Open access peer-reviewed chapter - ONLINE FIRST

Quasiconformal Reflections across Polygonal Lines

By Samuel L. Krushkal

Submitted: December 9th 2019Reviewed: April 9th 2020Published: May 20th 2020

DOI: 10.5772/intechopen.92441

Downloaded: 10

Abstract

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.

Keywords

  • Grunsky inequalities
  • univalent function
  • Beltrami coefficient
  • quasiconformal reflection
  • universal Teichmüller space
  • Fredholm eigenvalues
  • convex polygon
  • 2010 Mathematics Subject Classification: Primary: 30C55
  • 30C62
  • 30F60; Secondary: 31A35
  • 58B15

1. Quasiconformal reflections: general theory

1.1 Quasireflections and quasicurves

The classical Brouwer-Kerekjarto theorem ([12, 29], see also [88]) says that every periodic homeomorphism of the sphere S2S2is 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 Econsists 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 S2with ff=idand 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 LĈinterchanging its inner and outer domains.

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

wz¯=μzwz,zD,

where

z=12xiy,z=12xiy

are the distributional partial derivatives, μis a given function from LDwith μ<1, called the Beltrami coefficient (or complex dilatation) of the map w, and the quantity kw=μis 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., [5, 34, 64]).

Qualitatively, any quasicircle Lis characterized, due to [4], by uniform boundedness of the cross-ratios for all ordered quadruples z1z2z3z4of the distinct points on L; namely,

z1z2¯z1z3¯z3z4¯z2z4¯C<

for any quadruple of points z1,z2,z3,z4on Lfollowing 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

z2z1z3z1C.

This is shown in [4] by applying the properties of quasisymmetric maps. Ahlfors has established also that if a topological circle Ladmits quasireflections (i.e., is a quasicircle), then there exists a differentiable quasireflection across Lwhich is (euclidian) bilipschitz-continuous. This property is very useful in various applications. On its extension to hyperbolic M-bilipschitz reflections see [24].

Geometrically, a quasicircle is characterized by the property that, for any two points z1,z2on L, the ratio of the chordal distance z1z2to 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 [23, 66, 77]. We will not touch here the extension of this theory to higher dimensions.

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

One defines for each mirror Eits reflection coefficient

qE=infkf=infzf/z¯fE1

and quasiconformal dilatation

QE=1+qE/1qE1;

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

When E=Lis a quasicircle, the corresponding quantity

kE=infkf:fS1=EE2

and the reflection coefficient qEcan be estimated in terms of one another; moreover, due to [5, 55], we have

QE=KE1+kE1kE2.E3

The infimum in (2) is taken over all orientation-preserving quasiconformal automorhisms fcarrying 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,

QE=minQL:LEquasicircle.E4

The proof of this important theorem was given for finite sets E=z1znby Kühnau in [58], using Teichmüller’s theorem on extremal quasiconformal maps applied to the homotopy classes of homeomorphisms of the punctured spheres, and extended to arbitrary sets EhCby the author in [38].

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

QE=1+kE2/1kE2,

where kE=infz¯f/zfover all quasicircles LEand all orientation-preserving quasiconformal homeomorphisms f:ĈĈwith 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 EĈwe have QE=, unless Eis 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 [19]; 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 eigenvaluesρnof an oriented smooth closed Jordan curve Lon the Riemann sphere Ĉ=Care the eigenvalues of its double-layer potential, or equivalently, of the integral equation

uz+ρπLuζnζlog1ζzdsζ=hz,

which has has many applications (here nζis the outer normal and dsζis the length element at ζL).

The least positive eigenvalue ρL=ρ1plays a crucial role and is naturally connected with conformal and quasiconformal maps. It can be defined for any oriented closed Jordan curve Lby

1ρL=supDGuDGuDGu+DGu,

where Gand Gare, respectively, the interior and exterior of L;Ddenotes the Dirichlet integral, and the supremum is taken over all functions ucontinuous on Ĉand harmonic on GG. In particular, ρL=only for the circle.

An upper bound for ρLis given by Ahlfors’ inequality [3].

1ρLqL,E5

where qLdenotes 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 S1onto Lwhose Beltrami coefficients μfz=z¯f/zfhave 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 qLis equal to the minimum k0fof dilatations kf=μof quasiconformal extensions of the function f=fDinto 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 [41, 55, 59] and the references cited there.

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

1.3 The Grunsky and Milin inequalities

Let

D=z:z<1,D=zĈ=C:z>1.

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 Ĉin terms of an infinite system of the coefficient inequalities. In particular, his theorem for the canonical disk Dyields that a holomorphic function fz=z+const+Oz1in a neighborhood of z=can be extended to a univalent holomorphic function on the Dif and only if its Grunsky coefficients αmnsatisfy

m,n=1mnαmnxmxn1,E6

where αmnare defined by

logfzfζzζ=m,n=1αmnzmζn,zζD2,E7

the sequence x=xnruns over the unit sphere Sl2of the Hilbert space l2with norm x2=1xn2, and the principal branch of the logarithmic function is chosen (cf. [28]). The quantity

ϰf=supm,n=1mnαmnxmxn:x=xnSl21E8

is called the Grunsky norm of f.

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

m,n=1mnαmnxmxnkforanyx=xnSl2,E9

established first in [51] (see also [41, 45]). Then

ϰfkf,E10

where kfdenotes the Teichmüller norm of f, which is equal to the infimum of dilatations kwμ=μof quasiconformal extensions of fto Ĉ. Here wμdenotes a homeomorphic solution to the Beltrami equation z¯w=μzwon Cextending f.

Note that the Grunsky (matrix) operator

Gf=mnαmnfm,n=1

acts as a linear operator l2l2contracting the norms of elements xl2; the norm of this operator equals ϰf(cf. [26, 27]).

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 [41, 45, 50, 52, 53, 54]).

The method of Grunsky inequalities was generalized in several directions, even to bordered Riemann surfaces Xwith a finite number of boundary components (cf. [28, 64, 69, 77, 84]; see also [25]). In the general case, the generating function (7) must be replaced by a bilinear differential

logfzfζzζRXzζ=m,n=1βmnφmzφnζ:X×XC,E11

where the surface kernel RXzζrelates to the conformal map jθzζof Xonto the sphere Ĉslit along arcs of logarithmic spirals inclined at the angle θ0πto a ray issuing from the origin so that jθζζ=0and

jθz=zzθ1+const+O1/zzθaszzθ=jθ1

(in fact, only the maps j0and jπ/2are applied). Here φn1is a canonical system of holomorphic functions on Xsuch that (in a local parameter)

φnz=an,nzn+an+1,nzn+1+withan,n>0,n=1,2,,

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

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

logfzfζzζ=m,n=1βmnmnχzmχζn,E12

where χdenotes a conformal map of Donto the disk Dso that χ=,χ>0.

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

Sfz=fzf'z12fzfz2,zD.E13

These derivatives range over a bounded domain in the complex Banach space BDof hyperbolically bounded holomorphic functions φDwith norm

φB=supDλD2zφz,

where λDzdzdenotes the hyperbolic metric of Dof Gaussian curvature 4. This domain models the universal Teichmüller spaceTwith the base point χD(in holomorphic Bers’ embedding of T).

A theorem of Milin [69] extending the Grunsky univalence criterion for the disk Dto multiply connected domains Dstates that a holomorphic function fz=z+const+Oz1in a neighborhood of z=can be continued to a univalent function in the whole domain Dif and only if the coefficients βmnin (12) satisfy, similar to the classical case of the disk D, the inequality

m,n=1βmnxmxn1E14

for any point x=xnSl2. We call the quantity

ϰDf=supm,n=1βmnxmxn:x=xnSl2,E15

the generalized Grunsky norm of f. By (14), ϰDf1for any ffrom the class ΣDof univalent functions in Dwith hydrodynamical normalization

fz=z+b0+b1z1+nearz=.

The inequality ϰDf1is necessary and sufficient for univalence of fin D(see [28, 69, 77]).

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

ϰDfkf=tanhτT0SF,

where τTdenotes the Teichmüller distance on the universal Teichmüller space Twith 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 [46]. 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 Σand ϰf. We denote by Sthe canonical class of univalent functions Fz=z+a2z2+in the unit disk D.

The Grunsky norm of univalent functions FSis defined similar to (5), (6); so any such Fzand its inversion fz=1/F1/zunivalent in Dhave 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 Σis intrinsically connected with integrable holomorphic quadratic differentialsψzdz2on the complementary domain

D=Ĉ\D¯

(the elements of the subspace A1Dof L1Dformed by holomorphic functions), while the Grunsky norm naturally relates to the abelian structure determined by the set of quadratic differentials

A12D=ψA1D:ψ=ω2

having only zeros of even order on D.

We describe the general intrinsic features. Let Lbe a quasicircle passing through the points 0,1,, which is the common boundary of two domains Dand D. Let Lbe an oriented quasiconformal Jordan curve (quasicircle) on the Riemann sphere Ĉwith the interior and exterior domains Dand D. Denote by λDzdzthe hyperbolic metric of Dof Gaussian curvature 4and by δDz=distzDthe Euclidean distance from the point zDto the boundary. Then

14λDzδDz1,

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

Consider the unit ball of Beltrami coefficients supported on D,

BeltD1=μLC:μD=0μ<1

and take the corresponding quasiconformal automorphisms wμzof the sphere Ĉsatisfying on Cthe Beltrami equation z¯w=μzwpreserving the points 0,1,fixed. Recall that kw=μwis the dilatation of the map w.

Take the equivalence classes μand wμletting the coefficients μ1and μ2from BeltD1be equivalent if the corresponding maps wμ1and wμ2coincide on L(and hence on D¯). These classes are in one-to-one correspondence with the Schwarzians Swμon D, which fill a bounded domain in the space B2Dmodeling the universal Teichmüller space T=TDwith the base point D. The quotient map

ϕT:BeltD1T,ϕTμ=Swμ

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

τTϕTμϕTν=12inflogKwμwν1:μϕTμνϕTν,

It is the integral form of the infinitesimal Finsler metric

FTϕTμϕTμν=infν/1μ2:ϕTμν=ϕTμν

on the tangent bundle TTof T, which is locally Lipschitzian.

The Grunsky coefficients give rise to another Finsler structure Fxvon the bundle TT. It is dominated by the canonical Finsler structure FTxvand allows one to reconstruct the Grunsky norm along the geodesic Teichmüller disks in T(see [40]).

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

μ=infν:ϕTν=ϕTμ

and call μinfinitesimally extremal if

μ=infν:νLDϕT0ν=ϕT0μ.

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

μ=inf<μψ>D:ψAD:ψ=1

where ADis the space of the integrable holomorphic quadratic differentials on D(the subspace of L1Dformed by holomorphic functions on D) and the pairing

μψD=Dμzψzdxdy,μLD,ψL1Dz=x+iy.

Let w0wμ0be an extremal representative of its class w0with dilatation

kw0=μ0=infkwμ:wμL=w0L,

and assume that there exists in this class a quasiconformal map w1whose Beltrami coefficient μA1satisfies the inequality esssupArμw1z<kw0in some ring domain R=D\Gcomplement to a domain GD. Any such w1is called the frame map for the class w0, and the corresponding point in the universal Teichmüller space Tis 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[90].

(ii) The set of Strebel points is open and dense inT[63, 22].

The first assertion holds, for example, for asymptotically conformal curves L. Similar results hold also for arbitrary Riemann surfaces (cf. [18, 22]).

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

maxzLaz+zbab1asab0,

where zlies between a and b.

Such curves are quasicircles without corners and can be rather pathological (see, e.g., [78, 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 Hfadmits also a local version Hpfinvolving the Beltrami coefficients supported in the neighborhoods of a boundary point pD. Moreover (see, e.g., [22], Ch. 17), Hf=suppDHpf, and the points with Hpf=Hfare called substantial for fand for its equivalence class.

On the unique and non-unique extremality see, for example, [11, 17, 34, 68, 79, 89, 90, 97].

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 Tis the space of quasisymmetric homeomorphisms hof the unit circle S1=Dfactorized by Möbius transformations. Its topology and real geometry are determined by the Teichmüller metric, which naturally arises from extensions of these homeomorphisms hto 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 Tis defined by factorization of the ball of Beltrami coefficients

BeltD1=μLC:μD=0μ<1,

letting μ,νBeltD1be equivalent if the corresponding maps wμ,wνΣ0coincide on S1(hence, on D¯) and passing to Schwarzian derivatives Sfμ. The defining projection ϕT:μSwμis a holomorphic map from LDto B. The equivalence class of a map wμwill be denoted by wμ.

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

The space Tas 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 metricdTon Tis the largest pseudometric don Tdoes not get increased by holomorphic maps h:DTso that for any two points ψ1,ψ2T, we have

dTψ1ψ2infdD0t:h0=ψ1ht=ψ2,

where dDis the hyperbolic Poincaré metric on Dof Gaussian curvature 4, with the differential form

ds=λhypzdzdz/1z2.

The Carathéodory distance between ψ1and ψ2in Tis

cTψ1ψ2=supdDhψ1hψ2,

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 ψvof the tangent bundle TT, respectively, by

KTψv=inf1/r:r>0hHolDrTh0=ψdh0=v,CTψv=supdfψv:fHolTDfψ=0,

where HolXYdenotes the collection of holomorphic maps of a complex manifold Xinto Yand Dris 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 [17, 19, 22, 82]).

For the the universal Teichmüller space T, we have the following strengthened version of this theorem for universal Teichmüller space given in [36].

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

gTψ1ψ2=logtanhτTψ1ψ2=logkψ1ψ2,

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 curvatureκλof an upper semicontinuous Finsler metric ds=λtdtin a domain ΩCis defined by

κλt=Dlogλtλt2,

where Dis the generalized Laplacian

Dλt=4liminfr01r212π02πλt+reλt

(provided that λt<). Similar to C2functions, for which Dcoincides with the usual Laplacian, one obtains that λis subharmonic on Ωif and only if Dλt0; hence, at the points t0of local maximuma of λwith λt0>, we have Dλt00.

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

The holomorphic curvature of the Kobayashi metric Kxvof any complete hyperbolic manifold Xsatisfies κKX4at all points xvof the tangent bundle TXof X, and for the Carathéodory metric CXwe have κCxv4.

Finally, the pluricomplex Green function of a domain Xon a complex Banach space manifold Eis defined as gXxy=supuyxxyX, where supremum is taken over all plurisubharmonic functions uyx:X0satisfying uyx=logxy+O1in a neighborhood of the pole y. Here is the norm on Xand the remainder term O1is bounded from above. If Xis hyperbolic and its Kobayashi metric dXis logarithmically plurisubharmonic, then gXxy=logtanhdXxy, which yields the representation of gTin Theorem 3.

For details and general properties of invariant metrics, we refer to [14, 30] (see also [1, 41]).

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 uxon a domain Din a Banach space Xmeans that uxis upper continuous in Dand its restriction to the intersection of Dwith any complex line Lis subharmonic.

A deep Zhuravlev’s theorem implies that the intersection of the universal Teichmüller space Twith every complex line is a union of simply connected planar (moreover, this holds for any Teichmüller space); see ([49], pp. 75–82, [102]).

1.6 The Grunsky-Milin inequalities revised

Denote by Σ0Dthe subclass of ΣDformed by univalent Ĉ-holomorphic functions in Dwith expansions fz=z+b0+b1z1+near z=admitting quasiconformal extensions to Ĉ. It is dense in ΣDin the weak topology of locally uniform convergence on D.

Each Grunsky coefficient αmnfis a polynomial of a finite number of the initial coefficients b1,b2,,bm+n1of f; hence it depends holomorphically on Beltrami coefficients of extensions of fas well as on the Schwarzian derivatives SfB2D.

Consider the set

A12D=ψA1D:ψ=ω2

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

αDf=supμ0ψD:ψA12ψA1D=1.

The following theorem from [46] completely describes the relation between the Grunsky and Teichmüller norms (more special results were obtained in [33, 52]).

Theorem 4. For allfΣ0D,

ϰDfkk+αDf1+αDfk,k=kf,

and ϰDf<kunless

αDf=μ0,E16

where μ0is an extremal Beltrami coefficient in the equivalence class f. The last equality is equivalent to ϰDf=kf.

If ϰf=kfand the equivalence class of f(the collection of maps equal to fon S1=D) is a Strebel point, then the extremal μ0in this class is necessarily of the form

μ0=μ0ψ0/ψ0withψ0A12D.E17

Note that geometrically (16) means the equality of the Carathéodory and Teichmüller distances on the geodesic disk ϕTtμ0/μ0:tDin the universal Teichmüller space Tand that the mentioned above the strict inequality ϰf<kfis valid on the (open) dense subset of Σ0in 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=αmnSFis that these coefficients are holomorphic functions of the Schwarzians φ=Sfon the universal Teichmüller space T. Therefore, for every fΣ0and each x=xnSl2, the series

hxφ=m,n=1mnαmnφxmxnE18

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

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

m=jMn=lNmnαmnxmxn2m=jMxm2n=lNxn2

(for any finite M,N), which, in turn, are a consequence of the classical area theorem (see, e.g., [77], p. 61; [69], p. 193).

Using Parseval’s equality, one obtains that the elements of the distinguished set A12Dare represented in the form

ψz=1πm+n=2mnxmxnzm+n2E19

with x=xnl2so that xl2=ψA1(see [33]). This result extends to arbitrary domains D with quasiconformal boundaries but the proof is much more complicated (see [45]).

1.7 The first Fredholm eigenvalue and Grunsky norm

One of the basic tools in quantitative estimating the Freholm eigenvalues ρLof 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. [53, 83]).

Another important tool is the following Kühnau’s jump inequality [55]:

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

1ρL1α.E20

This implies the lower estimate for qLand 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=k0fhold for a given curve L(cf. [33, 54, 55, 56, 101]).

This was an open problem a long time even for the rectangles stated by R. Kühnau, after it was established only [55, 101] that the answer is in affirmative for the square and for close rectangles Rwhose moduli mRvary 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 [40].

The relation between the basic curvelinear functionals intrinsically connected with conformal and quasiconformal maps is described in Kühau’s paper [62].

1.8 Holomorphic motions

Let Ebe a subset of Ĉcontaining at least three points.

A holomorphic motion of Eis a function f:E×DĈsuch that:

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

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

  3. fz0=zfor all zE.

The remarkable lambda-lemma of Mañé, Sad, and Sullivan [67] yields that such holomorphic dependence on the time parameter provides quasiconformality of fin 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 zEor of the cross-ratios of the ordered quadruples of points of E.

The Slodkowski lifting theorem ([87], see also [8, 13, 15]) 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˜ztare holomorphic in tvia elements of LC, and Schwarz’s lemma yields

μf˜tt,

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., [8, 13, 17, 72, 73, 67, 42, 74, 86, 92, 93], and the references there).

There is an intrinsic connection between holomorphic motions and Teichmüller spaces, first mentioned by Bers and Royden in [10]. McMullen and Sullivan introduced in [74] the Teichmüller spaces for arbitrary holomorphic dynamical systems, and this approach is now one of the basic in complex dynamics.

Topics discussed in this section were studied in classic works [2, 20, 21, 31, 65, 71, 75, 76, 80, 81, 85, 91, 94, 95, 96] 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 ρLand allowed to establish it only for some special collections of curves and arcs.

In this section, we present, following [40, 47], 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 Tand the Finsler metrics on holomorphic disks in Tas 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

qL=1/ρL=ϰf=ϰf=k0f=k0f=1α,E21

where fand fdenote the appropriately normalized conformal maps DDand DD=Ĉ\D¯, respectively, k0fand k0fare the minimal dilatations of their quasiconformal extensions to Ĉ;ϰfand ϰfstand for their Grunsky norms, and παis the opening of the least interior angle between the boundary arcs LjL. Here 0<α<1if the corresponding vertex is finite and 1<α<0for 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 [40, 42]. 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 [52, 55, 59, 101]), 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. Let Lbe a closed unbounded curve with the convex interior, which is C1+δsmooth 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

    qL=1/ρL=1α.E22

  • More generally, assume that Lalso has a finite angle point z0with the angle opening πα0. Then, similar to (22),

    qL=1/ρL=max1α01α.

  • 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 [42].

    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 Pnbe a rectilinear polygon with the finite vertices A1,A2,,An1and with vertex A=, and let the interior angle at the vertex Ajbe equal to παjand at Abe equal to πα, where α<0and all aj1, so that α1++αn1+α=2. Let fnbe the conformal map of the upper half-plane U=z:Imz>0onto Pn, which without loss of generality, can be normalized by fnz=zi+Ozias zi(assuming that Pncontains the origin w=0).

    An important geometric characteristic of polynomials is the quantity

    1α=max1α11αn11α;E23

    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

    ϰfn=kfn>1a,E24

    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 f4of any rectangle has equal Grunsky and Teichmüller norms,

    ϰf4=kf4

    (see [39, 55, 101]).

    Using the Moebius map σ:z1/z, we transform the rectangle into a (nonconvex) circular quadrilateral σP4with angles π/2and 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

    kf̂4=ϰf̂4>1/2,E25

    where f̂4denotes the conformal map DσP4.

    Indeed, as was established by Kühnau [55], the quadrilaterals with the side ratios (conformal module) greater than 3.31have 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 σP4satisfying (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 kfand qPare lower continuous functionals under locally uniform convergence of quasiconformal maps (i.e., kfliminfkfnas fnfin 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. [47] LetPnbe a unbounded rectilinear polygon, neither convex nor concave, and hence contain the verticesAjwhose inner anglesπαjhave openingsπαjwith1<αj<2. Assume that all suchαjsatisfy

    αj1<1α,E26

    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

    ϰfnσ=kfn=qPn=1/ρPn=1α.E27

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

    fnζ=d10zξa1α11ξan1αn11+d0,E28

    where all aj=f1AjRand d0,d1are the corresponding complex constants. The logarithmic derivative bf=logf=f/fof this map has the form

    bfnz=1n1αj1zaj1.

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

    wzt:U×DαĈ,E29

    (containing fnas a fiber map), which is injective in the space coordinate zfor any fixed t, holomorphic in tfor a fixed zand wz0z.

    First observe that for real rIαthe solution Wrto the equation wz=rbf4zwzwith the initial conditions wri=i,wr=satisfies

    bWrz=1n1rαj1zaj=1nαjr1zaj,

    where

    αjr=rαj1+1.E30

    If the interior angles of the initial polygon Pnsatisfy the assumption (26), then all the functions Wrare represented by an integral of type (27) (replacing αjby αjr, and with suitable constants d0r,d1r).

    Geometrically this means that the exterior angle 2ππαjrat any finite vertex Ajrdecreases with r(but the value αjr1increases if 1<αj<2). Under the assumption (26), the admissible bounds for the possible values of angles ensure the univalence of this integral on Ufor every indicated t. This yields that every WrUalso is a polygon with the interior angles παjrfor r0, while W0U=U.

    Now we pass to the conformal map gnζ=fnσ0ζof the unit disk Donto Pn, using the function σ0ζ=1+ζ/1ζ. This map is represented similar to (28) by

    gnζ=d10ζ1nξejαj1+d0,

    where the points ejare the preimages of vertices ej=gn1Ajon the unit circle ζ=1. Pick d1to have gn0=1. For this function, we have a natural complex isotopy

    w˜tζ=1tgn:D×DC,E31

    with

    bw˜ζ=w˜tζw˜tζ=tggn=tbgn.E32

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

    w˜tζ=eWrσ0eζ.

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

    The corresponding Schwarzians Sw˜rζ=rbw˜rζr2bw˜rζ2/2fill a real analytic line Γin the universal Teichmüller space T(modeled as a bounded domain in the complex Banach space Bof hyperbolically bounded holomorphic functions on D). This line is located in the holomorphic disk Ω˜=bGT, where bdenotes the map tSw˜tand GIαis a simply connected planar domain.

    By Zhuravlev’s theorem (see [102, 49]), this domain contains for each rIαalso the points Sw˜twith tr(representing the curvelinear polygons with piecewise analytic boundaries).

    This generates the holomorphic motions (complex isotopies) w˜ζt:D×GĈand wzt:UĈwith wz1=fnz.

    The basic lambda-lemma for holomorphic motions implies that every fiber map wtzis the restriction to Uof a quasiconformal automorphism Ŵtzof the whole sphere Ĉ, and the Beltrami coefficients

    μzt=z¯Ŵtz/zŴtz,tDα,

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

    So we have a holomorphic map μtfrom the disk Dαinto the unit ball of Beltrami coefficients supported on U,

    BeltU1=μLC:μzU=0μ<1,

    and the classical Schwarz lemma implies the estimate

    kŴt=μŴt1αt.

    It follows that the extremal dilatation of the initial map fnz=Ŵ1zUsatisfies

    kfn1α.

    Hence, also qPn1αand by the inequality (10), ϰfn1α.

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

    1ρPn1α.

    Together with (5), this yields that the polygon Pnadmits 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 Lbe 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

    qL=1/ρL=1a,

    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 [9]:

    For an arbitrary finitely or infinitely generated Fuchsian group Γis the Bers embedding of its Teichmüller space TΓstarlike?

    Recall that in this embedding TΓis represented as a bounded domain formed by the Schwarzian derivatives Swof holomorphic univalent functions wzin the lower half-plane U=z:Imz<0(or in the disk) admitting quasiconformal extensions to the Riemann sphere Ĉ=Ccompatible with the group Γacting on U.

    It was shown in [35] that universal Teichmüller space T=T1has 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 [99] (see also [7]).

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

    φB=4supUy2φz

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

    Toki [100] extended the result on the nonstarlikeness of the space Tto 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 [100] is the same as in [35].

    On the other hand, it was established in [36] that also all finite dimensional Teichmüller spaces TΓof 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 Pnwith finite vertices A1,,An1and An=. Denote the exterior angles at Ajby παjso that π<αj<2π,j=1,,n1. Then, similar to (22), the conformal map fnof the lower half-plane H=z:Imz<0onto the complementary polygon Pn=Ĉ\Pn¯is represented by the Schwarz-Christoffel integral

    fnz=d10zξa1α11ξa2α21ξan1αn11+d0,

    with aj=fn1AjRand complex constants d0,d1; here fn1=. Its Schwarzian derivative is given by

    Sfnz=bfnz12bfn2z=1n1Cjzaj2j,l=1n1Cjlzajzal,E33

    where bf=f/fand

    Cj=αj1αj12/2<0,Cjl=αj1αl1>0.

    It defines a point of the universal Teichmüller space Tmodeled as a bounded domain in the space BHof hyperbolically bounded holomorphic functions on Hwith norm φBH=supHzz¯2φz.

    Denote by r0the positive root of the equation

    121n1αj12+j,l=1n1αj1αl1r21n1αj1r2=0,

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

    Theorem 8. [44] 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

    kwr=ϰwr=r2Sfn,r0BH.E34

    By the Ahlfors-Weill theorem [6], every φBHwith φBH<1/2is the Schwarzian derivative SWof a univalent function Win H, and this function has quasiconformal extension onto the upper half-plane H=z:Imz>0with Beltrami coefficient of the form

    μφz=2y2φz¯,φ=Sfz=x+iyH

    called harmonic. Theorem 7 yields that any wrwith r<r0does not admit extremal quasiconformal extensions of Teichmüller type, and in view of extremality of harmonic coefficients μSwrthe Schwarzians Swrfor some rbetween r0and 1must lie outside of the space T; so this space is not a starlike domain in BH.

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

    kfn=ϰfn=12SfnBHE35

    for all r1, completing the bounds (21).

    We illustrate this on the case of triangles. Let P3be a triangle with vertices A1,A2Rand A3=and exterior angles α1,α2,α3. The logarithmic derivative of conformal map f3:HP3has the form

    bf3z=α11za1+α21za2

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

    Sf3z=C1za12+C2za22C12za1za2

    with

    Cj=αj112αj12=αj2+12<0,j=1,2;C12=α11α21>0.

    If the angles of P3satisfy α1,α2<a3, where πα3is the angle at A3, the arguments from [44] yield that the harmonic Beltrami coefficient μSf3satisfies (35).

    Surprisingly, this construction is closely connected also with the weighted bounded rational approximation in sup norm [48, 103].

    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 Lwhose 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

    gcw=c1w+c2w¯+c3

    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 Da constant quasiconformal dilatation c, regarding such maps as the affine deformations and the collection of domains gcDas the affine class of D.

    If fis a quasiconformal automorphism of Ĉconformal in Dmapping the disk Donto a domain D, then for a fixed cthe maps gcDfand gcfDdiffer by a conformal map h:DgcDand hence have in the disk Dthe same Beltrami coefficient.

    Note that the inequality c<1equivalent to c2<c1follows 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 cby 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

    ϰgcf=kgcf.E36

    Theorems 9 essentially increases the set of quasicircles LĈfor which ρL=1/qLgiving 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 [40, 53, 54, 55] and for rectilinear or circular quadrilaterals having a common tangent circle [101]).

    5.2 Scheme of the proof of Theorem 9

    The proof follows the lines of Theorem 1.1 in [43] 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

    ftz=tfz/t=z+b0t+b1t2z1+b2t3z2+:D×DĈ.

    Then Sftz=t2Sft1zand this point-wise map determines a holomorphic map χft=Sft:DTso that the homotopy disks DSf=χfDfoliate the space T. Note also that

    αmnft=αmnftm+n,

    and if Fz=1/f1/zmaps the unit disk onto a convex domain, then all level lines fz=rfor zDare 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 f1of fto a maximal disk Db=zĈ:z>b,0<b<1and define

    fz=b1f1bzΣ0,z>1.

    Its Beltrami coefficient in Dis defined by holomorphic quadratic differentials ψA12of the form (19), and we have the holomorphic map, for a fixed xb=xnbl2,

    hxbSft=m,n=1mnαmnfxmbxnbbtm+nE37

    of the disk DSfinto 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/1t2to the disk DSF1(parametrized by t) and define on this disk the conformal metric ds=λh˜xtdtwith

    λh˜xbt=hxaχf1λD=h˜xbtdt1h˜xbt2.E38

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

    λϰtλϰt=limsupttλϰt

    (supporting means that λh˜xba=λϰaand λh˜xbt<λϰtin a neighborhood of a).

    The metric λϰtis logarithmically subharmonic on Dand its generalized Laplacian

    Δut=4liminfr01r212π02πut+reλt

    satisfies

    Δlogλϰ4λϰ2

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

    As was mentioned above, the Grunsky coefficients define on the tangent bundle TTa new Finsler structure Fϰφvdominated by the infinitesimal Teichmüller metric Fφv. This structure generates on any embedded holomorphic disk γDTthe corresponding Finsler metric λγt=Fϰγtγtand reconstructs the Grunsky norm by integration along the Teichmüller disks:

    Lemma 2. [43] On any extremal Teichmüller diskDμ0=ϕTtμ0:tD(and its isometric images inT), we have the equality

    tanh1ϰfrμ0=0rλϰtdt.

    Taking into account that the disk DSftouches at the point φ=Sfathe Teichmüller disk centered at the origin of Tand passing through this point and that the metric λϰdoes not depend on the tangent unit vectors whose initial points are the points of DSf, one obtains from Lemma 3 and the equality ϰfa=kfathat also

    λϰa=λKa.E39

    The following lemma is a needed reformulation of Theorem 3.

    Lemma 3. [43] 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˜xbwith λKusing Lemmas 2, 3, and Minda’s maximum principle given by.

    Lemma 4. [70] If a functionu:D+is upper semicontinuous in a domainDCand its (generalized) Laplacian satisfies the inequalityΔuzKuzwith some positive constantKat any pointzD, whereuz>, and if

    limsupzζuz0forallζD,

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

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

    ϰfr=kfr=m,n=1mnαmnF1rm+nxmrxnr

    for all r=t01(with xnrSl2depending on r) and that for any fΣ0with 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 fΣ0with equal Grunsky and Teichmüller norms.

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

    b1c=Sgc6=16limzw4Sgcw0,

    and for sufficiently small call composite maps

    Wf,cz=gcfz=z+b̂0c+b̂1cz1+,fΣ0,

    also satisfyb̂1c0.

    Finally, we use the following important result of Kühnau [53].

    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=λKon the disk DSWf,cand

    ϰWF,c=kWF,c.E40

    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,νfrom the unit ball in LC,

    wμwν=wτwithτ=ν+μ˜/1+ν¯μ˜

    and μ˜z=μwνzwzν¯/wzν(so for νfixed, τdepends holomorphically on μin Lnorm) and defining the corresponding functions (37), one gets now the holomorphic functions of cD. Then, constructing in a similar way the corresponding Finsler metrics

    λh˜xc=h˜xcdc/1h˜xc2,c<1.

    and taking their upper envelope λϰcand its upper semicontinuous regularization, one obtains a subharmonic metric of Gaussian curvature κλϰ4on 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

    Sg=hcHH2+SH,

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

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

    gcw=w+b0c+b1cw1+asw

    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

    ρPn<1/qPn.

    It follows simply from Theorem 8 that if a polygon Pn, whose edges are quasiconformal arcs, satisfies ρPn=1/qPnthen 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/qPnis 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/qP4holds in particular for quadrilaterals P4obtained 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

    ϰf=kf=ρP4=1/qP4,E41

    where F is the appropriately normalized conformal map ofDontoP4.

    The proof of this theorem essentially relies on Theorem 8 and on result of [40] that the equalities (41) are valid for all rectangles, and hence for their affine transformations.

    Fix such a quadrilateral P40=A10A20A30A40and consider the collection P0of quadrilaterals P4=A10A20A30A4with the same first three vertices and variable A4; the corresponding A4runs over a subset Eof the trice punctured sphere Ĉ\A10A20A30.

    The collection P0contains 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 Ta holomorphic disk Ωextending the real analytic curve filled by the Schwarzians, which correspond to the values t=Aon 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 Ωwith the dominant infinitesimal Teichmüller-Kobayashi metric λKof 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 Rstate that if the module μRsatisfies 1μR<1.037, then

    qR=1/ρR=1/2;

    if μR>2.76, then q∂ℛ>1/2(see [55, 101]).

    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 [37, 38]. 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 KCnwith the speed of its polynomial approximation. Another approach was applied by Kühnau in [56, 57, 60, 61]. In particular, using somewhat modification of Teichmüller’s Verschiebungssatz [98], he established in [61] the reflection coefficient of the set E, which consists of the interval 2i2iand a separate point t>0. All these results are presented in [42].

    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

    L=L1L2LN

    of smooth curvelinear quasiintervals (possibly mutually separated) such that Lcan be extended without adding new vertices (angular points) to a quasicircle L0Lcontaining z=and bounding a convex polygon PNthat satisfies the assumptions of Theorem 4 or a polygon considered in Theorem 7, then by these theorems, the reflection coefficient of the setLequals

    qL=1a,E42

    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 Ljcan be smoothly extended and then rounded off.

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

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    Samuel L. Krushkal (May 20th 2020). Quasiconformal Reflections across Polygonal Lines [Online First], IntechOpen, DOI: 10.5772/intechopen.92441. Available from:

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