aa r X i v : . [ m a t h . C V ] M a r REMOVABILITY AND NON-INJECTIVITY OF CONFORMALWELDING
MALIK YOUNSI
Abstract.
We construct a (non-removable) Jordan curve Γ and a non-M¨obiushomeomorphism of the Riemann sphere which is conformal on the complementof Γ and maps the curve Γ onto itself. The curve is flexible in the sense ofBishop and may be taken to have zero area. The existence of such curvesand conformal homeomorphisms is closely related to the non-injectivity ofconformal welding. Introduction
Let D be the open unit disk, let D ∗ := b C \ D be the complement of the closedunit disk in the Riemann sphere b C , and let T := ∂ D be the unit circle. Givena Jordan curve Γ, let f : D → Ω and g : D ∗ → Ω ∗ be conformal maps onto thebounded and unbounded complementary components of Γ respectively. Then f and g extend to homeomorphisms on the closure of their respective domains, sothat h Γ := g − ◦ f : T → T defines an orientation-preserving homeomorphism ofthe unit circle onto itself, called the conformal welding homeomorphism of Γ.Note that h Γ is uniquely determined by Γ up to pre- and post-composition byautomorphisms of the unit disk. Moreover, if T is a M¨obius transformation, then Γand T (Γ) have the same conformal welding homeomorphism, so that the conformalwelding correspondence W : [Γ] [ h Γ ]is well-defined, from the family of Jordan curves, modulo M¨obius equivalence, tothe family of orientation-preserving homeomorphisms of the circle, modulo pre- andpost-composition by automorphisms of the disk.This correspondence between Jordan curves and circle homeomorphisms has ap-peared over the years to be of central importance in a wide variety of areas ofmathematics and applications, such as Teichm¨uller theory, Kleinian groups, com-puter vision and numerical pattern recognition ([6],[30]), and so forth. For moreinformation on the applications of conformal welding, the interested reader mayconsult the survey article [9]. We also mention that recent years have witnesseda strong renewal of interest in conformal welding as other variants and generaliza-tions have been introduced and developed, such as generalized conformal welding([4], [8]), random conformal welding [1], conformal welding for finitely connected re-gions [20], conformal welding of random surfaces [29] and conformal laminations oftrees ([21], [27]), including applications to Shabat polynomials and Grothendieck’s Date : March 2, 2017.2010
Mathematics Subject Classification. primary 30C35; secondary 37E10, 30C85.
Key words and phrases.
Conformal welding, removability, flexible curves, capacity. dessins d’enfants . Conformal laminations are also related to the recent ground-breaking work of Miller and Sheffield on the relationship between the Brownianmap and Liouville Quantum Gravity ([22], [23], [24]).It is well-known that the conformal welding correspondence W is not surjective;in other words, there are orientation-preserving homeomorphisms of the circle (evenanalytic everywhere except at one point) which are not conformal welding homeo-morphisms. On the other hand, every quasisymmetric homeomorphism h : T → T is the conformal welding of some Jordan curve (in this case, a quasicircle). Here quasisymmetric means that adjacent arcs I, J ⊂ T of equal length are mapped by h onto arcs of comparable length : M − ≤ | h ( I ) || h ( J ) | ≤ M. The fact that every such h is a conformal welding homeomorphism is usually re-ferred to as the fundamental theorem of conformal welding and was first provedby Pfluger [26] in 1960. Another proof based on quasiconformal mappings waspublished shortly after by Lehto and Virtanen [17]. See also the papers of Bishop[4] and Schippers–Staubach [28]. We also mention that other sufficient conditionsfor a given circle homeomorphism h to be a welding were obtained by Lehto [16]and Vainio [31], in terms of h being sufficiently “nice”. In Section 2, we recall adeep theorem of Bishop [4] saying that on the other hand, any “wild” enough h is also the welding of some Jordan curve. Finding a complete characterization ofconformal welding homeomorphisms is most likely a very difficult problem.This paper, however, deals with the (non) injectivity of the welding correspon-dence W . Recall that if Γ is a Jordan curve, then T (Γ) has the same weldinghomeomorphism h Γ , for any M¨obius transformation T . Are these the only curveswith this property?If Γ and e Γ are two Jordan curves such that h Γ = h e Γ , then there are conformalmaps f : D → Ω, g : D ∗ → Ω ∗ , e f : D → e Ω, e g : D ∗ → e Ω ∗ such that g − ◦ f = e g − ◦ e f on T , i.e. e g ◦ g − = e f ◦ f − on Γ, where e Ω , e Ω ∗ are the bounded and unbounded complementary components ofthe curve e Γ. It follows that the map F : b C → b C defined by F ( z ) = (cid:26) ( e f ◦ f − )( z ) if z ∈ Ω ∪ Γ( e g ◦ g − )( z ) if z ∈ Ω ∗ is a homeomorphism conformal on b C \ Γ which maps the curve Γ onto the curve e Γ.This shows that if Γ is conformally removable, then Γ uniquely corresponds to itsconformal welding homeomorphism h Γ , up to M¨obius equivalence. Definition 1.1.
We say that a compact set E ⊂ C is conformally removable ifevery F ∈ CH( E ) is M¨obius, where CH( E ) is the collection of homeomorphisms F : b C → b C which are conformal on b C \ E .Single points, smooth curves and more generally, sets of σ -finite length, are allconformally removable. On the other hand, the theory of quasiconformal mappingscan be used to prove that sets of positive area are never conformally removable. The EMOVABILITY AND NON-INJECTIVITY OF CONFORMAL WELDING 3 converse is well-known to be false, and there exist nonremovable sets (even Jordancurves) of Hausdorff dimension one ([3], [15]) and removable sets of Hausdorff di-mension two ([18, Chapter V, Section 3.7]). In fact, no geometric characterizationof conformally removable sets is known. We also mention that this notion of remov-ability appears naturally in the study of various important problems in ComplexAnalysis and related areas, such as Koebe’s uniformization conjecture ([10], [33])and the MLC conjecture on the local connectivity of the Mandelbrot set ([11], [12],[13]). See also the survey article [32] for more information.Now, recall that if a Jordan curve Γ is conformally removable, then Γ uniquelycorresponds to its conformal welding homeomorphism, modulo M¨obius equivalence.The starting point of this paper is the following question.
Question 1.2.
Does the converse hold? Namely, if Γ is a non-removable Jordancurve, does there necessarily exist another curve having the same welding homeo-morphism, but which is not a M¨obius image of Γ ? Several papers in the literature seem to imply that the answer is trivially yes,either without proof or with the following argument : If Γ is not removable, then there exists a non-M¨obius F ∈ CH(Γ) . But then thecurve F (Γ) has the same welding homeomorphism as Γ , and is not M¨obius equiva-lent to it, since F is not a M¨obius transformation See e.g. [25, Lemma 2], [14, Corollary II.2], [8, Section 4], [2, Corollary 1], [3,p.324–325], [9, Section 3], [4, Remark 2], [1, Section 2.3], [19, Corollary 1.4]).Although it is true and easy to see that Γ and F (Γ) have the same welding home-omorphism, it is not clear at all that these two curves are not M¨obius equivalent,since there could a priori exist a M¨obius transformation T such that F (Γ) = T (Γ),even though F itself is not M¨obius. As far as we know, this remark first appeared inMaxime Fortier Bourque’s Master’s Thesis [7]. The question of whether such Γ , F and T as above actually exist was, however, left open. In this paper, we answerthat question in the affirmative. Theorem 1.3.
There exists a Jordan curve Γ and a non-M¨obius homeomorphism F : b C → b C conformal on b C \ Γ such that F (Γ) = Γ . Moreover, the curve Γ may betaken to have zero area. The construction is based on a result of Bishop [4] characterizing the conformalwelding homeomorphisms of so-called flexible curves.Theorem 1.3 shows that the above argument claiming to answer Question 1.2 is infact incorrect, and whether removability really characterizes injectivity of conformalwelding remains unknown.The remainder of the paper is organized as follows. In Section 2, we recallBishop’s characterization of the conformal welding homeomorphisms of flexiblecurves. Then, in Section 3, we use this result to prove Theorem 1.3. In Sec-tion 4, we discuss the non-injectivity of conformal welding for curves of positivearea. Finally, Section 5 contains some open problems related to Question 1.2.2.
Flexible curves and log-singular homeomorphisms
We first need the definition of logarithmic capacity, following [4]. For E ⊂ T Borel, let P ( E ) denote the collection of all Borel probability measures on E . M. YOUNSI
Definition 2.1.
Let µ ∈ P ( E ).(i) The energy of µ , noted I ( µ ), is given by I ( µ ) := Z Z log 2 | z − w | dµ ( z ) dµ ( w ) . (ii) The logarithmic capacity of E , noted cap( E ), is defined ascap( E ) := 1inf { I ( µ ) : µ ∈ P ( E ) } . It is well-known that logarithmic capacity is nonnegative, monotone and count-ably subadditive (see e.g. [5]). We will also need the simple fact that bi-H¨olderhomeomorphisms of the circle preserve sets of zero capacity.We can now define log-singular homeomorphisms.
Definition 2.2.
Let
I, J be two subarcs of the unit circle. An orientation-preservinghomeomorphism h : I → J is log-singular if there exists a Borel set E ⊂ I suchthat both E and h ( I \ E ) have zero logarithmic capacity.The following inductive construction of log-singular homeomorphisms was out-lined in [4, Remark 9]. We reproduce it here for the reader’s convenience. Proposition 2.3.
Let
I, J be two subarcs of T . Then there exists a log-singularhomeomorphism h : I → J .Proof. Start with any orientation-preserving linear homeomorphism h : I → J .At the first step, divide I into two subarcs, denoted by red and blue respectively,in such a way that the red subarc has small logarithmic capacity, say less than 2 − .Now, define a homeomorphism h on I which is linear on the red subarc and theblue subarc, and which satisfies h ( I ) = h ( I ) = J . We also construct h such thatit maps the blue subarc onto an arc of logarithmic capacity also less than 2 − .Now, at the n -th step, suppose that I has been divided into a finite number ofarcs { I k,n } , and that we have a homeomorphism h n : I → J which is linear on eachof those arcs. First, divide each I k,n into n arcs of equal length I k,n , I k,n , . . . , I k,nn .Then, divide each I k,nj into a red and a blue subarc, in such a way that the unionof all the red subarcs has logarithmic capacity less than 2 − n . Now, define a home-omorphism h n +1 : I → J which is linear on each of the red and blue subarcs,and which satisfies h n +1 ( I k,nj ) = h n ( I k,nj ) for each j, k . We also construct h n +1 sothat the union of all the images of the blue subarcs under the map has logarithmiccapacity less than 2 − n .It is not difficult to see that these maps h n converge to an orientation-preservinghomeomorphism h : I → J . Indeed, if ǫ >
0, then we can choose N sufficientlylarge so that for n ≥ N , the arcs h n ( I k,nj ) all have length less than ǫ . If m ≥ n and x ∈ I , then x belongs to one of the arcs I k,nj and h m ( x ) belongs to h n ( I k,nj ) byconstruction, thus | h n ( x ) − h m ( x ) | < ǫ. This shows that the sequence ( h n ) is uniformly Cauchy and therefore has a contin-uous limit h : I → J , which has to be an orientation-preserving homeomorphism,by construction. EMOVABILITY AND NON-INJECTIVITY OF CONFORMAL WELDING 5
Finally, the map h : I → J is log-singular. Indeed, if E is the set of points in I which belong to infinitely many of the red subarcs, thencap( E ) ≤ X n ≥ m − n ( m ∈ N )by the subadditivity of logarithmic capacity, so that cap( E ) = 0. On the otherhand, we have h ( T \ E ) ⊂ ∞ [ m =1 \ n ≥ m h n +1 ( B n ) , where B n is the union of all the blue subarcs constructed at the n -th step. It followsthat h ( T \ E ) is contained in a countable union of sets of zero logarithmic capacity,so that cap( h ( T \ E )) = 0, again by the subadditivity of logarithmic capacity. (cid:3) We will also need the following definition.
Definition 2.4.
A Jordan curve Γ ⊂ C is a flexible curve if the following twoconditions hold :(i) Given any Jordan curve e Γ and ǫ >
0, there exists F ∈ CH(Γ) such that d ( F (Γ) , e Γ) < ǫ, where d is the Hausdorff distance.(ii) Given points z , z in each complementary component of Γ and points w , w in each complementary component of e Γ, we can choose F above so that F ( z ) = w and F ( z ) = w .One can think of a flexible curve Γ as being “highly” non-removable, in the sensethat CH(Γ) is very large. Examples of flexible curves with any Hausdorff dimensionbetween 1 and 2 were constructed in [3].There is a close relationship between flexible curves and log-singular homeomor-phisms. Indeed, Bishop proved in [4] that an orientation-preserving homeomor-phism h : T → T is log-singular if and only if it is the conformal welding of aflexible curve Γ. In particular, this implies that every such h is the conformal weld-ing of a dense family of curves, since if F ∈ CH(Γ) is as in Definition 2.4, then h is also the conformal welding homeomorphism of F (Γ). We shall actually need thefollowing stronger result, see [4, Theorem 25]. Theorem 2.5 (Bishop) . Let h : T → T be an orientation-preserving log-singularhomeomorphism with h (1) = 1 , and let f , g be two conformal maps of D and D ∗ respectively onto disjoint domains. Then for any < r < and any ǫ > , thereare conformal maps f and g of D and D ∗ onto the two complementary componentsof a Jordan curve Γ satisfying the following conditions : (i) h = g − ◦ f on T ; (ii) | f ( z ) − f ( z ) | < ǫ for all z ∈ D with | z | ≤ r ; (iii) | g ( z ) − g ( z ) | < ǫ for all z ∈ D ∗ with | z | ≥ /r ;Moreover, the maps f, g may be constructed such that f (1) = g (1) = ∞ and suchthat the curve Γ has zero area.Remark. Since the last part of Theorem 2.5 was not stated explicitly in [4], let usbriefly explain how it follows from the construction. First, since h (1) = 1, condition M. YOUNSI ( i ) implies that f (1) = g (1). Composing f and g by a M¨obius transformation T ifnecessary, we can assume that f (1) = g (1) = ∞ . Note that if f and g approximate T − ◦ f and T − ◦ g on compact subsets of D and D ∗ respectively, then T ◦ f and T ◦ g approximate f and g .Now, to see that the curve Γ can be taken to have zero area, note that the mainpart of the proof in [4] is to construct quasiconformal mappings f : D → Ω and g : D → Ω ∗ onto the complementary components of a Jordan curve Γ satisfyingconditions (i), (ii) and (iii), and having quasiconstants close to 1. These maps f and g are obtained as limits of quasiconformal mappings f n : D → Ω n and g n : D ∗ → Ω ∗ n onto smooth Jordan domains with disjoint closures, with ∞ ∈ Ω ∗ n . Byconstruction, the curve Γ is contained in the topological annulus A n := b C \ (Ω n ∪ Ω ∗ n ),for each n . Moreover, the domains Ω n and Ω ∗ n are of the form Ω n := F n ( t D ) andΩ ∗ n := G n ( D ∗ /t ), where t < F n , G n are quasiconformalmappings of D , D ∗ onto the complementary components of a smooth Jordan curveΓ n . Clearly, by taking t closer to 1 if necessary, we can assume that the topologicalannulus A n has area as small as we want, say less than 2 − n . Then the curve Γ willhave zero area.Finally, the last part of the proof is to apply the measurable Riemann mappingtheorem to replace f and g by conformal maps Φ ◦ f and Φ ◦ g , where Φ is aquasiconformal mapping of the sphere. Since Φ preserves sets of area zero andsince it can be assumed to fix ∞ by composing with a M¨obius transformation, thenew conformal maps can also be taken so that they send 1 to ∞ and map T ontoa curve of zero area.3. Conformal homeomorphisms fixing a curve
We can now prove Theorem 1.3. Recall that we want to construct a Jordancurve Γ and a non-M¨obius homeomorphism F : b C → b C conformal on b C \ Γ suchthat F (Γ) = Γ.The idea of the construction is the following. Suppose that such a curve Γ andsuch a non-M¨obius homeomorphism F ∈ CH(Γ) exist, and suppose that F preservesthe orientation of Γ. Let f : D → Ω and g : D ∗ → Ω ∗ be conformal maps ontothe two complementary components of the curve. Then F ◦ f and F ◦ g are alsoconformal maps of D and D ∗ onto Ω and Ω ∗ respectively, so that F ◦ f = f ◦ σ and F ◦ g = g ◦ τ for some σ, τ ∈ Aut( D ). Note that neither σ nor τ is the identity, since otherwise F would also be the identity, contradicting the fact that it is not M¨obius. Now, since F is continuous on Γ, we must have f ◦ σ ◦ f − = g ◦ τ ◦ g − there, which can be rewritten as(1) W ◦ σ = τ ◦ W on T , where W := h Γ is the conformal welding of Γ. We thus obtain a functionalequation for the welding of the curve.The strategy now is to proceed backward. Start with an orientation-preservinghomeomorphism W : T → T satisfying the functional equation (1) for some σ, τ ∈ EMOVABILITY AND NON-INJECTIVITY OF CONFORMAL WELDING 7
Aut( D ). Suppose in addition that we can construct W so that it is the conformalwelding homeomorphism of some Jordan curve Γ, i.e. W = g − ◦ f where f and g areconformal maps of D and D ∗ respectively onto the two complementary componentsΩ, Ω ∗ of γ . Then we can define a map F conformal on both sides of γ by F ( z ) = (cid:26) ( f ◦ σ ◦ f − )( z ) if z ∈ Ω( g ◦ τ ◦ g − )( z ) if z ∈ Ω ∗ , and the fact that W = g − ◦ f satisfies Equation (1) implies that F extends to ahomeomorphism of the whole sphere. Clearly, the map F ∈ CH(Γ) fixes the curveΓ, and all that remains to prove is that we can choose σ, τ and W such that F isnot a M¨obius transformation.The main difficulty here is that if we choose W to be sufficiently nice, e.g.quasisymmetric, so that it is a conformal welding homeomorphism, then the curveΓ will be removable and F will necessarily be M¨obius. In order to circumvent thisdifficulty, a promising approach is to construct log-singular homeomorphic solutionsof the functional equation (1). Lemma 3.1.
Let a, b > , and let φ be a M¨obius transformation mapping the upperhalf-plane H onto the open unit disk D with φ ( ∞ ) = 1 , say φ ( z ) := z − iz + i . Define e σ, e τ ∈ Aut( H ) by e σ ( z ) := z + a and e τ ( z ) := z + b , and set σ := φ ◦ e σ ◦ φ − and τ := φ ◦ e τ ◦ φ − , so that σ, τ ∈ Aut( D ) . Then there exists a log-singular orientation-preserving home-omorphism W : T → T which satisfies W ◦ σ = τ ◦ W Proof.
Let I := φ ([0 , a )) and J := φ ([0 , b )), and let W : I → J be a log-singular orientation-preserving homeomorphism, as in Proposition 2.3. For n ∈ Z ,set I n := σ n ( I ) and J n := τ n ( J ). Note that the subarcs I n are pairwise disjointand that [ n ∈ Z I n = T \ { } , and similarly for the J n ’s. Now, extend W to all of T by setting W ( z ) := ( τ n ◦ W ◦ σ − n )( z ) ( z ∈ I n )and W (1) := 1. Clearly, the map W : T → T thereby obtained is an orientation-preserving homeomorphism. Moreover, if z ∈ I n for some n , then σ ( z ) ∈ I n +1 , sothat W ( σ ( z )) = ( τ n +1 ◦ W ◦ σ − ( n +1) )( σ ( z ))= ( τ ◦ ( τ n ◦ W ◦ σ − n ))( z )= ( τ ◦ W )( z ) , which shows that W satisfies Equation (1).It remains to prove that W : T → T satisfies the log-singular condition. Let E ⊂ I such that cap( E ) = 0 and cap( W ( I \ E )) = 0. For n ∈ Z , let E n := M. YOUNSI σ n ( E ) ⊂ I n . Note that cap( E n ) = 0 for all n , since bi-H¨older homeomorphismspreserve sets zero logarithmic capacity. It follows that E := ∪ n E n also has capacityzero, by subadditivity. Finally, we have W ( T \ E ) = [ n ∈ Z W ( I n \ E n ) ∪ { } = [ n ∈ Z W ( σ n ( I \ E )) ∪ { } = [ n ∈ Z τ n ( W ( I \ E )) ∪ { } , which shows that cap( W ( T \ E )) = 0, again by subadditivity and the fact thatbi-H¨older homeomorphisms preserve sets of zero logarithmic capacity.This completes the proof of the lemma. (cid:3) We can now proceed with the proof of Theorem 1.3.
Proof.
Let a, b > a = b , and let W : T → T , σ and τ be as in Lemma 3.1, sothat W ◦ σ = τ ◦ W. Also, let 0 < ǫ < | a − b | /
4, let z , z be points in the upper half-plane H and lowerhalf-plane H ∗ respectively, and let K ⊂ H and K ⊂ H ∗ be compact sets suchthat z , z + a ∈ K and z , z + b ∈ K . Lastly, take 0 < r < φ ( K ) ⊂ D (0 , r ) and φ ( K ) ⊂ C \ D (0 , /r ), where φ is the M¨obiustransformation of Lemma 3.1.By Theorem 2.5, we can write W := g − ◦ f , where f and g are conformal mapsof D and D ∗ onto the two complementary components Ω and Ω ∗ of a Jordan curveΓ, such that f (1) = g (1) = ∞ ,(2) | f ( z ) − φ − ( z ) | < ǫ ( | z | ≤ r )and(3) | g ( z ) − φ − ( z ) | < ǫ ( | z | ≥ /r ) . In particular, the above inequalities hold for z ∈ φ ( K ) and z ∈ φ ( K ) respectively.Now, define a map F by F ( z ) = (cid:26) ( f ◦ σ ◦ f − )( z ) if z ∈ Ω( g ◦ τ ◦ g − )( z ) if z ∈ Ω ∗ , so that F is conformal on b C \ Γ. Also, the equation W ◦ σ = τ ◦ W on T isequivalent to f ◦ σ ◦ f − = g ◦ τ ◦ g − on Γ, so that F extends to a homeomorphism of b C . Clearly, this map satisfies F (Γ) = Γ.It remains to prove that F is not a M¨obius transformation. Suppose, in order toobtain a contradiction, that F is M¨obius. First, note that F ( ∞ ) = ∞ , so that F has to be linear, say F ( z ) = cz + d . Now, the map F can be rewritten as F ( z ) = (cid:26) ( e f ◦ e σ ◦ e f − )( z ) if z ∈ Ω( e g ◦ e τ ◦ e g − )( z ) if z ∈ Ω ∗ , EMOVABILITY AND NON-INJECTIVITY OF CONFORMAL WELDING 9 where e f := f ◦ φ : H → Ω, e g := g ◦ φ : H ∗ → Ω ∗ , and e σ ( z ) := z + a , e τ ( z ) := z + b are as in Lemma 3.1. It is easy to see from this that F has only one fixed point, atinfinity, so that c = 1.Now, note that for z ∈ H , we have f ( φ ( z )) + d = e f ( z ) + d = F ( e f ( z )) = ( e f ◦ e σ )( z ) = e f ( z + a ) = f ( φ ( z + a )) , so that in particular, d − a = f ( φ ( z + a )) − f ( φ ( z )) − a = f ( φ ( z + a )) − ( z + a ) − f ( φ ( z )) + z . By Inequality (2) with z replaced by φ ( z ) , φ ( z + a ) ∈ φ ( K ), we get | d − a | ≤ ǫ + ǫ = 2 ǫ. Similarly, using the formula for F on Ω ∗ and Inequality (3), we get | d − b | ≤ ǫ, and combining the two inequalities yields | a − b | ≤ | a − d | + | d − b | ≤ ǫ, which contradicts the choice of ǫ . It follows that F is not a M¨obius transformation.Finally, we can take Γ to have zero area, by Theorem 2.5.This completes the proof of Theorem 1.3. (cid:3) Non-injectivity of conformal welding for curves of positive area
In this section, we mention that the argument described in the introduction,although incorrect in general, can nonetheless be made to work in the case ofcurves with positive area.
Theorem 4.1. If Γ is a Jordan curve with positive area, then there is anothercurve having the same welding homeomorphism, but which is not a M¨obius imageof Γ . The idea of the proof is quite simple. Since Γ has positive area, it is in particularnot removable, so there is a non-M¨obius homeomorphism F : b C → b C which isconformal off Γ. As already mentioned, the curve F (Γ) has the same welding as Γ,thus is a good candidate for the curve we want to construct. Unfortunately, as wesaw in Section 3, it may happen that this curve is a M¨obius image of the originalone, even though F itself is not M¨obius.However, in the positive-area case, it is easy to see using the measurable Riemannmapping theorem that the collection of non-M¨obius elements of CH(Γ) is infinite-dimensional, in some sense. A dimension argument relying on Ahlfors-Bers andBrouwer’s Invariance of Domain can then be applied to conclude that there mustbe at least one non-M¨obius F ∈ CH(Γ) such that F (Γ) = T (Γ) for every M¨obiustransformation T .As far as we know, Theorem 4.1 was first stated by Katznelson, Nag and Sullivanin [14]. See [7, Theorem 4.22] for a complete and detailed proof. It would be veryinteresting to find a more constructive proof though. Concluding remarks
In view of Theorem 4.1, Question 1.2 can be reduced to the following.
Question 5.1. If Γ is a non-removable Jordan curve with zero area, does therenecessarily exist another curve having the same conformal welding homeomorphism,but which is not a M¨obius image of Γ ? As observed in Section 4, the size of CH(Γ) may be relevant here.
Question 5.2. If Γ is a non-removable Jordan curve with zero area, is the collectionof non-M¨obius elements of CH(Γ) necessarily large, or infinite-dimensional, in somesense?
If Γ is non-removable, then there exists at least one non-M¨obius homeomorphism F : b C → b C conformal off Γ, but as far as we know, it is still open in general whetherthere must exist another non-M¨obius element of CH(Γ) which is not of the form T ◦ F , for T M¨obius.Finally, we conclude by mentioning that a positive answer to Question 5.1 wouldfollow if one could prove that there always exists a non-M¨obius homeomorphism of b C conformal off Γ which maps Γ onto a curve of positive area. Acknowledgments.
The author thanks Chris Bishop and Don Marshall for helpfuldiscussions.
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