A CHARACTERISATION OF PLANE QUASICONFORMAL MAPS USING TRIANGLES
Javier Aramayona and Peter Ha¨ıssinsky Abstract
We show that an injective continuous map between planar regions which distorts vertices of equilateral triangles by a small amount is quasiconformal.
Quasiconformal maps have become an important class of homeomor- phisms, for they arise in many fields of mathematics, such as pde’s, Teichm¨uller theory, hyperbolic geometry, complex dynamics, etc. Their involvement may be explained from the numerous characterisations of quasiconformality involving different flavours, which generally amount to loosening characterisations of conformal maps.
Let Ω⊂Cbe a domain in the plane, and let us first defineδΩ(z) = dist(z,C\Ω). Letf: Ω→Cbe an injective continuous map. Forz∈Ω andr∈(0, δΩ(z)), one may consider
Lf(z, r) = sup{|f(z)−f(w)|,|z−w|=r}, and ℓf(z, r) = inf{|f(z)−f(w)|, |z−w|=r}. Let us set Hf(z, r) =Lf(z, r)/ℓf(z, r) and
Hf(z) = lim sup
r→0
Hf(z, r)∈[1,∞].
The metric definition of F. W. Gehring asserts that f isK-quasiconfor- mal ifHf is finite everywhere, and ifHf≤K a.e. [3].
In his monograph [5], J. H. Hubbard proposes a new formulation of plane quasiconformal maps in terms of distortion of triangles (Defini- tion 4.5.1 therein). IfT is a Euclidean triangle inCwith verticesV(T) =
2000Mathematics Subject Classification. 30C62.
Key words. Planar quasiconformal maps, skewedness of triangles, conformal property.
{z1, z2, z3}, i.e., the convex hull ofV(T), one defines the skew ofT as skew(T) = skew(V(T))
= inf{L∈[1,∞],|zi−zj| ≤L|zi−zk|,{i, j, k}={1,2,3}}. He proves that if f: Ω →f(Ω) is a homeomorphism for which there is an increasing homeomorphismh:R+→R+ such that
skew(f(T)) = skew(f(V(T)))≤h(skew(T))
for any T ⊂ Ω, then f is quasiconformal. He then mentions that it suffices to control the distortion of triangles with skew bounded above by p
7/3 in order to establish the result, and he poses the question of whether it is enough to have control only on equilateral triangles (see the remark following Exercise 4.5.12 in [5]).
In this note, we prove
Theorem. There is a constant ε0 > 0 such that, if ε ∈ [0, ε0) and if f: Ω→f(Ω) is a homeomorphism such that
skew(f(T))≤1 +ε
for any equilateral triangle T ⊂ Ω, then f is K-quasiconformal for some K=K(ε).
This statement is essentially local. We were unable to prove or dis- prove the quasiconformality for arbitraryǫ >0. At first glance, imposing a uniform upper bound on the distortion of equilateral triangles looks like a slight variation of standard metric characterisations of quasicon- formal maps. However, it turns out to be a much weaker condition: for example, knowing the distance between the images of 0 and 1 only gives control on the images of the 6th roots of unity, saying nothing about the images of points closer to the origin. In fact, it is not at all clear that a map which distorts equilateral triangles by a uniform amount should even be continuous if this is not required a priori. This highlights some of the difficulties one would have to overcome in order to prove quasi- conformality in general, which does not seem obvious even for ε = 1.
The construction of potential counterexamples is also a delicate issue.
Outline of the paper. We first establish a local criterion of quasi- conformality (Corollary 1.2). We then prove the theorem under an ad- ditional property, but for any ε > 0 (Proposition 2.1). We prove this property is satisfied if ε is small enough (Proposition 2.2), completing the proof of our main result. In the Appendix, we will provide a short proof of the following fact, which improves the bound of p
7/3 given
in [5]: Letµ >0 and λ <∞. Iff: Ω→f(Ω) is a homeomorphism such that
skew(f(T))≤λ
for any triangleT ⊂Ω with skew(T)≤1+µ, thenfisK-quasiconformal for some K=K(λ, µ).
Classic references on quasiconformal maps include the monographs [1], [6], [7].
Acknowledgements. We would like to thank J. H. Hubbard for having drawn our attention to this problem. We are also grateful to the referee for her or his thorough work. We thank E. Maderna for providing a por- tion of the argument in the proof of Lemma 2.7. Finally, the first author thanks the University of Provence and the LATP for their hospitality.
1. A criterion of quasiconformality
Let Aut(C) denote the group of conformal automorphisms of the plane, i.e., affine maps z7→az+b, wherea∈C∗=C\ {0} andb∈C. Note the derivative of an affine map is constant. Givenz∈Candr >0, we denote byD(z, r) the open Euclidean disc centred atzof radius r.
Proposition 1.1. Let Ω⊂Cbe a domain, and F a family of injective continuous maps f: Ω→ Cwhich satisfies the following property. For any sequence (fn) of F, and for any αn ∈ Aut(C) such that αn(0) ∈ Ω and |α′n(0)| ≤ (1/3)δΩ(αn(0)), there are a subsequence (nk)k and a sequence (βk)k of elements ofAut(C)such that the sequence(βk◦fnk◦ αnk)k converges uniformly on the diskD(0,2)to an injective continuous map g:D(0,2)→C.
Then there is someK=K(F)such that each element ofF isK-qua- siconformal.
This proposition is similar to criteria given for global homeomor- phisms in [2, §2.2–2.5] and [4, Theorem 18 and Corollary 8].
Proof: Suppose, for contradiction, that the result were not true. Then one can find a sequence of maps (fn)n and a sequence of points (zn)n
in Ω such that Hfn(zn)≥2n. Let us considerrn ∈(0, δΩ(zn)/3) such that Hfn(zn, rn)≥n.
Let us consider αn(z) = zn+rnz. It follows that rn = |α′n(0)| ≤ (1/3)δΩ(αn(0)). Therefore, extracting a subsequence if necessary, one may assume that there is a sequence (βn)n of affine maps such that the maps gn=βn◦fn◦αn form a sequence which converges uniformly
on D(0,2) to an injective continuous map g: D(0,2) → C. It follows from the uniform convergence and the injectivity of gthat
limHgn(0,1) =Hg(0,1)<∞. ButHgn(0,1) =Hfn(zn, rn)≥n, a contradiction.
We derive a corollary as follows. Let P be a property which can be satisfied by a continuous complex-valued function defined on a planar re- gion. We say thatPis aconformal property if the following propositions hold:
(1) Iff satisfiesP, then eitherf is locally injective or constant.
(2) The propertyP is closed under uniform convergence.
(3) The property P is preserved under pre- and post-composition by affine maps.
We will denote by P(Ω) the set of continuous complex-valued func- tions defined on Ω which satisfy the property P. We also define its Schwarz class SP as the subset of injective functions ofP(D(0,3)) nor- malised byf(0) = 0 andf(1) = 1.
Corollary 1.2. Let P be a conformal property. Suppose that SP is equicontinuous at 0. Then SP is compact with respect to the topology of uniform convergence of compact subsets of D(0,3)and there is some finite constant K such that any non-constant map f on any domain Ω which satisfiesP is locally K-quasiconformal.
Given P,Ω, w, w′ ∈ Ω such that |w−w′| ≤ δΩ(w)/3, we define us- ing (3) the operatorTw,w′:P(Ω)→ P(D(0,3)) by
Tw,w′f(z) = f(w+ (w′−w)z)−f(w) f(w′)−f(w) .
Proof: We will first establish the corollary for SP. Thus we take Ω = D(0,3). We define, forz∈D(0,3),
Mz= sup{|f(z)|, f ∈ SP} ∈[0,∞].
We will prove that Mz is finite for allz. LetB be the set of pointsz∈ D(0,3) such thatMz is finite, andB′ those points of Bsuch that there is another pointw′ ∈ Bsuch that|w−w′| ≤δD(0,3)(w)/3.
By assumption, for any ε > 0, there is some ηε ∈ (0,1) such that Mz≤εwheneverz∈D(0, ηε). We letη =η1.
It follows that ifw∈ B′andw′∈ Bare as above, then|Tw,w′f(z)| ≤ε when|z| ≤ηε for anyf ∈ SP. Therefore, ifz∈D(w,|w−w′|ηε), then, for anyf ∈ SP,
|f(z)−f(w)| ≤ |f(w)−f(w′)|ε≤(Mw+Mw′)ε.
ThusSP is equicontinuous at every point ofB′ andB′ is an open set (it containsD(w,|w−w′|η)).
Let us now prove that B′ is closed in D(0,3). Let (wn)n be a se- quence in B′ which converges to w ∈ D(0,3). If n is large enough, thenδD(0,3)(wn)≥δD(0,3)(w)/2 andw∈D(wn, ηδD(0,3)(wn)/3). Hence w ∈ B. Taking n even larger implies thatwn ∈ D(w, δD(0,3)(w)/3) so that w∈ B′ (we may take such awn as second point).
Finally, we note that D(0, η) ⊂ B′ so that B′ = D(0,3) since it is a non-empty open and closed set. It follows that SP is equicontinuous at every point, so uniformly equicontinuous on every compact subset ofD(0,3).
Ascoli’s theorem implies that SP is a normal family. Moreover, from (1) and (2), any limit satisfies P and is not constant. Since it is a limit of injective functions, the limit is also injective, so it belongs to SP, andSP is compact.
Let us now consider sequences (fn)n ofSP and (αn)nof Aut(C) such that αn(0) ∈ D(0,3) and |α′n(0)| ≤δD(0,3)(αn(0))/3. By compactness of SP, the sequence (T0,1(fn◦αn)) has a convergent subsequence with an injective limit. It follows thatSP satisfies the assumptions of Propo- sition 1.1. Therefore there is some K such that every element ofSP is K-quasiconformal.
Let us fix a domain Ω, and letf ∈ P(Ω). IfDis a disc on whichf is injective, then one can find α ∈ Aut(C) such that α(D(0,3)) =D. It follows thatT0,1(f◦α)∈ SP andf|D isK-quasiconformal. Thus, maps in P(Ω) are locallyK-quasiconformal.
2. Skewed maps
Definition 1 (ε-skewed and (ε, ρ)-skewed map). Let ε > 0 and ρ ∈ [1/2,1). We say that a map f: Ω → C is ε-skewed if f is continuous and skew(f(T))≤1 +ε for every equilateral triangle T ⊂Ω. We say that f is (ε, ρ)-skewed if furthermore, for anyz, w ∈ Ω with|w−z| ≤ (1/3) min{δΩ(z), δΩ(w)},
f
z+w 2
−f(z)
≤ρ|f(z)−f(w)|.
The proof of the theorem will follow from the next three propositions.
Proposition 2.1. Letε >0andρ∈[1/2,1). The family of(ε, ρ)-skewed maps on D(0,3) which fix 0 and1is equicontinuous at the origin.
Proposition 2.2. There areε0 >0 and a function ρ: [0, ε0)→1 2,1 such that if0< ε < ε0then any injectiveε-skewed map is(ε, ρ(ε))-skewed.
Proposition 2.3. Both skewedness properties are conformal properties.
Using the three propositions above, we are able to prove our main theorem.
Proof of the theorem: First, both skewedness properties are conformal properties by Proposition 2.3. Therefore, it follows from Corollary 1.2 that it is enough to prove the equicontinuity at the origin. By Propo- sition 2.2, an ε-skewed map is (ε, ρ)-skewed for some ρ ∈ [1/2,1), so Proposition 2.1 implies the equicontinuity at the origin, and hence, by Corollary 1.2, the quasiconformality of injectiveε-skewed maps.
The proof of the propositions will follow after we establish several lemmas. Let us begin by giving a definition which will be convenient for our purposes.
Definition 2(n-connectivity). Letz1, z2, w1, w2∈Csuch thatz16=z2, w16=w2and|z1−z2|=|w1−w2|. We say that the segments [z1, z2] and [w1, w2] aren-connected if there exist nequilateral trianglesT1, . . . , Tn
of sidelength |z1−z2| such that [z1, z2] ⊂ T1, [w1, w2] ⊂ Tn, and Tj
andTj+1have exactly one edge in common, for allj= 1, . . . , n−1.
We state the following observation as a separate lemma, since we will make use of it later.
Lemma 2.4. Let ε >0 and let f:C→C be an ε-skewed map. If two segments[z1, z2] and[w1, w2]aren-connected, for some n∈N, then
1
(1 +ε)n|f(z1)−f(z2)| ≤ |f(w1)−f(w2)| ≤(1 +ε)n|f(z1)−f(z2)|. We start with Proposition 2.3.
Proof of Proposition 2.3: The fact that a non-constant ε-skewed or (ε, ρ)-skewed map is locally injective is the only non-trivial property to establish.
We let Ω be a domain, and we consider two distinct pointsz, w∈Ω.
Let Λz,w be the set of pointsz′∈Ω for which there isw′∈Ω such that the segments [z, w] and [z′, w′] are connected, in the sense of Definition 2, by a chain of triangles whose vertices all lie in Ω.
First, iff isε-skewed andf(z) =f(w) for some distinct points, then f is constant on Λz,w. Iff is not locally injective at a pointζ∈Ω, then one can find a sequence (zn, wn)n of distinct points which converge toζ such thatf(zn) =f(wn). Thus,f is constant on each Λn = Λzn,wn. As ntends to infinity, Λntends in the Hausdorff topology to Ω; furthermore, since (zn) tends to ζ, it follows from the continuity of f that f turns out to be constant equal to f(ζ). Thus, a non-constant ε-skewed or (ε, ρ)-skewed map is locally injective.
Actually, iff is (ε, ρ)-skewed in some discD(0, R) and is not constant, thenf is injective onD(0, R/2).
Lemma 2.5. Let ε >0 and ρ∈[1/2,1). For every (ε, ρ)-skewed map f:D(0,3)→C,
diamf([z, w])≤ |f(z)−f(w)| 1−ρ
for all z, w∈D(0,3) with|w−z| ≤(1/3) min{δD(0,3)(z), δD(0,3)(w)}. Proof: It is enough to show the result for z = 0 and w = 1. Let f:D(0,3) → C be an (ε, ρ)-skewed map. Consider, for x∈ [0,1], the binary expansion of x. Letxn be the truncation of the expansion of x after then-th digit. Then, for alln∈N,
|f(x)−f(0)| ≤ |f(x)−f(xn)|+ X
0≤k≤n−1
|f(xk)−f(xk+1)|.
Observe that|f(x)−f(xn)| →0 asn→ ∞and that|f(xk)−f(xk+1)| ≤ ρk|f(0)−f(1)|. Therefore
|f(x)−f(0)| ≤ |f(0)−f(1)| 1−ρ .
Lemma 2.6. Let ε >0 and ρ∈[1/2,1). For every (ε, ρ)-skewed map f:D(0,3)→ C, if T is an equilateral triangle with vertices z1, z2, z3 ∈ D(0,3)such that |zi−zj| ≤(1/3) min{δD(0,3)(zi), δD(0,3)(zj)} whenever i, j∈ {1,2,3}with i6=j, then
diamf(T)≤ 2(1 +ε)
1−ρ |f(z1)−f(z2)|. Proof: Letf:D(0,3)→Cbe an (ε, ρ)-skewed map.
We may assume thatf is not constant. Sincef is a continuous open mapping, the diameter off(T) is given by the diameter off(∂T).
Let z, w ∈ ∂T. Perhaps after relabeling the vertices of T we may assume that z ∈ [z1, z2] and w ∈ [z1, zj], for some j = 2,3. Then, by Lemma 2.5,
|f(z)−f(w)| ≤ |f(z1)−f(z)|+|f(z1)−f(w)|
≤ 1
1−ρ(|f(z1)−f(z2)|+|f(z1)−f(zj)|)
≤ 2
1−ρ max
k∈{2,3}|f(z1)−f(zk)|
≤ 2(1 +ε)
1−ρ |f(z1)−f(z2)|.
Proof of Proposition 2.1: Let f: D(0,3) →Cbe an (ε, ρ)-skewed map, normalised so that f(0) = 0 andf(1) = 1. Givenn∈N, let Hn be the regular hexagon centred at 0 and with vertex 21n. Divide Hn into six equilateral triangles, each of sidelength 21n, so that the vertices of each such triangle are 0 and two consecutive vertices ofHn. LetTnbe the set of these six triangles so obtained. We note that each edge of a triangle in Tn is at most 3-connected to [0,1/2n].
Then, by Lemma 2.6,
diamf(Hn)≤2 sup
T∈Tn
diamf(T)
≤ 4(1 +ε)4
1−ρ |f(0)−f(1/2n)|
≤ 4(1 +ε)4 1−ρ ρn. This proves the equicontinuity at the origin.
We now turn to the proof of Proposition 2.2. Let us start with the following lemma.
Lemma 2.7. There is someε1>0which satisfies the following property.
For all 0< ε < ε1 there exists cε>0 such that for every ε-skewed map f: Ω→C,
f
z+ (w−z)eiπ/3
−f
z+(w−z)e−iπ/3 ≥√
3(1−cε)|f(w)−f(z)|, for all z ∈ Ω and w ∈ D(z, δΩ(z)) such that f(z) 6= f(w). Moreover, cε can be chosen so thatcε→0 asε→0.
Proof: Letε >0 be given. Without loss of generality, we may assume that D(0,1) ⊂Ω, z = 0, w = 1, f(0) = 0 andf(1) = 1. In this case, z+ (w−z)eiπ/3=eiπ/3 andz+ (w−z)e−iπ/3=e−iπ/3. Let us denote the complex numbereiπ/3byα.
Since the triangles with vertices (0,1, α) and (0,1, α) are both equilat- eral, we have that skew(0,1, f(α))≤λand skew(0,1, f(α))≤λ, where λ= 1 +ε. Thus, there existsrε>0 such that
f(α), f(α)∈D(α, rε)∪D(α, rε)
andrε→0 asε→0. To conclude the proof, we only need to prove that each image is in a different disc.
Note that if ε > 0 is small enough then ε-skewed maps transform the vertices of equilateral triangles into three non-aligned points. There- fore, if{z1(t), z2(t), z3(t)}is a continuous motion of equilateral triangles in Ω, where tranges over some intervalI, then the vectors [(f(z2(t))− f(z1(t))),(f(z3(t))−f(z1(t)))],t∈I, form a continuous family of bases ofCoverRwith a fixed orientation. Applying this remark toz1(t) = 0, z2(t) = e−itπ/3 and z3(t) = ei(1−t)π/3 for t ∈ [0,1], we see that both bases [1, f(α)] and [f(α),1] have the same orientation, so the pointsf(α) andf(α) belong to two opposite discs.
Let us now state the following elementary lemma, omitting its proof.
Lemma 2.8. Letλ >1. The regionEλ={z∈C| 1λ|z| ≤ |z−1| ≤λ|z|}
is the complement of the union of two disjoint open discs in C, with centers λ2λ−12 and 1−λ1 2, respectively, and with radii both equal to λ2λ−1.
Ifw, w′∈C, we defineEλ(w, w′) =φ(Eλ) whereφ(z) = (w−w′)z+w′. Lemma 2.9. There is someε2>0which satisfies the following property.
For all 0< ε < ε2 there existssε>0 such that for every ε-skewed map f: Ω→C,
f
z+w 2
∈D
f(z) +f(w)
2 , sε|f(z)−f(w)|
,
for all z, w∈Ωsuch that|w−z| ≤(1/3) min{δΩ(z), δΩ(w)} andf(z)6= f(w). Moreover, sεcan be chosen so that sε→0 asε→0.
Proof: It suffices to show the result for z = 0 and w = 1. Letε > 0 and letf: Ω→Cbe anε-skewed map, normalised so thatf(0) = 0 and f(1) = 1. Setλ= 1 +εandα=eiπ3.
First, the segments 0,12
and 1
2,1
are 3-connected, as is shown by considering the three triangles with vertices 12(0,1, α), 12(1, α,1 +α) and
1
2(1,1 +α,2), respectively. Therefore,f 12
∈Eλ3(0,1), by Lemma 2.4.
By a similar argument, the segments α,12
and 1
2, α
are 3-connected and thusf 12
∈Eλ3(f(α), f(α)).
Now observe that the regionEλ is a neighbourhood of {ℜz= 1/2}, and asεtends to 0,Eλtends to this line on any compact set in the Haus- dorff topology. Therefore, for ε small enough, the region Eλ3(0,1)∩ Eλ3(α, α) is the union of two components: one compact, which con- tains 1/2 and the diameter of which tends to zero asεtends to zero, and the second is unbounded, but its diameter in the spherical metric tends to 0 as well with ε.
Thus, there exists′ε, R′ε>0 such that Eλ3(0,1)∩Eλ3(α, α)⊂D
1 2, s′ε
∪ {z∈C,|z| ≥R′ε} where s′ε→0 andR′ε→ ∞ asε→0.
Lemma 2.7 and its proof imply that we may as well assume, re- placing f by f, if necessary, that f(α) (resp. f(α)) lies in D(α, rε) (resp. in D(α, rε)) where rε tends to 0 with ε. It follows that there exist sε > 0 and Rε, depending only on s′ε, R′ε and rε, and therefore depending only on ε, such that
Eλ3(0,1)∩Eλ3(f(α), f(α))⊂D 1
2, sε
∪ {z∈C,|z| ≥Rε} where sε→0 andRε→ ∞ asε→0.
To conclude, we need to prove that f(1/2) belongs to the bounded component of Eλ3(0,1)∩Eλ3(f(α), f(α)). Letβ = (1/2)eiπ/3and γ= (1/2)(1 +eiπ/3). Note that the triangles{0,1/2, β} and{γ,1/2, β}are both equilateral and have one edge in common. Recalling that f(0) = 0 andf(1) = 1 and applying Lemma 2.7 for z=β andw= 1/2, one gets that that
|f(γ)| ≥√
3(1−cε)|f(β)−f(1/2)| and thus
|f(γ)| ≥
√3(1−cε)
λ2 |1−f(1/2)|
since the segments [β,1/2] and [1/2,1] are 2-connected. Also,
|f(γ)| ≤1 +|f(γ)−1| ≤1 +λ|1−f(1/2)|.
Combining these two inequalities we get that
|1−f(1/2)| ≤Bε= λ2
√3(1−cε)−λ3
which remains bounded asε→0 since this quantity tends to (√
3−1)−1 as εtends to 0.
Therefore, forεsmall enough, one has thatBε< Rεand thusf 12
∈ D 12, sε
, as desired.
Proof of Proposition 2.2: Let ε0 be such thatsε0 < 14, wheresε0 is the constant given by Lemma 2.9 forε0. Letε < ε0and letf be an injective ε-skewed map defined on a domain Ω. Pickz, w∈Ω such that|w−z| ≤ (1/3) min{δΩ(z), δΩ(w)}.
Using the transformation Tz,wf, we may as well assume thatz = 0, w = 1 and that f is normalised so that f(0) = 0 and f(1) = 1. Note that f 12
∈ D 12, sε
, by Lemma 2.9. The circle with centre 12 and radius sε intersects the segment 1
2,1
at a point uε. Let ρ(ε) = uε, noting that 12 < ρ(ε)<1. Then
f(0)−f 12
≤ |f(0)−uε| =|uε|= ρ(ε)|f(0)−f(1)|. The proposition follows.
Appendix A. Control of nearly equilateral triangles In this Appendix, we prove the following.
Proposition A.1. Let µ >0 and λ <∞. If f: Ω→f(Ω) is a homeo- morphism such that
skew(f(T))≤λ
for any triangleT ⊂Ωwithskew(T)≤1+µ, thenf isK-quasiconformal for someK=K(λ, µ).
Proof: Fixz0∈Ω andr∈(0, δΩ(z0)). We claim thatHf(z0, r)≤K for some constant K which depends only on λand µ. This will establish the proposition.
Without loss of generality, we may assume thatz0= 0 andr= 1. There is some η >0 of the formη =π/(3n),n≥1, such that skew(0,1, eit)≤ 1 +µfor all fort∈[π/3−η, π/3 +η].
Let us subdivide [0, π/3] intonintervalsIj= [(j−1)η, jη] of lengthη, and similarly [π/3,2π/3] byn intervalsJj = [π/3 + (j−1)η, π/3 +jη]
forj∈ {1, . . . , n}. Let us write, forj∈ {1, . . . , n}andk∈ {0,1}, zjk=ei(kπ/3+(j−1)η).
It follows that the skew of triangles with vertices 0,z0jandeitfort∈Jj
and of triangles with vertices 0,zj+11 andeitfort∈Ijis bounded by 1+µ.
Therefore,
λ−2n|f(0)−f(1)| ≤ |f(0)−f(eit)| ≤λ2n|f(0)−f(1)| fort∈[0,2π/3].
Finally, any segment of the form [0, eis] is at most 2-connected to a segment of the form [0, eit] witheitin someIj orJj, and thus the claim follows.
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[6] O. Lehto and K. I. Virtanen, “Quasiconformal mappings in the plane”, Second edition, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften126, Springer- Verlag, New York-Heidelberg, 1973.
[7] J. V¨ais¨al¨a,“Lectures onn-dimensional quasiconformal mappings”, Lecture Notes in Mathematics 229, Springer-Verlag, Berlin-New York, 1971.
Javier Aramayona:
Mathematics Institute University of Warwick Coventry CV4 7AL England
E-mail address:[email protected] Peter Ha¨ıssinsky:
LATP/CMI
Universit´e de Provence 39 rue Fr´ed´eric Joliot-Curie 13453 Marseille Cedex 13 France
E-mail address:[email protected]
Primera versi´o rebuda el 4 de juliol de 2007, darrera versi´o rebuda el 19 de desembre de 2007.
