Quasiconformal extension for harmonic mappings on finitely connected domains
QQUASICONFORMAL EXTENSION FOR HARMONICMAPPINGS ON FINITELY CONNECTED DOMAINS
IASON EFRAIMIDIS
Abstract.
We prove that a harmonic quasiconformal mapping defined on afinitely connected domain in the plane, all of whose boundary components areeither points or quasicircles, admits a quasiconformal extension to the wholeplane if its Schwarzian derivative is small. We also make the observation thata univalence criterion for harmonic mappings holds on uniform domains. Introduction
Let f be a harmonic mapping in a planar domain D and let ω = f z /f z beits dilatation. According to Lewy’s theorem the mapping f is locally univalentif and only if its Jacobian J f = | f z | − | f z | does not vanish. Duren’s book [4]contains valuable information about the theory of planar harmonic mappings.The Schwarzian derivative of f was defined by Hern´andez and Mart´ın [8] as S f = ρ zz − ( ρ z ) , where ρ = log J f . (1)When f is holomorphic this reduces to the classical Schwarzian derivative. An-other definition, introduced by Chuaqui, Duren and Osgood [3], applies to har-monic mappings which admit a lift to a minimal surface via the Weierstrass-Enneper formulas. However, focusing on the planar theory in this note we adoptthe definition (1).We assume that C \ D contains at least three points, so that D is equipped withthe hyperbolic metric, defined by λ D (cid:0) π ( z ) (cid:1) | π (cid:48) ( z ) || dz | = λ D ( z ) | dz | = | dz | − | z | , z ∈ D , where D is the unit disk and π : D → D is a universal covering map. The size ofthe Schwarzian derivative of a mapping f in D is measured by the norm (cid:107) S f (cid:107) D = sup z ∈ D λ D ( z ) − | S f ( z ) | . A domain D in C is a K -quasidisk if it is the image of the unit disk under a K -quasiconformal self-map of C , for some K ≥
1. The boundary of a quasidiskis called a quasicircle.According to a theorem of Ahlfors [1], if D is a K -quasidisk then there ex-ists a constant c >
0, depending only on K , such that if f is analytic in D with Mathematics Subject Classification.
Key words and phrases.
Harmonic mapping, Schwarzian derivative, quasiconformal exten-sion, quasiconformal decomposition. a r X i v : . [ m a t h . C V ] F e b I. EFRAIMIDIS (cid:107) S f (cid:107) D ≤ c then f is univalent in D and has a quasiconformal extension to C . Thishas been generalized by Osgood [12] to the case when D is a finitely connecteddomain whose boundary components are either points or quasicircles. Further,the univalence criterion was generalized to uniform domains (see Section 4 for adefinition) by Gehring and Osgood [7] and, subsequently, the quasiconformal ex-tension criterion was generalized to uniform domains by Astala and Heinonen [2].For harmonic mappings and the definition (1) of the Schwarzian derivative, aunivalence and quasiconformal extension criterion in the unit disk D was provedby Hern´andez and Mart´ın [9]. This was recently generalized to quasidisks bythe present author in [5]. Moreover, in [5] it was shown that if all boundarycomponents of a finitely connected domain D are either points or quasicirclesthen any harmonic mapping in D with sufficiently small Schwarzian derivative isinjective. The main purpose of this note is to prove the following theorem. Theorem 1.
Let D be a finitely connected domain whose boundary componentsare either points or quasicircles and let also d ∈ [0 , . Then there exists aconstant c > , depending only on the domain D and the constant d , such that if f is harmonic in D with (cid:107) S f (cid:107) D ≤ c and with dilatation ω satisfying | ω ( z ) | ≤ d for all z ∈ D then f admits a quasiconformal extension to C . As mentioned above, for the case when D is a (simply connected) quasidiskthis was shown in [5] while, on the other hand, for the case d = 0 (when f isanalytic) this was proved by Osgood in [12]. Osgood’s proof amounts to proving aunivalence criterion in f ( D ). Such an approach does not seem to work here sincefor a holomorphic φ on f ( D ) the composition φ ◦ f is not, in general, harmonic.Since isolated boundary points are removable for quasiconformal mappings (see[10, Ch.I, § ∂D consists of n non-degenerate quasicircles. Our proof will be based on the following theoremof Springer [13] (see also [10, Ch.II, § Theorem A ([13]) . Let D and D (cid:48) be two n -tuply connected domains whose bound-ary curves are quasicircles. Then every quasiconformal mapping of D onto D (cid:48) can be extended to a quasiconformal mapping of the whole plane. Hence, to prove Theorem 1 it suffices that we show that the boundary compo-nents of f ( D ) are quasicircles. We prove this in Section 3. It relies on Osgood’s[12] quasiconformal decomposition, which we briefly present in Section 2. InSection 4 we give a univalence criterion on uniform domains.2. Quasiconformal Decomposition
Let D be a domain in C . A collection D of domains ∆ ⊂ D is called a K -quasiconformal decomposition of D if each ∆ is a K -quasidisk and any two points z , z ∈ D lie in the closure of some ∆ ∈ D . This definition was introduced byOsgood in [12], along with the following lemma. Lemma B ([12]) . If D is a finitely connected domain and each component of ∂D is either a point or a quasicircle then D is quasiconformally decomposable. UASICONFORMAL EXTENSION FOR HARMONIC MAPPINGS 3
We now present, almost verbatim, the construction proving Lemma B. We focuson the parts of the construction we will be needing, maintaining the notation of[12] and skipping all the relevant proofs. The interested reader should consult[12] for further details.As we mentioned earlier, we may assume that ∂D consists of non-degeneratequasicircles C , C , . . . , C n − , for n ≥
2. Let F be a conformal mapping of D ontoa circle domain D (cid:48) . Then, with an application of Theorem A to F − , it will besufficient to find a quasiconformal decomposition of D (cid:48) . Hence we may assumethat D itself is a circle domain with boundary circles C j , j = 0 , . . . , n − n = 2 then we may assume that D is the annulus 1 < | z | < R . Then thedomains∆ = { z ∈ D : 0 < arg( z ) < π } , ∆ = e πi/ ∆ , ∆ = e πi/ ∆ make a quasiconformal decomposition of D .Let n ≥
3. Then there exists a conformal mapping Ψ of the circle domain D onto a domain D (cid:48) consisting of the entire plane minus n finite rectilinear slitslying on rays emanating from the origin. The mapping can be chosen so thatno two distinct slits lie on the same ray. The boundary behavior of Ψ is thefollowing: it can be analytically extended to D , and the two endpoints of the slit C (cid:48) j = Ψ( C j ) correspond to two points on the circle C j which partition C j intotwo arcs, each of which is mapped onto C (cid:48) j in a one-to-one fashion.Let ξ j be the endpoint of C (cid:48) j furthest from the origin and let Q (cid:48) j be the part ofthe ray that joins ξ j to infinity. Let also S (cid:48) j be the sector between C (cid:48) j and C (cid:48) j +1 .Let ω (cid:48) j be the midpoint of C (cid:48) j and let P (cid:48) j be a polygonal arc joining ω (cid:48) j to ω (cid:48) j +1 that, except for its endpoints, lies completely in S (cid:48) j . Then P (cid:48) = n − (cid:91) j =0 P (cid:48) j is a closed polygon separating 0 from ∞ that does not intersect any of the Q (cid:48) j .Let G (cid:48) and G (cid:48) be the components of D (cid:48) \ P (cid:48) that contain 0 and ∞ , respectively.Now define ∆ (cid:48) j = G (cid:48) ∪ S (cid:48) j , ∆ (cid:48) j = D (cid:48) \ ∆ (cid:48) j and D (cid:48) = { ∆ (cid:48) j : j = 0 , , . . . , n − } ∪ { ∆ (cid:48) j : j = 0 , , . . . , n − } . This collection has the covering property for D (cid:48) .We denote the various parts of D corresponding under Ψ − to those of D (cid:48) bythe same symbol without the prime. Then D = { ∆ j : j = 0 , , . . . , n − } ∪ { ∆ j : j = 0 , , . . . , n − } is a quasiconformal decomposition of D . I. EFRAIMIDIS
Figure 1.
The slit C (cid:48) j and the three distinguished domains.3. Proof of Theorem 1
Let f be a mapping in D as in Theorem 1. By Theorem 2 in [5], f is injectiveif c is sufficiently small. Also, f extents continuously to ∂D since every boundarypoint of D belongs to ∂ ∆ for some ∆ in the collection D and, by Theorem 1 in[5], the restriction of f on ∆ admits a homeomorphic extension to C .Let Ψ be a conformal mapping of D onto the slit domain D (cid:48) of the previoussection. Let C j be a boundary quasicircle of D . We first prove that f ( C j ) is aJordan curve. The slit C (cid:48) j is divided by its midpoint ω (cid:48) j into two line segments,which we denote by Σ (cid:48) j ( m ) , m = 1 ,
2, so thatΣ (cid:48) j (1) = { z ∈ C (cid:48) j : | z | ≤ | ω (cid:48) j |} and Σ (cid:48) j (2) = { z ∈ C (cid:48) j : | z | ≥ | ω (cid:48) j |} . Let Σ (cid:48) j ( m ) ± denote the two sides of Σ (cid:48) j ( m ), so that a point z on Σ (cid:48) j ( m ) − isreached only by points z ∈ S (cid:48) j − , meaning that arg z → (arg z ) − when z → z . Similarly, a point z on Σ (cid:48) j ( m ) + is reached only by points z ∈ S (cid:48) j , so thatarg z → (arg z ) + when z → z . Corresponding under Ψ − are four disjoint -except for their endpoints- arcs on the quasicircle C j , denoted without the primeby Σ j ( m ) ± , m = 1 ,
2. Now consider the domains ∆ ,j − , ∆ j and ∆ k in thecollection D , for some k (cid:54) = j − , j ; see Figure 1 for their images under Ψ. ByTheorem 1 in [5] f is injective up to the boundary of each ∆ ∈ D . Note thatthe arcs Σ j (1) − , Σ j (1) + and Σ j (2) − are subsets of ∂ ∆ ,j − , so that their imagesunder f , except for their endpoints, are disjoint. It remains to show that theimages of these three arcs under f are not intersected by the remaining image f (Σ j (2) + ). Note that the arcs Σ j (1) − , Σ j (1) + and Σ j (2) + are subsets of ∂ ∆ j ,so that f (Σ j (2) + ) does not intersect f (Σ j (1) − ) nor f (Σ j (1) + ). What remains tobe seen is that f (Σ j (2) − ) and f (Σ j (2) + ) are disjoint and this follows from thefact that the arcs Σ j (2) − and Σ j (2) + are subsets of ∂ ∆ k .To see that the Jordan curve f ( C j ) is actually a quasicircle note that eachpoint of f ( C j ) belongs to some open subarc of f ( C j ) which is entirely includedin the boundary of either f (∆ ,j − ) , f (∆ j ) or f (∆ k ). These three domains arequasidisks by Theorem 3 in [5]. Now the assertion that f ( C j ) is a quasicirclefollows by an application of Theorem 8.7 in [10, Ch.II, § UASICONFORMAL EXTENSION FOR HARMONIC MAPPINGS 5 Remarks on uniform domains
A domain D in C is called uniform if there exist positive constants a and b such that each pair of points z , z ∈ D can be joined by an arc γ ⊂ D so thatfor each z ∈ γ it holds (cid:96) ( γ ) ≤ a | z − z | and min j =1 , (cid:96) ( γ j ) ≤ b dist( z, ∂D ) , where γ , γ are the components of γ \{ z } , dist( z, ∂D ) denotes the euclidean dis-tance from z to the boundary of D and (cid:96) ( · ) denotes euclidean length. Uniformdomains were introduced by Martio and Sarvas [11]; see also, e.g. , [7] for thisequivalent definition. In [11] it was shown that all boundary components of auniform domain are either points or quasicircles. The converse of this is alsotrue for finitely connected domains, but not, in general, for domains of infiniteconnectivity; see [6, § Theorem C ([11],[7]) . If D is a uniform domain then there exists a constant c > such that every analytic function f in D with (cid:107) S f (cid:107) D ≤ c is injective. Gehring and Osgood [7] gave a different proof of Theorem C by providing acharacterization of uniform domains. They showed that a domain D is uniformif and only if it is quasiconformally decomposable in the following weaker (thanthe one we saw in Section 2) sense: there exists a constant K with the propertythat for each z , z ∈ D there exists a K -quasidisk ∆ ⊂ D for which z , z ∈ ∆.Note that, in contrast to Osgood’s [12] decomposition, here ∆ depends on thepoints z , z . However, this can readily be used to generalize the implication (i) ⇒ (iii) of Theorem 2 in [5], according to which a univalence criterion for harmonicmappings holds on finitely connected uniform domains. The following theoremextends it to all uniform domains. Theorem 2.
Let D be a uniform domain in C . Then there exists a constant c > such that if f is harmonic in D with (cid:107) S f (cid:107) D ≤ c then f is injective.Proof. Assume that there exist distinct points z , z ∈ D for which f ( z ) = f ( z ).By [7], there exists a K -quasidisk ∆ ⊂ D for which z , z ∈ ∆. The domainmonotonicity for the hyperbolic metric shows that (cid:107) S f (cid:107) ∆ ≤ (cid:107) S f (cid:107) D ≤ c. But the homeomorphic extension of Theorem 1 in [5] shows that if c is sufficientlysmall then f is injective up to the boundary of ∆, a contradiction. (cid:3) Regarding quasiconformal extension, Astala and Heinonen [2] proved the fol-lowing theorem.
Theorem D ([2]) . If D is a uniform domain then there exists a constant c > such that every analytic function f in D with (cid:107) S f (cid:107) D ≤ c admits a quasiconformalextension to C . I. EFRAIMIDIS
This evidently implies Theorem C and was also proved in substantially greatergenerality, but we omit it here. It is not clear how to generalize Theorem D tothe setting of harmonic mappings. Therefore, we propose the following problem.
Problem.
Let D be a uniform domain. Does there exist a constant c > f is harmonic in D with (cid:107) S f (cid:107) D ≤ c and with dilatation ω satisfyingsup z ∈ D | ω ( z ) | < f admits a quasiconformal extension to C ? References [1] L. Ahlfors, Quasiconformal reflections,
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