Quasiconformal Reflections across Polygonal Lines

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.


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 S 2 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 S 2 with f ∘ f ¼ id and are called topological reflections.
We shall consider here quasiconformal reflections or quasireflections on the Riemann sphere ¼  ∪ ∞ f g ¼ S 2 , that is, the orientation reversing quasiconformal automorphisms of order 2 (involutions) of the sphere with f ∘ f ¼ id. The topological circles admitting such reflections are called quasicircles. Such circles are locally quasiintervals, that is, the images of straight line segments under the infimum in (1) is taken over all quasireflections across E, provided those exist, and is attained by some quasireflection f 0 .
When E ¼ L is a quasicircle, the corresponding quantity and the reflection coefficient q E 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 k f * Theorem 1. For any set E ⊂ which admits quasireflections, there is a quasicircle L ⊃ E with the same reflection coefficient; therefore, The proof of this important theorem was given for finite sets E ¼ z 1 , … , z n f g 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 E ⊂ hC by the author in [14].
Theorem 1 yields, in particular, that similar to (3) for any set E ⊂, its quasiconformal dilatation satsfies where k E ¼ inf ∥∂ z f =∂ z f ∥ ∞ over all quasicircles L ⊃ E and all orientationpreserving quasiconformal homeomorphisms f : ! with f À Á ¼ 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 z ↦ z within conjugation by transformations g ∈ PSL 2,  ð Þ. 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 E ⊂ we have Q E ¼ ∞, 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.

Fredholm eigenvalues
Recall that the Fredholm eigenvalues ρ n of an oriented smooth closed Jordan curve L on the Riemann sphere ¼  ∪ ∞ f g are the eigenvalues of its double-layer potential, or equivalently, of the integral equation which has has many applications (here n ζ is the outer normal and ds ζ is the length element at ζ ∈ L).
The least positive eigenvalue ρ L ¼ ρ 1 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 and harmonic on G ∪ G * . In particular, ρ L ¼ ∞ only for the circle.
An upper bound for ρ L is given by Ahlfors' inequality [17].
where q L 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 PSL 2,  ð Þ= AE 1, it suffices to consider the quasiconformal homeomorphisms of the sphere carrying S 1 onto L whose Beltrami coefficients μ f z The inequality (5) serves as a background for defining the value ρ L , 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.

The Grunsky and Milin inequalities
Let 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 in terms of an infinite system of the coefficient inequalities. In particular, his theorem for the canonical disk  * yields that a holomorphic where α mn are defined by the sequence x ¼ x n ð Þ runs over the unit sphere S l 2 À Á of the Hilbert space l 2 with x n j j 2 , 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 k f ð Þ denotes the Teichmüller norm of f , which is equal to the infimum of dilatations k w μ ð Þ ¼ ∥μ∥ ∞ of quasiconformal extensions of f to. Here w μ denotes a homeomorphic solution to the Beltrami equation ∂ z w ¼ μ∂ z w on  extending f .
Note that the Grunsky (matrix) operator acts as a linear operator l 2 ! l 2 contracting the norms of elements x ∈ l 2 ; the norm of this operator equals ϰ f ð Þ (cf. [23,24]). For most functions f , we have in (10) the strong inequality ϰ f ð Þ< k f ð Þ (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 R X z, ζ ð Þ relates to the conformal map j θ z, ζ ð Þ of X onto the sphere slit along arcs of logarithmic spirals inclined at the angle θ ∈ 0, π ½ Þto a ray issuing from the origin so that j θ ζ, ζ ð Þ ¼ 0 and (in fact, only the maps j 0 and j π=2 are applied).
Here φ n f g ∞ 1 is a canonical system of holomorphic functions on X such that (in a local parameter) φ n z ð Þ ¼ a n,n z n þ a nþ1,n z nþ1 þ … with a n,n > 0, n ¼ 1, 2, … , and the derivatives (linear holomorphic differentials) φ 0 n form a complete orthonormal system in H 2 X ð Þ. We shall deal only with simply connected domains X ¼ D * ∍∞ with quasiconformal boundaries (quasidisks). For any such domain, the kernel R D vanishes identically on D * Â D * , and the expansion (11) assumes the form where χ denotes a conformal map of D * onto the disk (12) is represented as a polynomial of a finite number of the initial coefficients b 1 , b 2 , … , b s 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 B D * ð Þ of hyperbolically bounded holomorphic functions φ ∈  * with norm where λ D * z ð Þ|dz| denotes the hyperbolic metric of D * of Gaussian curvature À4. This domain models the universal Teichmüller space T with the base point χ 0 ∞ ð ÞD * (in holomorphic Bers' embedding of T).
A theorem of Milin [29] extending the Grunsky univalence criterion for the disk  * to multiply connected domains D * states that a holomorphic function f z ð Þ in a neighborhood of z ¼ ∞ can be continued to a univalent function in the whole domain D * if and only if the coefficients β mn in (12) satisfy, similar to the classical case of the disk  * , the inequality for any point x ¼ x n ð Þ∈ S l 2 À Á . We call the quantity the generalized Grunsky norm of f . By (14), ϰ D * f ð Þ≤ 1 for any f from the class Σ D * ð Þ of univalent functions in D * with hydrodynamical normalization The inequality ϰ D * f ð Þ≤ 1 is necessary and sufficient for univalence of f in D * (see [11,20,29]).
The norm (15) also is dominated by the Teichmüller norm k f ð Þ of this map. Similar to (10), where τ T 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 * ¼  * , β mn ¼ ffiffiffiffiffiffi ffi mn p α mn ; for this disk, we shall use the notations Σ and ϰ f ð Þ. We denote by S the canonical class of univalent functions The Grunsky norm of univalent functions F ∈ S is defined similar to (5), (6); so any such F z ð Þ and its inversion f z ð Þ ¼ 1=F 1=z ð Þ univalent in D * have the same Grunsky coefficients α mn . Technically it is more convenient to deal with functions univalent in  * .

Extremal quasiconformality
A crucial point here is that the Teichmüller norm on Σ is intrinsically connected with integrable holomorphic quadratic differentials ψ z ð Þdz 2 on the complementary domain D ¼nD * (the elements of the subspace A 1 D ð Þ of L 1 D ð Þ 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 with the interior and exterior domains D and D * . Denote by λ D z ð Þ|dz| the hyperbolic metric of D of Gaussian curvature À4 and by δ D z ð Þ ¼ dist z, ∂D ð Þthe Euclidean distance from the point z ∈ D to the boundary. Then where the right-hand inequality follows from the Schwarz lemma and the left from Koebe's 1 4 theorem. Consider the unit ball of Beltrami coefficients supported on D, and take the corresponding quasiconformal automorphisms w μ z ð Þ of the spherê  satisfying on  the Beltrami equation ∂ z w ¼ μ∂ z w preserving the points 0, 1, ∞ fixed. Recall that k w ð Þ ¼ ∥μ w ∥ ∞ is the dilatation of the map w. Take the equivalence classes μ ½ and w μ ½ letting the coefficients μ 1 and μ 2 from Belt D * ð Þ 1 be equivalent if the corresponding maps w μ 1 and w μ 2 coincide on L (and hence on D). These classes are in one-to-one correspondence with the Schwarzians S w μ on D * , which fill a bounded domain in the space B 2 D * ð Þmodeling the universal It is the integral form of the infinitesimal Finsler metric on the tangent bundle T T of T, which is locally Lipschitzian. The Grunsky coefficients give rise to another Finsler structure F x, v ð Þ on the bundle T T. It is dominated by the canonical Finsler structure F T x, v ð Þ and allows one to reconstruct the Grunsky norm along the geodesic Teichmüller disks in T (see [33]).
We call the Beltrami coefficient μ ∈ Belt D * ð Þ 1 extremal (in its class) 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

Conformal Maps
Let w 0 ≔ w μ 0 be an extremal representative of its class w 0 ½ with dilatation and assume that there exists in this class a quasiconformal map w 1 whose Beltrami coefficient μ A 1 satisfies the inequality ess sup A r |μ w 1 z ð Þ| < k w 0 ð Þ in some ring domain R ¼ D * nG complement to a domain G ⊃ D * . Any such w 1 is called the frame map for the class w 0 ½ , 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 class f ½ has a frame map, then the extremal map f 0 in this class (minimizing the dilatation ∥μ∥ ∞ ) is unique and either a conformal or a Teichmüller map with Beltrami coefficient μ 0 ¼ k|ψ 0 |=ψ 0 on D, defined by an integrable holomorphic quadratic differential ψ 0 on D and a constant k ∈ 0, 1 ð Þ [34]. [35,36]. The first assertion holds, for example, for asymptotically conformal curves L. Similar results hold also for arbitrary Riemann surfaces (cf. [36,37]).

(ii) The set of Strebel points is open and dense in T
Recall that a Jordan curve L ⊂  is called asymptotically conformal if for any pair of points a, b ∈ L, 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 C 1 -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 H f ð Þ admits also a local version H p f ð Þ involving the Beltrami coefficients supported in the neighborhoods of a boundary point p ∈ ∂D. Moreover (see, e.g., [36], Ch. 17), H f ð Þ ¼ sup p ∈ ∂D H p f ð Þ, and the points with H p f ð Þ ¼ H f ð Þ 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.

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 S 1 ¼ ∂ 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, ν ∈ Belt  ð Þ 1 be equivalent if the corresponding maps w μ , w ν ∈ Σ 0 coincide on S 1 (hence, on  * ) and passing to Schwarzian derivatives S f μ . The defining projection ϕ T : μ ! S w μ is a holomorphic map from L ∞  ð Þ to B. The equivalence class of a map w μ will be denoted by w μ ½ . An intrinsic complete metric on the space T is the Teichmüller metric, defined above in Section 1.4, with its infinitesimal Finsler form (structure) 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 metric d T on T is the largest pseudometric d on T does not get increased by holomorphic maps h :  ! T so that for any two points ψ 1 , ψ 2 ∈ T, we have where d  is the hyperbolic Poincaré metric on  of Gaussian curvature À4, with the differential form The Carathéodory distance between ψ 1 and ψ 2 in T is where the supremum is taken over all holomorphic maps h :  ! T.
The corresponding differential (infinitesimal) forms of the Kobayashi and Carathéodory metrics are defined for the points ψ, v ð Þ of the tangent bundle T T ð Þ, respectively, by where Hol X, Y ð Þdenotes the collection of holomorphic maps of a complex manifold X into Y and  r is the disk jzj < r f g. 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.
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 metric τ T ψ 1 , ψ 2 ð Þof either of the spaces T or T  * ð Þ is plurisubharmonic separately in each of its arguments; hence, the pluricomplex Green function of T equals where k is the norm of extremal Beltrami coefficient defining the distance between the points ψ 1 , ψ 2 in T (and similar for the space T  * ð Þ).
Þof metric τ T and has constant holomorphic sectional curvature κ K ψ, v ð Þ ¼ À4 on the tangent bundle T T ð Þ. In other words, the Teichmüller-Kobayashi metric is the largest invariant plurisubharmonic metric on T.
The generalized Gaussian curvature κ λ of an upper semicontinuous Finsler where  is the generalized Laplacian . Similar to C 2 functions, for which  coincides with the usual Laplacian, one obtains that λ is subharmonic on Ω if and only if λ t ð Þ ≥ 0; hence, at the points t 0 of local maximuma of λ with λ t 0 ð Þ> À ∞, we have λ t 0 ð Þ≤ 0. 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 K x, v ð Þ of any complete hyperbolic manifold X satisfies κ K X ≥ À 4 at all points x, v ð Þ of the tangent bundle T X ð Þ of X, and for the Carathéodory metric C X we have κ C x, v ð Þ≤ À 4. Finally, the pluricomplex Green function of a domain X on a complex Banach space manifold E is defined as g X x, y ð Þ ¼ supu y x ð Þ x, y ∈ X ð Þ , where supremum is taken over all plurisubharmonic functions u y x ð Þ : ð Þ in a neighborhood of the pole y. Here ∥ Á ∥ is the norm on X and the remainder term O 1 ð Þ is bounded from above. If X is hyperbolic and its Kobayashi metric d X is logarithmically plurisubharmonic, then g X x, y ð Þ ¼ log tanh d X x, y ð Þ, which yields the representation of g T 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 u x ð Þ on a domain D in a Banach space X means that u x ð Þ 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]).

The Grunsky-Milin inequalities revised
Denote by Σ 0 D * ð Þ the subclass of Σ D * ð Þ formed by univalent-holomorphic functions in D * with expansions f z Each Grunsky coefficient α mn f ð Þ is a polynomial of a finite number of the initial coefficients b 1 , b 2 , … , b mþnÀ1 of f ; hence it depends holomorphically on Beltrami coefficients of extensions of f as well as on the Schwarzian derivatives S f ∈ B 2 D * ð Þ.

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]).
where μ 0 is an extremal Beltrami coefficient in the equivalence class f ½ . The last equality is equivalent to Note that geometrically (16) means the equality of the Carathéodory and Teichmüller distances on the geodesic disk ϕ T tμ 0 =∥μ 0 ∥ ð Þ: t ∈  f g in the universal Teichmüller space T and that the mentioned above the strict inequality ϰ f ð Þ< k f ð Þ is valid on the (open) dense subset of Σ 0 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 α mn f ð Þ ¼ α mn S F ð Þ is that these coefficients are holomorphic functions of the Schwarzians φ ¼ S f on the universal Teichmüller space T. Therefore, for every f ∈ Σ 0 and each defines a holomorphic map of the space T into the unit disk , and  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 ffiffiffiffiffiffi ffi mn p x m x n z mþnÀ2 (19) with x ¼ x n ð Þ∈ l 2 so that ∥x∥ l 2 ¼ ∥ψ∥ A 1 (see [52]). This result extends to arbitrary domains D with quasiconformal boundaries but the proof is much more complicated (see [22]).

The first Fredholm eigenvalue and Grunsky norm
One of the basic tools in quantitative estimating the Freholm eigenvalues ρ L of quasicircles is given by the classical Kühnau-Schiffer theorem mentioned above. This theorem states that the value ρ L is reciprocal to the Grunsky norm ϰ f ð Þ of the Riemann mapping function of the exterior domain of L (see. [27,53]).
Another important tool is the following Kühnau's jump inequality [12]: If a closed curve L ⊂ contains two analytic arcs with the interior intersection angle πα 0 , then This implies the lower estimate for q L and 1=ρ L . By approximation, this inequality extends to smooth arcs.
One of the standard ways of establishing the reflection coefficients q L (respectively, the Fredholm eigenvalues ρ L ) consists of verifying wether the equality in (5) or the equality ϰ f * ð Þ ¼ k 0 f * ð Þ 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 m R ð Þ vary in the interval 1 ≤ m R ð Þ< 1:037; moreover, in this case q L ¼ 1=ρ L ¼ 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].

Holomorphic motions
Let E be a subset of containing at least three points. A holomorphic motion of E is a function f : E Â  ! such that: a. for every fixed z ∈ E, the function t ↦ f z, t ð Þ : E Â  ! is holomorphic in ; b. for every fixed t ∈ , the map f z, t 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) f extends to a holomorphic motion of the closure E of E; (ii) each f t z ð Þ ¼ f t, z ð Þ : E ! is quasiconformal; (iii) f is jointly continuous in z, t ð Þ. Quasiconformality here means, in the general case, the boundedness of the distortion of the circles centered at the points z ∈ E or of the cross-ratios of the ordered quadruples of points of E.
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].

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 ρ L 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 domain D ⊂  with piecewise C 1þδ -smooth boundary L δ > 0 ð Þ(and all its fractional linear images), we have the equalities where f and f * denote the appropriately normalized conformal maps  ! D and  * ! D * ¼nD, respectively, k 0 f ð Þ and k 0 f * ð Þ are the minimal dilatations of their quasiconformal extensions to; ϰ f ð Þ and ϰ f * ð Þ stand for their Grunsky norms, and π|α| is the opening of the least interior angle between the boundary arcs L j ⊂ L. Here 0 < α < 1 if the corresponding vertex is finite and À1 < 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 by |β| À 1, where π|β| is the opening of the largest interior angle of D.
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.

Examples
1. Let L be a closed unbounded curve with the convex interior, which is C 1þδ 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 2. More generally, assume that L also has a finite angle point z 0 with the angle opening πα 0 . 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].

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.

Main theorem
Let P n be a rectilinear polygon with the finite vertices A 1 , A 2 , … , A nÀ1 and with vertex A ∞ ¼ ∞, and let the interior angle at the vertex A j be equal to πα j and at A ∞ be equal to πα ∞ , where α ∞ < 0 and all a j 6 ¼ 1, so that α 1 þ … þ α nÀ1 þ α ∞ ¼ 2. Let f n be the conformal map of the upper half-plane U ¼ z : Im z > 0 f gonto P n, which without loss of generality, can be normalized by f n z ð Þ ¼ z À i þ O z À i ð Þas z ! i (assuming that P n contains the origin w ¼ 0).
An important geometric characteristic of polynomials is the quantity it valuates the local boundary quasiconformal dilatation of P n . Using this quantity, we first prove that an assertion similar to Theorem 4 fails for the generic rectilinear polygons. Theorem 6. There exist rectilinear polygons P n whose conformal mapping functions f n satisfy where a is defined via (23). Proof. We shall use the rectangles P 4 ; in this case all α j ¼ 1=2. It is known that the mapping function f 4 of any rectangle has equal Grunsky and Teichmüller norms, (see [12,55,87]). Using the Moebius map σ : z ↦ 1=z, we transform the rectangle into a (nonconvex) circular quadrilateral σ P 4 ð Þ 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 wheref 4 denotes the conformal map  ! σ P 4 ð Þ. Indeed, as was established by Kühnau [12], the quadrilaterals with the side ratios (conformal module) greater than 3:31 have the reflection coefficient q ∂P 4 > 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 σ P 4 ð Þ satisfying (25) can be approximated by appropriate rectilinear polygons P n . 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 k f ð Þ and q ∂P are lower continuous functionals under locally uniform convergence of quasiconformal maps (i.e., k f ð Þ≤ lim inf k f n À Á as f n ! f in the indicated topology, and similarly for the reflection coefficient). This contradiction proves the theorem.

The main result
The main result of this section is Theorem 7. [86] Let P n be a unbounded rectilinear polygon, neither convex nor concave, and hence contain the vertices A j whose inner angles πα j have openings πα j with 1 < α j < 2. Assume that all such α j satisfy where α is given by (23) (which means that the maximal value in (22) is attained at some vertex A j with 0 < |a j | < 1).
For any such polygon, taking appropriate Moebius map σ :  ! U, we have the equalities Proof. Let P n be an unbounded rectilinear polygon. Its conformal mapping function f n : U ! P n fixing the infinite point and with f n i ð Þ ¼ 0 is represented by the Schwarz-Christoffel integral where all a j ¼ f À1 * A j À Á ∈  and d 0 , d 1 are the corresponding complex constants.
jtj < 1=j1Àjαk f g , we construct for f n an ambient complex isotopy (holomorpic motion) (containing f n as a fiber map), which is injective in the space coordinate z for any fixed t, holomorphic in t for a fixed z and w z, 0 ð Þ z. First observe that for real r ∈ I α the solution W r to the equation If the interior angles of the initial polygon P n satisfy the assumption (26), then all the functions W r are represented by an integral of type (27) (replacing α j by α j r ð Þ, and with suitable constants d 0r , d 1r ). Geometrically this means that the exterior angle 2π À πα j r ð Þ at any finite vertex A j r ð Þ decreases with r (but the value α j r ð Þ À 1 increases if 1 < α j < 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 W r U ð Þ also is a polygon with the interior angles πα j r ð Þ for r 6 ¼ 0, while W 0 U ð Þ ¼ U. Now we pass to the conformal map g n ζ ð Þ ¼ f n ∘ σ 0 ζ ð Þ of the unit disk  onto P n , using the function σ 0 ζ ð Þ ¼ 1 þ ζ ð Þ= 1 À ζ ð Þ. This map is represented similar to (28) by where the points e j are the preimages of vertices e j ¼ g À1 n A j À Á on the unit circle jζj¼ 1 f g. Pick d 1 to have g 0 n 0 ð Þ ¼ 1. For this function, we have a natural complex isotopyw Following (31), we set for t ¼ re iθ , The relations (32) yield that this function also is univalent in .
The corresponding Schwarzians Sw r ζ ð Þ ¼ rb 0w r ζ ð Þ À r 2 bw r ζ ð Þ 2 =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 ). This line is located in the holomorphic diskΩ ¼ b G ð Þ⊂ T, where b denotes the map t ↦ Sw t and G ⊃ I α is a simply connected planar domain. By Zhuravlev's theorem (see [50,51]), this domain contains for each r ∈ I α also the points Sw t with |t| ≤ r (representing the curvelinear polygons with piecewise analytic boundaries).
This generates the holomorphic motions (complex isotopies)w ζ, t ð Þ :  Â G !  and w z, t ð Þ : U ! with w z, 1 ð Þ ¼ f n z ð Þ. The basic lambda-lemma for holomorphic motions implies that every fiber map w t z ð Þ is the restriction to U of a quasiconformal automorphismŴ t z ð Þ of the whole sphere, and the Beltrami coefficients in the lower half-plane U * ¼ z : Im z < 0 f gdepend holomorphically on t as elements of the space L ∞ U * ð Þ. So we have a holomorphic map μ Á, t ð Þ from the disk  α 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 f n z ð Þ ¼Ŵ 1 z ð Þ|U satisfies k f n À Á ≤ |1 À |αk: Hence, also q ∂P n ≤ |1 À |αk and by the inequality (10), ϰ f n À Á ≤ |1 À |αk. On the other hand, Kühnau's lower bound (20) implies Together with (5), this yields that the polygon P n admits all equalities (27) completing the proof of the theorem.

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 quasicircle L of the indicated form, its quasireflection coefficient q L and Fredholm eigenvalue ρ L are given by where |α| is defined similar to (22) by angles between the subintervals of L. The same is valid for images γ L ð Þ under the Moebius maps γ ∈ PSL 2,  ð Þ.

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 T Γ ð Þ starlike? Recall that in this embedding T Γ ð Þ is represented as a bounded domain formed by the Schwarzian derivatives S w of holomorphic univalent functions w z ð Þ in the lower half-plane U * ¼ z : Im z < 0 f g(or in the disk) admitting quasiconformal extensions to the Riemann sphere ¼  ∪ ∞ f g compatible with the group Γ acting on U * .
It was shown in [89] that universal Teichmüller space T ¼ T 1 ð Þ 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 φ ∈ T for which the line interval tφ : 0 < t < 1 f g contains the points from BnS, where B ¼ B U * ð Þis 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 T Γ ð Þ 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 P n with finite vertices A 1 , … , A nÀ1 and A n ¼ ∞. Denote the exterior angles at A j by πα j so that π < α j < 2π, j ¼ 1, … , n À 1. Then, similar to (22), the conformal map f n of the lower half-plane H * ¼ z : Im z < 0 f gonto the complementary polygon P * n ¼nP n is represented by the Schwarz-Christoffel integral Its Schwarzian derivative is given by where b f ¼ f 00 = f 0 and It defines a point of the universal Teichmüller space T modeled as a bounded domain in the space B H * ð Þ of hyperbolically bounded holomorphic functions on Denote by r 0 the positive root of the equation If the angles of P * 3 satisfy α 1 , α 2 < |a 3 |, where Àπα 3 is the angle at A 3 , the arguments from [93] yield that the harmonic Beltrami coefficient μ S f 3 satisfies (35). Surprisingly, this construction is closely connected also with the weighted bounded rational approximation in sup norm [95,96].

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, ρ L < 1=q L .
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 k f ð Þ, ϰ f ð Þ, q L , ρ L . One of the interesting questions is whether the equality of Teichmüller and Grunsky norms is preserved under the affine deformations with c ¼ c 2 =c 1 , |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 g c , which are conformal in the complementary domain D * ¼nD and have in D a constant quasiconformal dilatation c, regarding such maps as the affine deformations and the collection of domains g c D ð Þ as the affine class of D.
If f is a quasiconformal automorphism of conformal in  * mapping the disk  onto a domain D, then for a fixed c the maps g c |D∘ f and g c ∘ f ð Þ| differ by a conformal map h : D ! g c D ð Þ and hence have in the disk  the same Beltrami coefficient. Note that the inequality |c| < 1 equivalent to |c 2 | < |c 1 | 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 function f ∈ Σ 0 with ϰ f ð Þ ¼ k f ð Þ mapping the disk  * onto the complement of a bounded domain (quasidisk) D and any affine deformation g c of this domain (with |qc| < 1), we have the equality Theorems 9 essentially increases the set of quasicircles L ⊂ for which ρ L ¼ 1=q L 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]).

Scheme of the proof of Theorem 9
The proof follows the lines of Theorem 1.1 in [97] and is divided into several lemmas.
where H w ð Þ ¼ F À1 w ð Þ, on the domain F  * ð Þ. Then, for any c ∈ , the composition g c ∘ F also satisfies ϰ g c ∘ F À Á ¼ k g c ∘ F À Á . Note that by the lambda lemma for holomorphic motions, the map h determines a holomorphic disk in the ball of Beltrami coefficients on F  ð Þ, which yields, together with assumptions of the theorem, that for small |c|, This was an essential point in the proof.

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 polygons P n with sufficiently large number of sides such that ρ ∂P n < 1=q ∂P n : It follows simply from Theorem 8 that if a polygon P n , whose edges are quasiconformal arcs, satisfies ρ ∂P n ¼ 1=q ∂P n 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 ρ ∂P n ¼ 1=q ∂P n 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 ρ ∂P 4 ¼ 1=q ∂P 4 holds in particular for quadrilaterals P 4 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 quadrilateral P 4 , we have where F is the appropriately normalized conformal map of  * onto P * 4 . 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 P 0 4 ¼ A 0 1 A 0 2 A 0 3 A 0 4 and consider the collection P 0 of quadrilaterals P 4 ¼ A 0 1 A 0 2 A 0 3 A 4 with the same first three vertices and variable A 4 ; the corresponding A 4 runs over a subset E of the trice punctured spheren A 0 1 , A 0 2 , A 0 3 È É . The collection P 0 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 ¼ A on E. On this disk, one can construct, similar to (38), the corresponding metric λ ϰ . Lemmas 4-6 again imply that this metric must coincide at all points of Ω with the dominant infinitesimal Teichmüller-Kobayashi metric λ K 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 P 0 4 ).

An open problem here is the following question of Kühnau (personal communication)
Question: Does the reflection coefficient of a rectangle R be a monotone nondecreasing function of its conformal module μ R (the ratio of the vertical and horizontal side lengths)?
The results of Kühnau and Werner for the rectangles R state that if the module μ R ð Þ satisfies 1 ≤ μ R ð Þ< 1:037, then if μ R ð Þ> 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.

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 K⋐ n 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 À2i, 2i ½ 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.

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 L 0 ⊃ L containing z ¼ ∞ and bounding a convex polygon P N that satisfies the assumptions of Theorem 4 or a polygon considered in Theorem 7, then by these theorems, the reflection coefficient of the set L equals 27 Quasiconformal Reflections across Polygonal Lines DOI: http://dx.doi.org/10.5772/intechopen.92441 where α is defined for L 0 similar 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 L j can be smoothly extended and then rounded off.
Note also that adding to L a finite number of appropriately located isolated points z 1 , … z m does not change the reflection coefficient (42).