, while for any symmetric convex body the function defines a norm on . Terms such as ellipse and polygon should be understood as including the interior of the respective objects.
For a convex body and a linear transformation , let . Define the infimum taken over all . Notice that by John’s theorem. Expressed in different terms, is the (multiplicative) Banach-Mazur distance between and the closed Euclidean unit ball in .
It is easy to verify that there is a matrix such that attains . Consider the family . By John’s theorem, restricting to does not affect the infimal value of . For such a map
we must have and . This is seen by looking at the action of on the test points and . We also have . Hence the set is compact as a subset of and it follows that a nonzero map minimizing must exist.
Let be a symmetric convex body and a linear map such that . Then the image of satisfies .
Let . Without loss of generality we can assume that the outer radius satisfies . Let . Then from John’s theorem it follows that . Where convenient, we will use complex notation for points in . For , let denote the unique point in . By rotating if necessary, we will assume that .
We first need a fact about the existence of contact points with the circles and . Specifically, we claim that there exist values such that and . Suppose this does not hold. Then there exist such that , whenever , and whenever or . Observe that if are such that and , then . In particular, and .
Consider now a small linear stretch in the direction . Expressed in a suitable orthonormal basis for , this linear stretch takes the form for some sufficiently small parameter .
Let . For sufficiently small , consider the function
We obtain the functions by considering the (Euclidean) norm of the image of the points (the numerator) and (the denominator), as expressed relative to the basis . Then is an upper bound for for sufficiently small and satisfies . In particular, . We compute
Since , we see that . This contradicts the minimality of . The existence of the desired values now follows.
We now estimate from above. Write . By covering with the triangles , and the set
along with their reflections about the origin, we obtain
See Figure 0(a). Observe that , so the right inequality holds for . Next, compute . Since this satisfies when , we obtain holds for all .
We can estimate from below using the polygons , , and the set
See Figure 0(b). This gives
Now , so the left inequality holds when . Since , we obtain for all . This completes the proof. ∎
By taking and to be the identity map, we see that Lemma 2.1 is sharp.
The proof of the previous lemma allows us show the next fact, that the linear map which attains is unique up to a conformal transformation.
Let be a symmetric convex body, and let be linear maps such that . Then for some and orthogonal transformation .
Let and , where we have chosen and so that and that ; here and throughout this proof denotes the unique point in . It suffices to prove the lemma with in place of . Note that after making this reduction we must have .
Let denote the closed Euclidean unit ball in . The conclusion of the lemma holds if and only if ; thus it suffices to show that , where . The set is an ellipse , whose boundary consists of those points satisfying the equation for some with and . Let and let .
Recall the assumption that ; this implies that . In particular, if is such that , then by the second inclusion. On the other hand, if is such that , then by the first inclusion. It must follow that , , , and , where satisfy and as in the proof of Lemma 2.1. Then must have at least three critical points on the interval , unless in fact is identically equal to . This leads to a contradiction, since (being derived from the equation for an ellipse, or by inspection) cannot have more than two critical points over the interval . The result follows. ∎
One view on John’s theorem is that it provides one with a canonical choice of ellipse associated to a convex body , namely the ellipse maximizing area. The point of the previous lemma is to justify a different notion of canonical ellipse for a convex body, related to minimizing distance to the Euclidean ball in the sense of the Banach-Mazur distance. This ellipse is obtained by taking for any such that . Lemma 2.2 shows that this ellipse is independent of the choice of .
3. Proof of main theorem
This section gives the proof of Theorem 1.2. It is a modification of the proof which comprises Section 14 of [Raj16]. As such, we will highlight the modifications while referring the reader to [Raj16] for additional details. We will also follow the notation found there where convenient.
For a Lipschitz function into a metric space , we use to denote the metric differential of Kirchheim [Kirc:94] at the point , which exists for a.e. . As explained in [Raj16, Lem. 14.1, 14.2], for every quasiconformal map there exist disjoint measurable sets () covering up to a set of measure zero such that is -Lipschitz. The map can be extended to a Lipschitz map . Then for all and a.e , is a non-zero norm on . This notation will be used in the following proof.
Recall that we are assuming the existence of a quasiconformal homeomorphism . For a.e. , we obtain a non-zero norm on from the metric derivative of the function described above, where is such that . For each such norm , the set is a symmetric convex body in .
Let be an invertible linear mapping for which . Let ; this gives an ellipse field on defined for a.e . As we have seen from Lemma 2.2, the ellipse does not depend on our choice of . Setting for the remaining points in gives an ellipse field defined on all . The associated complex dilatation is measurable and has a uniform bound less than 1. Observe that the analogous ellipse field in Rajala’s proof was obtained using John’s theorem instead.
Applying the measurable Riemann mapping theorem gives a quasiconformal mapping such that
for a.e. and some . Let , observing that differs from by a scaling factor and orthogonal transformation.
Define . As above we obtain Lipschitz pieces for disjoint sets . Then there exists such that , for a.e. , where is such that and . Hence for a.e. , the metric derivative satisfies , and the Jacobian is given by . By Lemma 2.1 we see that
Following [Raj16], this suffices to show the inequality for all curve families in .
Similarly we have
valid for a.e . This gives by Lemma 2.1
which suffices to show that for all curve families in . ∎
Acknowledgments. The author thanks Jeremy Tyson, Kai Rajala and Marius Junge for conversations related to the topic of this paper.