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Planar Quasiconformal Mappings; Deformations and Interactions

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Planar quasiconformal mappings; deformations and interactions Kari Astala To F.W. Gehring on his 70th birthday 1 Introduction The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Basically, this is not due to the history of the topic but rather since planar quasiconformal mappings admit flexible methods (so far) not available in space. In this expository paper we wish to describe some recent trends and activities in quasiconformal theory peculiar to the plane. It is obvious, though, that not all topics can be covered no matter which point of view is taken; many important advances and connections must necessarily be bypassed. Therefore we concentrate on a specific theme, a property that singles out the difference between mappings in plane and in space: Planar quasiconformal mappings admit an effective deformation theory, any such mapping can be deformed to the identity, within the family of all quasiconformal mappings. Our goal is to discuss different aspects of these parametric representations of planar qua- siconformal mappings, with a few occasional words of the background of the topics, to see how and why these mappings can be deformed, and above all which are the results and con- sequences obtained, which are the questions still open. In particular, in many respects it is the deformation properties which are the "cause" of the many connections and applications of quasiconformality. Thus we try to give a particular attention to discuss how this field of ideas interacts with other parts of mathematical analysis. It will be no surprise to see the strong influence of Gehring also within these contexts. By the analytic definition a homeomorphism f :Ω ! Ω0 between planar domains Ω and Ω0 1;2 is called K-quasiconformal if it is contained in the Sobolev class Wloc (Ω) and its directional derivatives satisfy maxαj@αf(x)j ≤ Kminαj@αf(x)j a:e: x 2 Ω: A corresponding definition works of course for higher dimensional mappings as well, but what is special to the mappings in plane is the well known Beltrami differential equation, that the inequality is equivalent to @f(z) = µ(z) @f(z); (1) K−1 where µ is the complex dilatation or the Beltrami coefficient of f with kµk1 = K+1 < 1. This is a first order linear differential equation, solvable by singular integrals. Note also that even if the recent work of Donaldson and Sullivan [12], developed also by Iwaniec and Martin [20], found a counterpart of (1) in even dimensional spaces this machinery still lacks the required properties for finding parametric representations in space. However, in the planar case (1) always does admit quasiconformal solutions: by the measurable Riemann 1 1 1;2 mapping theorem for any µ 2 L (C) with kµk1 < 1 we have a homeomorphic Wloc -solution of (1), see [2] [9] [21], unique up to postcomposing with a M¨obiustransformation. This result is the starting point of the deformation theory for planar quasiconformal map- pings. It turns out that using (1) we can actually construct two natural and similar but in many respects different ways of parametrizing the mappings. To be more precise, the basic idea for the first deformation is that we may embedd our mapping f to a flow fftgt2R of a (nonautonomous) vector field v. In fact, by careful choices obtain a geodesic in the Teichm¨ullermetric, −1 log K(ft ◦ fs ) = jt − sj; f0 = id: For this approach, to control the flow we must find the infinitesimal generating vector field v for which @tft(x) = v(t; ft(x)); ft0 = f: (2) Thus the questions here are how to find v, what are its properties and, especially, how to characterize vector fields generating quasiconformal flows. The second approach arises in connection with deformations of complex analytic structures, where the holomorphicity of the parametrization is an essential feature required. Then one must abandon the vector field setting and more directly use the properties of the equation (1). Namely, if µ 2 L1(C) has a compact support, let f = f µ be the solution of (1) with f(z) = z + O(jzj−1). Then, if µ varies holomorphically; so does f µ(z); (3) thus giving us holomorphic ways of deforming f to the identity. Indeed, as is well known this holomorphic dependence follows with the help of the classical Beurling operator, the two dimensional counterpart of the Hilbert transform, 1 Z !(ζ) dm(ζ) S!(z) = − : π (ζ − z)2 C It defines an isometry in L2(C), and moreover, what makes it useful for the Beltrami equation is that S transforms @-derivatives to @-derivatives, S(@u) = @u; 8u 2 W 1;2(C): (4) Using this identity for u(z) = f(z) − z and combining with the equation @f(z) = µ(z)@f(z) we have @f(z) = 1 + (I − Sµ)−1S(µ): (5) 2 Since as an operator on L (C), the norm of g 7! S(µg) is at most kµk1kSkL2 < 1, the expression is well defined and shows that @f(z) depends holomorphically on µ. Since by the Cauchy formula 1 Z µ∂f(ζ) dm(ζ) f(z) = z − ; π (ζ − z) C we have the same conclusion for f(z). 2 Flows of quasiconformal mappings The parametric representations in terms of flows of quasiconformal mappings was originally derived by Shah Dao-Shing [32]. However Gehring and Reich [17] were the first who sys- tematically developed and utilized the method. Their goal was to find a new approach to 2 quasiconformal distortion problems. We return to these applications below after first analysing the basic properties and later consequences of their construction. To build up the flow, let us agree first on the notation fµ for the mapping with complex dilatation µ, normalized so that 0, 1 and 1 are fixed. To embedd a given K-quasiconformal mapping f to a continuous flow, we may assume that f = fµ and define then a further family of dilatations by µ(z) t µ (z) = tanh( tanh−1 jµ(z)j);T = log K: t jµ(z)j T According to the measurable Riemann mapping theorem for each t 2 R there exists a unique normalized mapping ft = fµt . Geometrically, µt(z) is chosen to be the point on the radius through µ(z) such that in the unit disk the hyperbolic distance from 0 to µt(z) is equal to t=T times the distance between 0 and µ(z). From the analytic view, or by the transformation formulas of the complex dilatation for a composition of two quasiconformal mappings [21], the µt(z)'s are constructed precisely so that we have 1 log K(f ◦ f −1) = jt − sj: t s T 1 We have made the above choice T for the speed of the flow for later purposes, c.f. (6), (7). Sometimes it is also convenient to have a finite version of the flow. That is, considering t only a discrete set of parameter values and noticing that K(fµt ) ≤ e we have the factorization theorem: For any > 0, all quasiconformal mappings in C can be written as a composition of a finite number of (1 + )-quasiconformal mappings. The generating vectorfield can next be given simply as −1 v(z; t) = @tft ◦ ft (z): (6) In other words, the equation @tft(x) = v(t; ft(x)), fT = f is satisfied directly by the definition. Less obvious from here is to see any specific properties and, in particular, any explicit character- izations of vector fields that generate quasiconformal flows. However, by a clever combination of (1) and (2) Gehring and Reich found two interesting identities 1 µ(z) @zft @zv ◦ ft = (7) 2 jµ(z)j @zft and @tJft = 2 Re[@v ◦ ft]Jft (8) for the vector field v and the Jacobian determinant Jft of ft. The first of these answer imme- diately our question. Corollary 2.1 The vector field v generating the above flow of normalized quasiconformal map- pings fftgt2R satisfies 1 k@vk ≤ : 1 2 Indeed, as shown later by Reimann [28] and Semenov [33] the corollary admits a direct converse. 1;1 Theorem 2.2 Suppose we have a continuous vector field v 2 Wloc (C) such that v = O(jzj log jzj) as jzj ! 1 and k@vk1 ≤ C < 1: Then the flow fgtgt2R of v consists of quasiconformal mappings with Ct K(gt) ≤ e : 3 The exact value of the constant C counts, of course, only for the speed of the flow. Remarkably this direction holds in any dimensions ([28]),([33]). We only need to reinterpret the boundedness of j@v(z)j in terms of the differential operator 1 1 ΣV (x) = [DV (x)t + DV (x)] − (T rDV (x))I; x 2 Rn; 2 n in the case of vectorfields in Rn. As shown by Ahlfors [3], in two dimensions jΣV (z)j = j@V (z)j. To indicate the usefulness of this approach let us recall another closely related argument of Reimann [30]: Suppose we have a quasiconformal mapping f on the sphere S2 which defines an isomorphisms between two groups of M¨obiustransformations G and Γ, i.e. f −1 ◦ Γ ◦ f = G . Since the M¨obiusgroups themselves extend canonically to the unit ball B3 we may ask if there is a way to extend the isomorphism as well. For this [30] shows first that the generating vector field v constructed above from the (two dimensional) mapping f is G-invariant in an appropriate sense. Moreover, in all dimensions an M¨obius-equivariant extension of vector fields is easily found by a Poisson-type integraloperator.
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