Open access peer-reviewed chapter

Quasiconformal Reflections across Polygonal Lines

Written By

Samuel L. Krushkal

Submitted: 09 December 2019 Reviewed: 09 April 2020 Published: 20 May 2020

DOI: 10.5772/intechopen.92441

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Structure Topology and Symplectic Geometry

Edited by Kamal Shah and Min Lei

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An important open problem in geometric complex analysis is to establish algorithms for explicit determination of the basic curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals.


  • Grunsky inequalities
  • univalent function
  • Beltrami coefficient
  • quasiconformal reflection
  • universal Teichmüller space
  • Fredholm eigenvalues
  • convex polygon

1. Quasiconformal reflections: general theory

1.1 Quasireflections and quasicurves

The classical Brouwer-Kerekjarto theorem ([1, 2], see also [3]) says that every periodic homeomorphism of the sphere S2 is topologically equivalent to a rotation, or to a product of a rotation and a reflection across a diametral plane. The first case corresponds to orientation-preserving homeomorphisms (and then E consists of two points), the second one is orientation reversing, and then either the fixed point set E is empty (which is excluded in our situation) or it is a topological circle.

We are concerned with homeomorphisms reversing orientation. Such homeomorphisms of order 2 are topological involutions of S2 with ff=id and are called topological reflections.

We shall consider here quasiconformal reflections or quasireflections on the Riemann sphere Ĉ=C=S2, that is, the orientation reversing quasiconformal automorphisms of order 2 (involutions) of the sphere with ff=id. The topological circles admitting such reflections are called quasicircles. Such circles are locally quasiintervals, that is, the images of straight line segments under quasiconformal maps of the sphere S2. Any quasireflection preserves pointwise fixed a quasicircle LĈ interchanging its inner and outer domains.

Under quasiconformal mapwz of a domain DĈ, we understand an orientation-preserving generalized solution of the Beltrami equation (uniformly elliptic system of the first order)




are the distributional partial derivatives, μ is a given function from LD with μ<1, called the Beltrami coefficient (or complex dilatation) of the map w, and the quantity kw=μ is the (quasiconformal) dilatation of this map. There are some equivalent analytic and geometric definitions of such maps.

Quasiconformality preserves (up to bounded perturbations) the main intrinsic properties of conformal maps (see, e.g., [4, 5, 6]).

Qualitatively, any quasicircle L is characterized, due to [7], by uniform boundedness of the cross-ratios for all ordered quadruples z1z2z3z4 of the distinct points on L; namely,


for any quadruple of points z1,z2,z3,z4 on L following this order. Using a fractional linear transformation, one can send one of the points, for example, z4, to infinity; then the above inequality assumes the form


This is shown in [7] by applying the properties of quasisymmetric maps. Ahlfors has established also that if a topological circle L admits quasireflections (i.e., is a quasicircle), then there exists a differentiable quasireflection across L which is (euclidian) bilipschitz-continuous. This property is very useful in various applications. On its extension to hyperbolic M-bilipschitz reflections see [8].

Geometrically, a quasicircle is characterized by the property that, for any two points z1,z2 on L, the ratio of the chordal distance z1z2 to the diameters of the corresponding subarcs with these endpoints is uniformly bounded. Note also that every quasicircle has zero two-dimensional Lebesgue measure.

Other characterizations of quasicircles are given, for example, in [9, 10, 11]. We will not touch here the extension of this theory to higher dimensions.

Quasireflections across more general sets EĈ also appear in certain questions and are of independent interest. Those sets admitting quasireflections are called quasiconformal mirrors.

One defines for each mirror E its reflection coefficient


and quasiconformal dilatation


the infimum in (1) is taken over all quasireflections across E, provided those exist, and is attained by some quasireflection f0.

When E=L is a quasicircle, the corresponding quantity


and the reflection coefficient qE can be estimated in terms of one another; moreover, due to [4, 12], we have


The infimum in (2) is taken over all orientation-preserving quasiconformal automorhisms f carrying the unit circle onto L, and kf=z¯f/zf.

Theorem 1. For any setEĈwhich admits quasireflections, there is a quasicircleLEwith the same reflection coefficient; therefore,


The proof of this important theorem was given for finite sets E=z1zn by Kühnau in [13], using Teichmüller’s theorem on extremal quasiconformal maps applied to the homotopy classes of homeomorphisms of the punctured spheres, and extended to arbitrary sets EhC by the author in [14].

Theorem 1 yields, in particular, that similar to (3) for any set EĈ, its quasiconformal dilatation satsfies


where kE=infz¯f/zf over all quasicircles LE and all orientation-preserving quasiconformal homeomorphisms f:ĈĈ with fR̂=L.

This theorem implies various quantitative consequences. A new its application will be given in the last section.

We point out that the conformal symmetry on the extended complex plane is strictly rigid and reduces to reflection zz¯ within conjugation by transformations gPSL2C. The quasiconformal symmetry avoids such rigidity and is possible over quasicircles. Theorem 1 shows that, in fact, this case is the most general one, since for any set EĈ we have QE=, unless E is a subset of a quasicircle with the same reflection coefficient.

Let us mention also that a somewhat different construction of quasiconformal reflections across Jordan curves has been provided in [15]; it relies on the conformally natural extension of homeomorphisms of the circle introduced by Douady and Earle [16].

The quasireflection coefficients of curves are closely connected with intrinsic functionals of conformal and quasiconformal maps such as their Teichmüller and Grunsky norms and the first Fredholm eigenvalue, which imply a deep quantitative characterization of the features of these maps.

One of the main problem here, important also in applications of geometric complex analysis, is to establish the algorithms for explicit determination of these quantities for individual quasicircles or quasiintervals. This was remains open a long time even for generic quadrilaterals.

1.2 Fredholm eigenvalues

Recall that the Fredholm eigenvaluesρn of an oriented smooth closed Jordan curve L on the Riemann sphere Ĉ=C are the eigenvalues of its double-layer potential, or equivalently, of the integral equation


which has has many applications (here nζ is the outer normal and dsζ is the length element at ζL).

The least positive eigenvalue ρL=ρ1 plays a crucial role and is naturally connected with conformal and quasiconformal maps. It can be defined for any oriented closed Jordan curve L by


where G and G are, respectively, the interior and exterior of L;D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on Ĉ and harmonic on GG. In particular, ρL= only for the circle.

An upper bound for ρL is given by Ahlfors’ inequality [17].


where qL denotes the minimal dilatation of quasireflections across L.

In view of the invariance of all quantities in (5) under the action of the Möbius group PSL2C/±1, it suffices to consider the quasiconformal homeomorphisms of the sphere carrying S1 onto L whose Beltrami coefficients μfz=z¯f/zf have support in the unit disk D=z<1, and fDz=z+b0+b1z1+, where D=Ĉ\D¯ (or in the upper half-plane U=Imz>0). Then qL is equal to the minimum k0f of dilatations kf=μ of quasiconformal extensions of the function f=fD into D.

The inequality (5) serves as a background for defining the value ρL, being combined with the features of Grunsky inequalities given by the classical Kühnau-Schiffer theorem. The related results can be found, for example, in surveys [12, 18, 19] and the references cited there.

In the following sections, we provide a new general approach.

1.3 The Grunsky and Milin inequalities



In 1939, Grunsky discovered the necessary and sufficient conditions for univalence of a holomorphic function in a finitely connected domain on the extended complex plane Ĉ in terms of an infinite system of the coefficient inequalities. In particular, his theorem for the canonical disk D yields that a holomorphic function fz=z+const+Oz1 in a neighborhood of z= can be extended to a univalent holomorphic function on the D if and only if its Grunsky coefficients αmn satisfy


where αmn are defined by


the sequence x=xn runs over the unit sphere Sl2 of the Hilbert space l2 with norm x2=1xn2, and the principal branch of the logarithmic function is chosen (cf. [20]). The quantity


is called the Grunsky norm of f.

For the functions with k-quasiconformal extensions (k<1), we have instead of (8) a stronger bound


established first in [21] (see also [18, 22]). Then


where kf denotes the Teichmüller norm of f, which is equal to the infimum of dilatations kwμ=μ of quasiconformal extensions of f to Ĉ. Here wμ denotes a homeomorphic solution to the Beltrami equation z¯w=μzw on C extending f.

Note that the Grunsky (matrix) operator


acts as a linear operator l2l2 contracting the norms of elements xl2; the norm of this operator equals ϰf (cf. [23, 24]).

For most functions f, we have in (10) the strong inequality ϰf<kf (moreover, the functions satisfying this inequality form a dense subset of Σ), while the functions with the equal norms play a crucial role in many applications (see [18, 22, 25, 26, 27, 28]).

The method of Grunsky inequalities was generalized in several directions, even to bordered Riemann surfaces X with a finite number of boundary components (cf. [6, 11, 20, 29, 30]; see also [31]). In the general case, the generating function (7) must be replaced by a bilinear differential


where the surface kernel RXzζ relates to the conformal map jθzζ of X onto the sphere Ĉ slit along arcs of logarithmic spirals inclined at the angle θ0π to a ray issuing from the origin so that jθζζ=0 and


(in fact, only the maps j0 and jπ/2 are applied). Here φn1 is a canonical system of holomorphic functions on X such that (in a local parameter)


and the derivatives (linear holomorphic differentials) φn form a complete orthonormal system in H2X.

We shall deal only with simply connected domains X=D with quasiconformal boundaries (quasidisks). For any such domain, the kernel RD vanishes identically on D×D, and the expansion (11) assumes the form


where χ denotes a conformal map of D onto the disk D so that χ=,χ>0.

Each coefficient αmnf in (12) is represented as a polynomial of a finite number of the initial coefficients b1,b2,,bs of f; hence it depends holomorphically on Beltrami coefficients of quasiconformal extensions of f as well as on the Schwarzian derivatives


These derivatives range over a bounded domain in the complex Banach space BD of hyperbolically bounded holomorphic functions φD with norm


where λDzdz denotes the hyperbolic metric of D of Gaussian curvature 4. This domain models the universal Teichmüller spaceT with the base point χD (in holomorphic Bers’ embedding of T).

A theorem of Milin [29] extending the Grunsky univalence criterion for the disk D to multiply connected domains D states that a holomorphic function fz=z+const+Oz1 in a neighborhood of z= can be continued to a univalent function in the whole domain D if and only if the coefficients βmn in (12) satisfy, similar to the classical case of the disk D, the inequality


for any point x=xnSl2. We call the quantity


the generalized Grunsky norm of f. By (14), ϰDf1 for any f from the class ΣD of univalent functions in D with hydrodynamical normalization


The inequality ϰDf1 is necessary and sufficient for univalence of f in D (see [11, 20, 29]).

The norm (15) also is dominated by the Teichmüller norm kf of this map. Similar to (10),


where τT denotes the Teichmüller distance on the universal Teichmüller space T with the base point D, and for the most of univalent functions, we also have here the strict inequality.

The quasiconformal theory of generic Grunsky coefficients has been developed in [32]. This technique is a powerful tool in geometric complex analysis having fundamental applications in the Teichmüller space theory and other fields.

Note that in the case D=D, βmn=mnαmn; for this disk, we shall use the notations Σ and ϰf. We denote by S the canonical class of univalent functions Fz=z+a2z2+ in the unit disk D.

The Grunsky norm of univalent functions FS is defined similar to (5), (6); so any such Fz and its inversion fz=1/F1/z univalent in D have the same Grunsky coefficients αmn. Technically it is more convenient to deal with functions univalent in D.

1.4 Extremal quasiconformality

A crucial point here is that the Teichmüller norm on Σ is intrinsically connected with integrable holomorphic quadratic differentialsψzdz2 on the complementary domain


(the elements of the subspace A1D of L1D formed by holomorphic functions), while the Grunsky norm naturally relates to the abelian structure determined by the set of quadratic differentials


having only zeros of even order on D.

We describe the general intrinsic features. Let L be a quasicircle passing through the points 0,1,, which is the common boundary of two domains D and D. Let L be an oriented quasiconformal Jordan curve (quasicircle) on the Riemann sphere Ĉ with the interior and exterior domains D and D. Denote by λDzdz the hyperbolic metric of D of Gaussian curvature 4 and by δDz=distzD the Euclidean distance from the point zD to the boundary. Then


where the right-hand inequality follows from the Schwarz lemma and the left from Koebe’s 14 theorem.

Consider the unit ball of Beltrami coefficients supported on D,


and take the corresponding quasiconformal automorphisms wμz of the sphere Ĉ satisfying on C the Beltrami equation z¯w=μzw preserving the points 0,1, fixed. Recall that kw=μw is the dilatation of the map w.

Take the equivalence classes μ and wμ letting the coefficients μ1 and μ2 from BeltD1 be equivalent if the corresponding maps wμ1 and wμ2 coincide on L (and hence on D¯). These classes are in one-to-one correspondence with the Schwarzians Swμ on D, which fill a bounded domain in the space B2D modeling the universal Teichmüller space T=TD with the base point D. The quotient map


is holomorphic (as the map from LD to B2D). Its intrinsic Teichmüller metric is defined by


It is the integral form of the infinitesimal Finsler metric


on the tangent bundle TT of T, which is locally Lipschitzian.

The Grunsky coefficients give rise to another Finsler structure Fxv on the bundle TT. It is dominated by the canonical Finsler structure FTxv and allows one to reconstruct the Grunsky norm along the geodesic Teichmüller disks in T (see [33]).

We call the Beltrami coefficient μBeltD1extremal (in its class) if


and call μinfinitesimally extremal if


Any infinitesimally extremal Beltrami coefficient μ is globally extremal (and vice versa), and by the basic Hamilton-Krushkal-Reich-Strebel theorem the extremality of μ is equivalent to the equality


where AD is the space of the integrable holomorphic quadratic differentials on D (the subspace of L1D formed by holomorphic functions on D) and the pairing


Let w0wμ0 be an extremal representative of its class w0 with dilatation


and assume that there exists in this class a quasiconformal map w1 whose Beltrami coefficient μA1 satisfies the inequality esssupArμw1z<kw0 in some ring domain R=D\G complement to a domain GD. Any such w1 is called the frame map for the class w0, and the corresponding point in the universal Teichmüller space T is called the Strebel point.

These points have the following important properties.

Theorem 2. (i) If a classfhas a frame map, then the extremal mapf0in this class (minimizing the dilatationμ) is unique and either a conformal or a Teichmüller map with Beltrami coefficientμ0=kψ0/ψ0onD, defined by an integrable holomorphic quadratic differentialψ0onDand a constantk01 [34].

(ii) The set of Strebel points is open and dense inT [35, 36].

The first assertion holds, for example, for asymptotically conformal curves L. Similar results hold also for arbitrary Riemann surfaces (cf. [36, 37]).

Recall that a Jordan curve LC is called asymptotically conformal if for any pair of points a,bL,


where z lies between a and b.

Such curves are quasicircles without corners and can be rather pathological (see, e.g., [38, p. 249]). In particular, all C1-smooth curves are asymptotically conformal.

The polygonal lines are not asymptotically conformal, and the presence of angles causes non-uniqueness of extremal quasireflections.

The boundary dilatation Hf admits also a local version Hpf involving the Beltrami coefficients supported in the neighborhoods of a boundary point pD. Moreover (see, e.g., [36], Ch. 17), Hf=suppDHpf, and the points with Hpf=Hf are called substantial for f and for its equivalence class.

On the unique and non-unique extremality see, for example, [5, 34, 39, 40, 41, 42, 43, 44].

The extremal quasiconformality is naturally connected with extremal quasireflections.

1.5 Complex geometry and basic Finsler metrics on universal Teichmüller space

Recall that the universal Teichmüller space T is the space of quasisymmetric homeomorphisms h of the unit circle S1=D factorized by Möbius transformations. Its topology and real geometry are determined by the Teichmüller metric, which naturally arises from extensions of these homeomorphisms h to the unit disk. This space admits also the complex structure of a complex Banach manifold (and this is valid for all Teichmüller spaces).

One of the fundamental notions of geometric complex analysis is the invariant Kobayashi metric on hyperbolic complex manifolds, even in the infinite dimensional Banach or locally convex complex spaces.

The canonical complex Banach structure on the space T is defined by factorization of the ball of Beltrami coefficients


letting μ,νBeltD1 be equivalent if the corresponding maps wμ,wνΣ0 coincide on S1 (hence, on D¯) and passing to Schwarzian derivatives Sfμ. The defining projection ϕT:μSwμ is a holomorphic map from LD to B. The equivalence class of a map wμ will be denoted by wμ.

An intrinsic complete metric on the space T is the Teichmüller metric, defined above in Section 1.4, with its infinitesimal Finsler form (structure) FTϕTμϕTμν,μBeltD1;ν,νLC.

The space T as a complex Banach manifold also has invariant metrics. Two of these (the largest and the smallest metrics) are of special interest. They are called the Kobayashi and the Carathéodory metrics, respectively, and are defined as follows.

The Kobayashi metricdT on T is the largest pseudometric d on T does not get increased by holomorphic maps h:DT so that for any two points ψ1,ψ2T, we have


where dD is the hyperbolic Poincaré metric on D of Gaussian curvature 4, with the differential form


The Carathéodory distance between ψ1 and ψ2 in T is


where the supremum is taken over all holomorphic maps h:DT.

The corresponding differential (infinitesimal) forms of the Kobayashi and Carathéodory metrics are defined for the points ψv of the tangent bundle TT, respectively, by


where HolXY denotes the collection of holomorphic maps of a complex manifold X into Y and Dr is the disk z<r.

The Schwarz lemma implies that the Carathéodory metric is dominated by the Kobayashi metric (and similarly for their infinitesimal forms). We shall use here mostly the Kobayashi metric.

Due to the fundamental Gardiner-Royden theorem, the Kobayashi metric on any Teichmüller spaces is equal to its Teichmüller metric (see [15, 36, 40, 45]).

For the the universal Teichmüller space T, we have the following strengthened version of this theorem for universal Teichmüller space given in [46].

Theorem 3. The Teichmüller metricτTψ1ψ2of either of the spacesTorTDis plurisubharmonic separately in each of its arguments; hence, the pluricomplex Green function ofTequals


wherekis the norm of extremal Beltrami coefficient defining the distance between the pointsψ1,ψ2inT(and similar for the spaceTD).

The differential (infinitesimal) Kobayashi metricKTψvon the tangent bundleTTofTis logarithmically plurisubharmonic inψT, equals the infinitesimal Finsler formFTψvof metricτTand has constant holomorphic sectional curvatureκKψv=4on the tangent bundleTT.

In other words, the Teichmüller-Kobayashi metric is the largest invariant plurisubharmonic metric on T.

The generalized Gaussian curvatureκλ of an upper semicontinuous Finsler metric ds=λtdt in a domain ΩC is defined by


where D is the generalized Laplacian


(provided that λt<). Similar to C2 functions, for which D coincides with the usual Laplacian, one obtains that λ is subharmonic on Ω if and only if Dλt0; hence, at the points t0 of local maximuma of λ with λt0>, we have Dλt00.

The sectional holomorphic curvature of a Finsler metric on a complex Banach manifold X is defined in a similar way as the supremum of the curvatures over appropriate collections of holomorphic maps from the disk into X for a given tangent direction in the image.

The holomorphic curvature of the Kobayashi metric Kxv of any complete hyperbolic manifold X satisfies κKX4 at all points xv of the tangent bundle TX of X, and for the Carathéodory metric CX we have κCxv4.

Finally, the pluricomplex Green function of a domain X on a complex Banach space manifold E is defined as gXxy=supuyxxyX, where supremum is taken over all plurisubharmonic functions uyx:X0 satisfying uyx=logxy+O1 in a neighborhood of the pole y. Here is the norm on X and the remainder term O1 is bounded from above. If X is hyperbolic and its Kobayashi metric dX is logarithmically plurisubharmonic, then gXxy=logtanhdXxy, which yields the representation of gT in Theorem 3.

For details and general properties of invariant metrics, we refer to [47, 48] (see also [18, 49]).

Theorem 3 has various applications in geometric function theory and in complex geometry Teichmüller spaces. Its proof involves the technique of the Grunsky coefficient inequalities.

Plurisubharmonicity of a function ux on a domain D in a Banach space X means that ux is upper continuous in D and its restriction to the intersection of D with any complex line L is subharmonic.

A deep Zhuravlev’s theorem implies that the intersection of the universal Teichmüller space T with every complex line is a union of simply connected planar (moreover, this holds for any Teichmüller space); see ([50], pp. 75–82, [51]).

1.6 The Grunsky-Milin inequalities revised

Denote by Σ0D the subclass of ΣD formed by univalent Ĉ-holomorphic functions in D with expansions fz=z+b0+b1z1+ near z= admitting quasiconformal extensions to Ĉ. It is dense in ΣD in the weak topology of locally uniform convergence on D.

Each Grunsky coefficient αmnf is a polynomial of a finite number of the initial coefficients b1,b2,,bm+n1 of f; hence it depends holomorphically on Beltrami coefficients of extensions of f as well as on the Schwarzian derivatives SfB2D.

Consider the set


consisting of the integrable holomorphic quadratic differentials on D having only zeros of even order and put


The following theorem from [32] completely describes the relation between the Grunsky and Teichmüller norms (more special results were obtained in [26, 52]).

Theorem 4. For allfΣ0D,


and ϰDf<k unless


where μ0 is an extremal Beltrami coefficient in the equivalence class f. The last equality is equivalent to ϰDf=kf.

If ϰf=kf and the equivalence class of f (the collection of maps equal to f on S1=D) is a Strebel point, then the extremal μ0 in this class is necessarily of the form


Note that geometrically (16) means the equality of the Carathéodory and Teichmüller distances on the geodesic disk ϕTtμ0/μ0:tD in the universal Teichmüller space T and that the mentioned above the strict inequality ϰf<kf is valid on the (open) dense subset of Σ0 in both strong and weak topologies (i.e., in the Teichmüller distance and in locally uniform convergence on D).

An important property of the Grunsky coefficients αmnf=αmnSF is that these coefficients are holomorphic functions of the Schwarzians φ=Sf on the universal Teichmüller space T. Therefore, for every fΣ0 and each x=xnSl2, the series


defines a holomorphic map of the space T into the unit disk D, and ϰDF=supxhxSF.

The convergence and holomorphy of the series (18) simply follow from the inequalities


(for any finite M,N), which, in turn, are a consequence of the classical area theorem (see, e.g., [11], p. 61; [29], p. 193).

Using Parseval’s equality, one obtains that the elements of the distinguished set A12D are represented in the form


with x=xnl2 so that xl2=ψA1 (see [52]). This result extends to arbitrary domains D with quasiconformal boundaries but the proof is much more complicated (see [22]).

1.7 The first Fredholm eigenvalue and Grunsky norm

One of the basic tools in quantitative estimating the Freholm eigenvalues ρL of quasicircles is given by the classical Kühnau-Schiffer theorem mentioned above. This theorem states that the valueρLis reciprocal to the Grunsky normϰfof the Riemann mapping function of the exterior domain ofL (see. [27, 53]).

Another important tool is the following Kühnau’s jump inequality [12]:

If a closed curve LĈ contains two analytic arcs with the interior intersection angle πα, then


This implies the lower estimate for qL and 1/ρL. By approximation, this inequality extends to smooth arcs.

One of the standard ways of establishing the reflection coefficients qL (respectively, the Fredholm eigenvalues ρL) consists of verifying wether the equality in (5) or the equality ϰf=k0f hold for a given curve L (cf. [12, 28, 52, 54, 55]).

This was an open problem a long time even for the rectangles stated by R. Kühnau, after it was established only [12, 55] that the answer is in affirmative for the square and for close rectangles R whose moduli mR vary in the interval 1mR<1.037; moreover, in this case qL=1/ρL=1/2. The method exploited relied on an explicit construction of an extremal reflection. The complete answer was given in [33].

The relation between the basic curvelinear functionals intrinsically connected with conformal and quasiconformal maps is described in Kühau’s paper [56].

1.8 Holomorphic motions

Let E be a subset of Ĉ containing at least three points.

A holomorphic motion of E is a function f:E×DĈ such that:

  1. for every fixed zE, the function tfzt:E×DĈ is holomorphic in D;

  2. for every fixed tD, the map fzt=ftz:EĈ is injective;

  3. fz0=z for all zE.

The remarkable lambda-lemma of Mañé, Sad, and Sullivan [57] yields that such holomorphic dependence on the time parameter provides quasiconformality of f in the space parameter z. Moreover: (i)fextends to a holomorphic motion of the closureE¯ofE;

(ii) eachftz=ftz:E¯Ĉis quasiconformal; (iii)fis jointly continuous inzt.

Quasiconformality here means, in the general case, the boundedness of the distortion of the circles centered at the points zE or of the cross-ratios of the ordered quadruples of points of E.

The Slodkowski lifting theorem ([58], see also [59, 60, 61]) solves the problem of Sullivan and Thurston on the extension of holomorphic motions from any set to a whole sphere:

Extended lambda-lemma: Any holomorphic motionf:E×DĈcan be extended to a holomorphic motionf˜:Ĉ×DĈ, withf˜E×D=f.

The corresponding Beltrami differentials μf˜tz=z¯f˜zt/zf˜zt are holomorphic in t via elements of LC, and Schwarz’s lemma yields


or equivalently, the maximal dilatations Kf˜t1+t/1t. This bound cannot be improved in the general case.

Holomorphic motions have been important in the study of dynamical systems, Kleinian groups, holomorphic families of conformal maps and of Riemann surfaces as well as in many other fields (see, e.g., [40, 57, 59, 60, 62, 63, 64, 65, 66, 67, 68], and the references there).

There is an intrinsic connection between holomorphic motions and Teichmüller spaces, first mentioned by Bers and Royden in [69]. McMullen and Sullivan introduced in [65] the Teichmüller spaces for arbitrary holomorphic dynamical systems, and this approach is now one of the basic in complex dynamics [70].

Topics discussed in this section were studied in classic works [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85] as well as other references.


2. Unbounded convex polygons

2.1 Main theorem

The inequalities (5), (20) have served a long time as the main tool for establishing the exact or approximate values of the Fredholm value ρL and allowed to establish it only for some special collections of curves and arcs.

In this section, we present, following [33, 86], a new method that enables us to solve the indicated problems for large classes of convex domains and of their fractional linear images. This method involves in an essential way the complex geometry of the universal Teichmüller space T and the Finsler metrics on holomorphic disks in T as well as the properties of holomorphic motions on such disks.

It is based on the following general theorem for unbounded convex domains giving implies an explicit representation of the main associated curvelinear and analytic functionals invariants by geometric characteristics of these domains solving the problem for unbounded convex domains completely.

Theorem 5. For every unbounded convex domainDCwith piecewiseC1+δ-smooth boundaryLδ>0(and all its fractional linear images), we have the equalities


where f and f denote the appropriately normalized conformal maps DD and DD=Ĉ\D¯, respectively, k0f and k0f are the minimal dilatations of their quasiconformal extensions to Ĉ;ϰf and ϰf stand for their Grunsky norms, and πα is the opening of the least interior angle between the boundary arcs LjL. Here 0<α<1 if the corresponding vertex is finite and 1<α<0 for the angle at the vertex at infinity.

The same is true also for the unbounded concave domains (the complements of convex ones) which do not contain; for those one must replace the last term byβ1, whereπβis the opening of the largest interior angle ofD.

The proof of Theorem 5 is outlined in [33, 64]. In the next section we provide an extension of this important theorem to nonconvex polygons giving the detailed proof.

The equalities of type (21) were known earlier only for special closed curves (see [12, 19, 26, 55]), for example, for polygons bounded by circular arcs with a common inner tangent circle. The proof of Theorem 4 involves a completely different approach; it relies on the properties of holomorphic motions.

Let us mention also that the geometric assumptions of Theorem 4 are applied in the proof in an essential way. Its assertion extends neither to the arbitrary unbounded nonconvex or nonconcave domains nor to the arbitrary bounded convex domains.

This theorem has various important consequences. It distinguishes a broad class of domains, whose geometric properties provide the explicit values of intrinsic conformal and quasiconformal characteristics of these domains.

2.2 Examples

  1. 1. Let L be a closed unbounded curve with the convex interior, which is C1+δ smooth at all finite points and has at infinity the asymptotes approaching the interior angle πα<0. For any such curve, Theorem 4 yields the equalities


  1. 2. More generally, assume that L also has a finite angle point z0 with the angle opening πα0. Then, similar to (22),


Simultaneously this quantity gives the exact value of the reflection coefficient for any convex curvelinear lune bounded by two smooth arcs with the common endpoints a,b, because any such lune is a Moebius image of the exterior domain for the above curve L.

Other quantitative examples illustrating Theorem 5 are presented in [64].


3. Extension to unbounded non-convex polygons

3.1 An open question

An open question is to establish the extent in which Theorem 5 can be prolonged to arbitrary unbounded polygons

Our goal is to show that this is possible for unbounded rectilinear polygons for which the extent of deviation from convexity is sufficiently small.

This extension essentially increases the collections of individual polygonal curves and arcs with explicitly established Fredholm eigenvalues and reflection coefficients.

3.2 Main theorem

Let Pn be a rectilinear polygon with the finite vertices A1,A2,,An1 and with vertex A=, and let the interior angle at the vertex Aj be equal to παj and at A be equal to πα, where α<0 and all aj1, so that α1++αn1+α=2. Let fn be the conformal map of the upper half-plane U=z:Imz>0 onto Pn, which without loss of generality, can be normalized by fnz=zi+Ozi as zi (assuming that Pn contains the origin w=0).

An important geometric characteristic of polynomials is the quantity


it valuates the local boundary quasiconformal dilatation of Pn.

Using this quantity, we first prove that an assertion similar to Theorem 4 fails for the generic rectilinear polygons.

Theorem 6. There exist rectilinear polygonsPnwhose conformal mapping functionsfnsatisfy


whereais defined via (23).

Proof. We shall use the rectangles P4; in this case all αj=1/2. It is known that the mapping function f4 of any rectangle has equal Grunsky and Teichmüller norms,


(see [12, 55, 87]).

Using the Moebius map σ:z1/z, we transform the rectangle into a (nonconvex) circular quadrilateral σP4 with angles π/2 and mutually orthogonal edges so that two unbounded edges from these are rectilinear and two bounded are circular, and note that for sufficiently long rectangles must be


where f̂4 denotes the conformal map DσP4.

Indeed, as was established by Kühnau [12], the quadrilaterals with the side ratios (conformal module) greater than 3.31 have the reflection coefficient qP4>1/2 (the last inequality follows also from the fact that the long rectangles give in the limit a half-strip with two unbounded parallel sides. Such a domain is not a quasidisk, so its reflection coefficient equals 1); this proves (25).

Any circular quadrilateral σP4 satisfying (25) can be approximated by appropriate rectilinear polygons Pn. Assuming now that the equalities of type (21) or (24) are valid for all such polygons, one obtains a contradiction with (25), because both dilatation kf and qP are lower continuous functionals under locally uniform convergence of quasiconformal maps (i.e., kfliminfkfn as fnf in the indicated topology, and similarly for the reflection coefficient). This contradiction proves the theorem.

3.3 The main result

The main result of this section is

Theorem 7. [86] LetPnbe a unbounded rectilinear polygon, neither convex nor concave, and hence contain the verticesAjwhose inner anglesπαjhave openingsπαjwith1<αj<2. Assume that all suchαjsatisfy


whereαis given by (23) (which means that the maximal value in (22) is attained at some vertexAjwith0<aj<1).

For any such polygon, taking appropriate Moebius map σ:DU, we have the equalities


Proof. Let Pn be an unbounded rectilinear polygon. Its conformal mapping function fn:UPn fixing the infinite point and with fni=0 is represented by the Schwarz-Christoffel integral


where all aj=f1AjR and d0,d1 are the corresponding complex constants. The logarithmic derivative bf=logf=f/f of this map has the form


Letting Iα=tR:1/1α<t<1/1α,Dα=tC:t<1/1α, we construct for fn an ambient complex isotopy (holomorpic motion)


(containing fn as a fiber map), which is injective in the space coordinate z for any fixed t, holomorphic in t for a fixed z and wz0z.

First observe that for real rIα the solution Wr to the equation wz=rbf4zwz with the initial conditions wri=i,wr= satisfies




If the interior angles of the initial polygon Pn satisfy the assumption (26), then all the functions Wr are represented by an integral of type (27) (replacing αj by αjr, and with suitable constants d0r,d1r).

Geometrically this means that the exterior angle 2ππαjr at any finite vertex Ajr decreases with r (but the value αjr1 increases if 1<αj<2). Under the assumption (26), the admissible bounds for the possible values of angles ensure the univalence of this integral on U for every indicated t. This yields that every WrU also is a polygon with the interior angles παjr for r0, while W0U=U.

Now we pass to the conformal map gnζ=fnσ0ζ of the unit disk D onto Pn, using the function σ0ζ=1+ζ/1ζ. This map is represented similar to (28) by


where the points ej are the preimages of vertices ej=gn1Aj on the unit circle ζ=1. Pick d1 to have gn0=1. For this function, we have a natural complex isotopy




Following (31), we set for t=re,


The relations (32) yield that this function also is univalent in D.

The corresponding Schwarzians Sw˜rζ=rbw˜rζr2bw˜rζ2/2 fill a real analytic line Γ in the universal Teichmüller space T (modeled as a bounded domain in the complex Banach space B of hyperbolically bounded holomorphic functions on D). This line is located in the holomorphic disk Ω˜=bGT, where b denotes the map tSw˜t and GIα is a simply connected planar domain.

By Zhuravlev’s theorem (see [50, 51]), this domain contains for each rIα also the points Sw˜t with tr (representing the curvelinear polygons with piecewise analytic boundaries).

This generates the holomorphic motions (complex isotopies) w˜ζt:D×GĈ and wzt:UĈ with wz1=fnz.

The basic lambda-lemma for holomorphic motions implies that every fiber map wtz is the restriction to U of a quasiconformal automorphism Ŵtz of the whole sphere Ĉ, and the Beltrami coefficients


in the lower half-plane U=z:Imz<0 depend holomorphically on t as elements of the space LU.

So we have a holomorphic map μt from the disk Dα into the unit ball of Beltrami coefficients supported on U,


and the classical Schwarz lemma implies the estimate


It follows that the extremal dilatation of the initial map fnz=Ŵ1zU satisfies


Hence, also qPn1α and by the inequality (10), ϰfn1α.

On the other hand, Kühnau’s lower bound (20) implies


Together with (5), this yields that the polygon Pn admits all equalities (27) completing the proof of the theorem.

3.4 Some applications

Theorem 6 widens the collections of curves with explicitly given Fredholm eigenvalues and reflection coefficients.

For example, let L be a saw-tooth quasicircle with a finite number of triangular and trapezoidal teeth joined by rectilinear segments. We assume that the angles of these teeth satisfy the condition (26). Then we have the following consequence of Theorem 7.

Corollary 1. For any quasicircleLof the indicated form, its quasireflection coefficientqLand Fredholm eigenvalueρLare given by


whereαis defined similar to (22) by angles between the subintervals ofL. The same is valid for imagesγLunder the Moebius mapsγPSL2C.


4. Connection with complex geometry of universal Teichmüller space

4.1 Introductory remarks

Another reason why the convex polygons are interesting for quasiconformal theory is their close geometric connection with the geometry of universal Teichmüller space.

There is an interesting still unsolved completely question on shape of holomorphic embeddings of Teichmüller spaces stated in [88]:

For an arbitrary finitely or infinitely generated Fuchsian group Γ is the Bers embedding of its Teichmüller space TΓ starlike?

Recall that in this embedding TΓ is represented as a bounded domain formed by the Schwarzian derivatives Sw of holomorphic univalent functions wz in the lower half-plane U=z:Imz<0 (or in the disk) admitting quasiconformal extensions to the Riemann sphere Ĉ=C compatible with the group Γ acting on U.

It was shown in [89] that universal Teichmüller space T=T1 has points that cannot be joined to a distinguished point even by curves of a considerably general form, in particular, by polygonal lines with the same finite number of rectilinear segments. The proof relies on the existence of conformally rigid domains established by Thurston in [90] (see also [91]).

This implies, in particular, that universal Teichmüller space is not starlike with respect to any of its points, and there exist points φT for which the line interval :0<t<1 contains the points from B\S, where B=BU is the Banach space of hyperbolically bounded holomorphic functions in the half-plane U with norm


and S denotes the set of all Schwarzian derivatives of univalent functions on U. These points correspond to holomorphic functions on U which are only locally univalent.

Toki [92] extended the result on the nonstarlikeness of the space T to Teichmüller spaces of Riemann surfaces that contain hyperbolic disks of arbitrary large radius, in particular, for the spaces corresponding to Fuchsian groups of second kind. The crucial point in the proof of [92] is the same as in [89].

On the other hand, it was established in [46] that also all finite dimensional Teichmüller spaces TΓ of high enough dimensions are not starlike.

The nonstarlikeness causes obstructions to some problems in the Teichmüller space theory and its applications to geometric complex analysis.

The argument exploited in the proof of Theorems 4 and 5 provide much simpler constructive proof that the universal Teichmüller space is not starlike, representing explicitly the functions, which violate this property. It reveals completely different underlying geometric features.

Pick unbounded convex rectilinear polygon Pn with finite vertices A1,,An1 and An=. Denote the exterior angles at Aj by παj so that π<αj<2π,j=1,,n1. Then, similar to (22), the conformal map fn of the lower half-plane H=z:Imz<0 onto the complementary polygon Pn=Ĉ\Pn¯ is represented by the Schwarz-Christoffel integral


with aj=fn1AjR and complex constants d0,d1; here fn1=. Its Schwarzian derivative is given by


where bf=f/f and


It defines a point of the universal Teichmüller space T modeled as a bounded domain in the space BH of hyperbolically bounded holomorphic functions on H with norm φBH=supHzz¯2φz.

Denote by r0 the positive root of the equation


and put Sfn,t=tbfn'bfn2/2,t>0. Then for appropriate αj, we have.

Theorem 8. [93] For any convex polygonPn, the SchwarziansrSfn,r0define for any0<r<r0a univalent functionwr:HCwhose harmonic Beltrami coefficientνrz=r/2y2Sfn,r0z¯inHis extremal in its equivalence class, and


By the Ahlfors-Weill theorem [94], every φBH with φBH<1/2 is the Schwarzian derivative SW of a univalent function W in H, and this function has quasiconformal extension onto the upper half-plane H=z:Imz>0 with Beltrami coefficient of the form


called harmonic. Theorem 7 yields that any wr with r<r0 does not admit extremal quasiconformal extensions of Teichmüller type, and in view of extremality of harmonic coefficients μSwr the Schwarzians Swr for some r between r0 and 1 must lie outside of the space T; so this space is not a starlike domain in BH.

4.2 There are unbounded convex polygons Pn for which the equalities (33) are valid in the strengthened form


for all r1, completing the bounds (21).

We illustrate this on the case of triangles. Let P3 be a triangle with vertices A1,A2R and A3= and exterior angles α1,α2,α3. The logarithmic derivative of conformal map f3:HP3 has the form


with aj=f31AjR,j=1,2, and similar to (34),




If the angles of P3 satisfy α1,α2<a3, where πα3 is the angle at A3, the arguments from [93] yield that the harmonic Beltrami coefficient μSf3 satisfies (35).

Surprisingly, this construction is closely connected also with the weighted bounded rational approximation in sup norm [95, 96].


5. Quasiconformal features and fredholm eigenvalues of bounded convex polygons

5.1 Affine deformations and Grunsky norm

As it was mentioned above, there exist bounded convex domains even with analytic boundaries L whose conformal mapping functions have different Grunsky and Teichmüller norms, and therefore, ρL<1/qL.

The aim of this chapter is to provide the classes of bounded convex domains, especially polygons, for which these norms are equal and give explicitly the values of the associate curve functionals kf,ϰf,qL,ρL.

One of the interesting questions is whether the equality of Teichmüller and Grunsky norms is preserved under the affine deformations


with c=c2/c1,c<1 (as well as of more general maps) of quasidisks.

In the case of unbounded convex domains, this follows from Theorem 4. We establish this here for bounded domains D.

More precisely, we consider the maps gc, which are conformal in the complementary domain D=Ĉ\D¯ and have in D a constant quasiconformal dilatation c, regarding such maps as the affine deformations and the collection of domains gcD as the affine class of D.

If f is a quasiconformal automorphism of Ĉ conformal in D mapping the disk D onto a domain D, then for a fixed c the maps gcDf and gcfD differ by a conformal map h:DgcD and hence have in the disk D the same Beltrami coefficient.

Note that the inequality c<1 equivalent to c2<c1 follows immediately from the orientation preserving under this map and its composition with conformal map by forming the corresponding affine deformation (which arises after extension the constant Beltrami coefficient c by zero to the complementary domain).

The following theorem solves the problem positively.

Theorem 9. For any functionfΣ0withϰf=kfmapping the diskDonto the complement of a bounded domain (quasidisk)Dand any affine deformationgcof this domain (withqc<1), we have the equality


Theorems 9 essentially increases the set of quasicircles LĈ for which ρL=1/qL giving simultaneously the explicit values of these curve functionals. Even for quadrilaterals, this fact was known until now only for some special types of them (for rectangles [12, 27, 28, 33] and for rectilinear or circular quadrilaterals having a common tangent circle [55]).

5.2 Scheme of the proof of Theorem 9

The proof follows the lines of Theorem 1.1 in [97] and is divided into several lemmas.

First, we establish some auxiliary results characterizing the homotopy disk of a map with ϰf=kf.

Take the generic homotopy function


Then Sftz=t2Sft1z and this point-wise map determines a holomorphic map χft=Sft:DT so that the homotopy disks DSf=χfD foliate the space T. Note also that


and if Fz=1/f1/z maps the unit disk onto a convex domain, then all level lines fz=r for zD are starlike.

Lemma 1. If the homotopy functionftoffΣ0satisfyϰft0=kft0for some0<t0<1, then the equalityϰft=kftholds for alltt0and the homotopy diskDSfthas no critical pointstwith0<t<t0.

Take the univalent extension f1 of f to a maximal disk Db=zĈ:z>b,0<b<1 and define


Its Beltrami coefficient in D is defined by holomorphic quadratic differentials ψA12 of the form (19), and we have the holomorphic map, for a fixed xb=xnbl2,


of the disk DSf into D. In view of our assumption on f, the series (37) is convergent in some wider disk t<aa>1.

Using the map (37), we pull back the hyperbolic metric λDt=dt/1t2 to the disk DSF1 (parametrized by t) and define on this disk the conformal metric ds=λh˜xtdt with


of Gaussian curvature 4 at noncritical points. In fact, this is the supporting metric at t=a for the upper envelope λϰ=supxSl2λh˜xbt of metrics (38) followed by its upper semicontinuous regularization


(supporting means that λh˜xba=λϰa and λh˜xbt<λϰt in a neighborhood of a).

The metric λϰt is logarithmically subharmonic on D and its generalized Laplacian




(while for λh˜xb we have at its noncritical points Δlogλh˜xb=4λh˜xb2).

As was mentioned above, the Grunsky coefficients define on the tangent bundle TT a new Finsler structure Fϰφv dominated by the infinitesimal Teichmüller metric Fφv. This structure generates on any embedded holomorphic disk γDT the corresponding Finsler metric λγt=Fϰγtγt and reconstructs the Grunsky norm by integration along the Teichmüller disks:

Lemma 2. [97] On any extremal Teichmüller diskDμ0=ϕTtμ0:tD(and its isometric images inT), we have the equality


Taking into account that the disk DSf touches at the point φ=Sfa the Teichmüller disk centered at the origin of T and passing through this point and that the metric λϰ does not depend on the tangent unit vectors whose initial points are the points of DSf, one obtains from Lemma 3 and the equality ϰfa=kfa that also


The following lemma is a needed reformulation of Theorem 3.

Lemma 3. [97] The infinitesimal formsKTφvandFTφvof both Kobayashi and Teichmüller metrics on the tangent bundleTTofTare continuous logarithmically plurisubharmonic inφTand have constant holomorphic sectional curvatureκKφv=4.

We compare the metric λh˜xb with λK using Lemmas 2, 3, and Minda’s maximum principle given by.

Lemma 4. [98] If a functionu:D+is upper semicontinuous in a domainDCand its (generalized) Laplacian satisfies the inequalityΔuzKuzwith some positive constantKat any pointzD, whereuz>, and if


then eitheruz<0for allzDor elseuz=0for allzΩ.

Lemma 4 and the equality (39) imply that the metrics λh˜xb,λϰ,λK must be equal in the entire disk DSF, which yields by Lemma 2 the equality


for all r=t01 (with xnrSl2 depending on r) and that for any fΣ0 with ϰf=kfits homotopy diskDSFhas only a singularity at the origin ofT.

We may now investigate the action of affine deformations on the set of functions fΣ0 with equal Grunsky and Teichmüller norms.

Lemma 5. For any affine deformationgcof a convex domainDwith expansiongcw=w+b0c+b1cw1+nearw=, we have


and for sufficiently small c all composite maps


also satisfyb̂1c0.

Finally, we use the following important result of Kühnau [27].

Lemma 6. For any functionfz=z+b0+b1z1+Σ0withb10, the extremal quasiconformal extensions of the homotopy functionsfttoDare defined for sufficiently smalltr0=r0fr0>0by nonvanishing holomorphic quadratic differentials, and therefore,ϰft=kft.

Using these lemmas, one establishes the equalities λϰ=λK on the disk DSWf,c and


The final step of the proof is to extend the last equality to all c with c<1.

Applying again the chain rule for Beltrami coefficients μ,ν from the unit ball in LC,


and μ˜z=μwνzwzν¯/wzν (so for ν fixed, τ depends holomorphically on μ in L norm) and defining the corresponding functions (37), one gets now the holomorphic functions of cD. Then, constructing in a similar way the corresponding Finsler metrics


and taking their upper envelope λϰc and its upper semicontinuous regularization, one obtains a subharmonic metric of Gaussian curvature κλϰ4 on the nonsingular disk c<1. One can repeat for this metric all the above arguments using the already established equality (40) for small c.

5.3 Generalization

The arguments in the proof of Theorem 9 are extended almost straightforwardly to more general case:

Theorem 10. LetFΣ0andϰF=kF. Lethbe a holomorphic mapDTwithout critical points inDandh0=SF. Denote bygcthe univalent solution of the Schwarzian equation


whereHw=F1w, on the domainFD. Then, for anycD, the compositiongcFalso satisfiesϰgcF=kgcF.

Note that by the lambda lemma for holomorphic motions, the map h determines a holomorphic disk in the ball of Beltrami coefficients on FD, which yields, together with assumptions of the theorem, that for small c,


with b1c0. This was an essential point in the proof.

5.4 Bounded polygons

The case of bounded convex polygons has an intrinsic interest, in view of the following negative fact underlying the features and contrasting Theorem 5.

Theorem 11. There exist bounded rectilinear convex polygonsPnwith sufficiently large number of sides such that


It follows simply from Theorem 8 that if a polygon Pn, whose edges are quasiconformal arcs, satisfies ρPn=1/qPn then this equality is preserved for all its affine images. In particular, this is valid for all rectilinear polygons obtained by affine maps from polygons with edges having a common tangent ellipse (which includes the regular n-gons).

Theorem 10 naturally gives rise to the question whether the property ρPn=1/qPn is valid for all bounded convex polygons with sufficiently small number of sides.

In the case of triangles this immediately follows from Theorem 7 as well as from Werner’s result.

Noting that the affinity preserves parallelism and moves the lines to lines, one concludes from Theorem 8 that the equality ρP4=1/qP4 holds in particular for quadrilaterals P4 obtained by affine transformations from quadrilaterals that are symmetric with respect to one of diagonals and for quadrilaterals whose sides have common tangent outwardly ellipse (in particular, for all parallelograms and trapezoids). For the same reasons, it holds also for hexagons with axial symmetry having two opposite sides parallel to this axes.

In fact, Theorem 8 allows us to establish much stronger result answering the question positively for quadrilaterals.

Theorem 12. For every rectilinear convex quadrilateralP4, we have


where F is the appropriately normalized conformal map ofDontoP4.

The proof of this theorem essentially relies on Theorem 8 and on result of [33] that the equalities (41) are valid for all rectangles, and hence for their affine transformations.

Fix such a quadrilateral P40=A10A20A30A40 and consider the collection P0 of quadrilaterals P4=A10A20A30A4 with the same first three vertices and variable A4; the corresponding A4 runs over a subset E of the trice punctured sphere Ĉ\A10A20A30.

The collection P0 contains the trapezoids, for which we have the equalities (41) by Theorem 8 (and consequently, the infinitesimal equality (39) at the corresponding points a).

Similar to the proof of Theorem 6, one obtains in the universal Teichmüller space T a holomorphic disk Ω extending the real analytic curve filled by the Schwarzians, which correspond to the values t=A on E. On this disk, one can construct, similar to (38), the corresponding metric λϰ. Lemmas 4–6 again imply that this metric must coincide at all points of Ω with the dominant infinitesimal Teichmüller-Kobayashi metric λK of T. Together with Lemma 2, this provides the global equalities (41) for all points of the disk Ω (and hence for the prescribed quadrilateral P40).

5.5 An open problem here is the following question of Kühnau (personal communication)

Question: Does the reflection coefficient of a rectangleRbe a monotone nondecreasing function of its conformal moduleμR(the ratio of the vertical and horizontal side lengths)?

The results of Kühnau and Werner for the rectangles R state that if the module μR satisfies 1μR<1.037, then


if μR>2.76, then q∂ℛ>1/2 (see [12, 55]).

On the other hand, the reflection coefficients of long rectangles are close to 1, because the limit half-strip is not a quasidisk.


6. Reflections across finite collections of quasiintervals

6.1 General comments

There are only a few exact estimates of the reflection coefficients of quasiconformal arcs (quasiintervals) and some their sharp upper bounds presented in [14, 99]. The most of these bounds have been obtained using the classical Bernstein-Walsh-Siciak theorem, which quantitatively connects holomorphic extension of a function defined on a compact KCn with the speed of its polynomial approximation. Another approach was applied by Kühnau in [54, 100, 101, 102]. In particular, using somewhat modification of Teichmüller’s Verschiebungssatz [103], he established in [102] the reflection coefficient of the set E, which consists of the interval 2i2i and a separate point t>0. All these results are presented in [64].

Theorems 4 and 6 open a new way in solving this problem following the lines of the first example after Theorem 4.

6.2 Reflections across the finite collections of quasiintervals

Theorems 5 and 7 open a new way in solving this problem following the lines of the first example after Theorem 5. Namely, given a finite union


of smooth curvelinear quasiintervals (possibly mutually separated) such that L can be extended without adding new vertices (angular points) to a quasicircle L0L containing z= and bounding a convex polygon PN that satisfies the assumptions of Theorem 4 or a polygon considered in Theorem 7, then by these theorems, the reflection coefficient of the setLequals


whereαis defined forL0similar to (23).

The main point here is to get a convex (or sufficiently close to convex, as in Theorem 7) polygon, because the initial and final arcs of components Lj can be smoothly extended and then rounded off.

Note also that adding to L a finite number of appropriately located isolated points z1,zm does not change the reflection coefficient (42).


Additional Classification

2010 Mathematics Subject Classification: Primary: 30C55, 30C62, 30F60; Secondary: 31A35, 58B15


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Written By

Samuel L. Krushkal

Submitted: 09 December 2019 Reviewed: 09 April 2020 Published: 20 May 2020