An important open problem in geometric complex analysis is to establish algorithms for explicit determination of the basic curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals.
- Grunsky inequalities
- univalent function
- Beltrami coefficient
- quasiconformal reflection
- universal Teichmüller space
- Fredholm eigenvalues
- convex polygon
- 2010 Mathematics Subject Classification: Primary: 30C55
- 30F60; Secondary: 31A35
1. Quasiconformal reflections: general theory
1.1 Quasireflections and quasicurves
The classical Brouwer-Kerekjarto theorem ([12, 29], see also ) says that every periodic homeomorphism of the sphere S2is 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 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 with and are called topological reflections.
We shall consider here quasiconformal reflections or quasireflections on the Riemann sphere , that is, the orientation reversing quasiconformal automorphisms of order 2 (involutions) of the sphere with . 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 . Any quasireflection preserves pointwise fixed a quasicircle interchanging its inner and outer domains.
Under quasiconformal mapof a domain , we understand an orientation-preserving generalized solution of the Beltrami equation (uniformly elliptic system of the first order)
are the distributional partial derivatives, is a given function from with , called the Beltrami coefficient (or complex dilatation) of the map , and the quantity is the (quasiconformal) dilatation of this map. There are some equivalent analytic and geometric definitions of such maps.
Qualitatively, any quasicircle is characterized, due to , by uniform boundedness of the cross-ratios for all ordered quadruples of the distinct points on ; namely,
for any quadruple of points on following this order. Using a fractional linear transformation, one can send one of the points, for example, , to infinity; then the above inequality assumes the form
This is shown in  by applying the properties of quasisymmetric maps. Ahlfors has established also that if a topological circle admits quasireflections (i.e., is a quasicircle), then there exists a differentiable quasireflection across which is (euclidian) bilipschitz-continuous. This property is very useful in various applications. On its extension to hyperbolic -bilipschitz reflections see .
Geometrically, a quasicircle is characterized by the property that, for any two points on , the ratio of the chordal distance to the diameters of the corresponding subarcs with these endpoints is uniformly bounded. Note also that every quasicircle has zero two-dimensional Lebesgue measure.
Quasireflections across more general sets also appear in certain questions and are of independent interest. Those sets admitting quasireflections are called quasiconformal mirrors.
One defines for each mirror its reflection coefficient
and quasiconformal dilatation
the infimum in (1) is taken over all quasireflections across , provided those exist, and is attained by some quasireflection .
When is a quasicircle, the corresponding quantity
The infimum in (2) is taken over all orientation-preserving quasiconformal automorhisms carrying the unit circle onto , and .
Theorem 1.For any setwhich admits quasireflections, there is a quasicirclewith the same reflection coefficient; therefore,
The proof of this important theorem was given for finite sets by Kühnau in , using Teichmüller’s theorem on extremal quasiconformal maps applied to the homotopy classes of homeomorphisms of the punctured spheres, and extended to arbitrary sets by the author in .
Theorem 1 yields, in particular, that similar to (3) for any set , its quasiconformal dilatation satsfies
where over all quasicircles and all orientation-preserving quasiconformal homeomorphisms with .
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 within conjugation by transformations . 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 we have , unless 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 ; it relies on the conformally natural extension of homeomorphisms of the circle introduced by Douady and Earle .
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 eigenvaluesof an oriented smooth closed Jordan curve on the Riemann sphere are the eigenvalues of its double-layer potential, or equivalently, of the integral equation
which has has many applications (here is the outer normal and is the length element at ).
The least positive eigenvalue plays a crucial role and is naturally connected with conformal and quasiconformal maps. It can be defined for any oriented closed Jordan curve by
where and are, respectively, the interior and exterior of denotes the Dirichlet integral, and the supremum is taken over all functions continuous on and harmonic on . In particular, only for the circle.
An upper bound for is given by Ahlfors’ inequality .
where denotes the minimal dilatation of quasireflections across .
In view of the invariance of all quantities in (5) under the action of the Möbius group , it suffices to consider the quasiconformal homeomorphisms of the sphere carrying onto whose Beltrami coefficients have support in the unit disk , and , where (or in the upper half-plane ). Then is equal to the minimum of dilatations of quasiconformal extensions of the function into .
The inequality (5) serves as a background for defining the value , 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
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 function in a neighborhood of can be extended to a univalent holomorphic function on the if and only if its Grunsky coefficients satisfy
where are defined by
the sequence runs over the unit sphere of the Hilbert space with norm , and the principal branch of the logarithmic function is chosen (cf. ). The quantity
is called the Grunsky norm of f.
For the functions with k-quasiconformal extensions (), we have instead of (8) a stronger bound
where denotes the Teichmüller norm of , which is equal to the infimum of dilatations of quasiconformal extensions of to . Here denotes a homeomorphic solution to the Beltrami equation on extending .
Note that the Grunsky (matrix) operator
For most functions , we have in (10) the strong inequality (moreover, the functions satisfying this inequality form a dense subset of ), 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 with a finite number of boundary components (cf. [28, 64, 69, 77, 84]; see also ). In the general case, the generating function (7) must be replaced by a bilinear differential
where the surface kernel relates to the conformal map of onto the sphere slit along arcs of logarithmic spirals inclined at the angle to a ray issuing from the origin so that and
(in fact, only the maps and are applied). Here is a canonical system of holomorphic functions on such that (in a local parameter)
and the derivatives (linear holomorphic differentials) form a complete orthonormal system in .
We shall deal only with simply connected domains with quasiconformal boundaries (quasidisks). For any such domain, the kernel vanishes identically on , and the expansion (11) assumes the form
where denotes a conformal map of onto the disk so that .
Each coefficient in (12) is represented as a polynomial of a finite number of the initial coefficients of ; hence it depends holomorphically on Beltrami coefficients of quasiconformal extensions of as well as on the Schwarzian derivatives
These derivatives range over a bounded domain in the complex Banach space of hyperbolically bounded holomorphic functions with norm
where denotes the hyperbolic metric of of Gaussian curvature . This domain models the universal Teichmüller spacewith the base point (in holomorphic Bers’ embedding of ).
A theorem of Milin  extending the Grunsky univalence criterion for the disk to multiply connected domains states that a holomorphic function in a neighborhood of can be continued to a univalent function in the whole domain if and only if the coefficients in (12) satisfy, similar to the classical case of the disk , the inequality
for any point . We call the quantity
the generalized Grunsky norm of . By (14), for any from the class of univalent functions in with hydrodynamical normalization
The norm (15) also is dominated by the Teichmüller norm of this map. Similar to (10),
where denotes the Teichmüller distance on the universal Teichmüller space with the base point , 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 . 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 , ; for this disk, we shall use the notations and . We denote by the canonical class of univalent functions in the unit disk .
The Grunsky norm of univalent functions is defined similar to (5), (6); so any such and its inversion univalent in have the same Grunsky coefficients . Technically it is more convenient to deal with functions univalent in .
1.4 Extremal quasiconformality
A crucial point here is that the Teichmüller norm on is intrinsically connected with integrable holomorphic quadratic differentialson the complementary domain
(the elements of the subspace of 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 be a quasicircle passing through the points , which is the common boundary of two domains and . Let be an oriented quasiconformal Jordan curve (quasicircle) on the Riemann sphere with the interior and exterior domains and . Denote by the hyperbolic metric of of Gaussian curvature and by the Euclidean distance from the point to the boundary. Then
where the right-hand inequality follows from the Schwarz lemma and the left from Koebe’s theorem.
Consider the unit ball of Beltrami coefficients supported on ,
and take the corresponding quasiconformal automorphisms of the sphere satisfying on the Beltrami equation preserving the points fixed. Recall that is the dilatation of the map .
Take the equivalence classes and letting the coefficients and from be equivalent if the corresponding maps and coincide on (and hence on ). These classes are in one-to-one correspondence with the Schwarzians on , which fill a bounded domain in the space modeling the universal Teichmüller space with the base point . The quotient map
is holomorphic (as the map from to ). Its intrinsic Teichmüller metric is defined by
It is the integral form of the infinitesimal Finsler metric
on the tangent bundle of , which is locally Lipschitzian.
The Grunsky coefficients give rise to another Finsler structure on the bundle . It is dominated by the canonical Finsler structure and allows one to reconstruct the Grunsky norm along the geodesic Teichmüller disks in (see ).
We call the Beltrami coefficient extremal (in its class) if
and call infinitesimally extremal if
Any infinitesimally extremal Beltrami coefficient is globally extremal (and vice versa), and by the basic Hamilton-Krushkal-Reich-Strebel theorem the extremality of is equivalent to the equality
where is the space of the integrable holomorphic quadratic differentials on (the subspace of formed by holomorphic functions on ) and the pairing
Let be an extremal representative of its class with dilatation
and assume that there exists in this class a quasiconformal map whose Beltrami coefficient satisfies the inequality in some ring domain complement to a domain . Any such is called the frame map for the class , and the corresponding point in the universal Teichmüller space is called the Strebel point.
These points have the following important properties.
Theorem 2. (i) If a classhas a frame map, then the extremal mapin this class (minimizing the dilatation) is unique and either a conformal or a Teichmüller map with Beltrami coefficienton, defined by an integrable holomorphic quadratic differentialonand a constant.
Recall that a Jordan curve is called asymptotically conformal if for any pair of points ,
where lies between a and b.
Such curves are quasicircles without corners and can be rather pathological (see, e.g., [78, p. 249]). In particular, all -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 admits also a local version involving the Beltrami coefficients supported in the neighborhoods of a boundary point . Moreover (see, e.g., , Ch. 17), , and the points with are called substantial for and for its equivalence class.
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 is the space of quasisymmetric homeomorphisms of the unit circle factorized by Möbius transformations. Its topology and real geometry are determined by the Teichmüller metric, which naturally arises from extensions of these homeomorphisms 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 is defined by factorization of the ball of Beltrami coefficients
letting be equivalent if the corresponding maps coincide on (hence, on ) and passing to Schwarzian derivatives . The defining projection is a holomorphic map from to . The equivalence class of a map will be denoted by .
An intrinsic complete metric on the space is the Teichmüller metric, defined above in Section 1.4, with its infinitesimal Finsler form (structure) .
The space 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 metricon is the largest pseudometric on does not get increased by holomorphic maps so that for any two points , we have
where is the hyperbolic Poincaré metric on of Gaussian curvature , with the differential form
The Carathéodory distance between and in is
where the supremum is taken over all holomorphic maps .
The corresponding differential (infinitesimal) forms of the Kobayashi and Carathéodory metrics are defined for the points of the tangent bundle , respectively, by
where denotes the collection of holomorphic maps of a complex manifold into and is the disk .
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 , we have the following strengthened version of this theorem for universal Teichmüller space given in .
Theorem 3. The Teichmüller metricof either of the spacesoris plurisubharmonic separately in each of its arguments; hence, the pluricomplex Green function ofequals
whereis the norm of extremal Beltrami coefficient defining the distance between the pointsin(and similar for the space).
The differential (infinitesimal) Kobayashi metricon the tangent bundleofis logarithmically plurisubharmonic in, equals the infinitesimal Finsler formof metricand has constant holomorphic sectional curvatureon the tangent bundle.
In other words, the Teichmüller-Kobayashi metric is the largest invariant plurisubharmonic metric on T.
The generalized Gaussian curvatureof an upper semicontinuous Finsler metric in a domain is defined by
where is the generalized Laplacian
(provided that ). Similar to functions, for which coincides with the usual Laplacian, one obtains that is subharmonic on if and only if ; hence, at the points of local maximuma of with , we have .
The sectional holomorphic curvature of a Finsler metric on a complex Banach manifold is defined in a similar way as the supremum of the curvatures over appropriate collections of holomorphic maps from the disk into for a given tangent direction in the image.
The holomorphic curvature of the Kobayashi metric of any complete hyperbolic manifold satisfies at all points of the tangent bundle of , and for the Carathéodory metric we have .
Finally, the pluricomplex Green function of a domain on a complex Banach space manifold is defined as , where supremum is taken over all plurisubharmonic functions satisfying in a neighborhood of the pole . Here is the norm on and the remainder term is bounded from above. If is hyperbolic and its Kobayashi metric is logarithmically plurisubharmonic, then , which yields the representation of in Theorem 3.
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 on a domain in a Banach space means that is upper continuous in and its restriction to the intersection of with any complex line is subharmonic.
A deep Zhuravlev’s theorem implies that the intersection of the universal Teichmüller space with every complex line is a union of simply connected planar (moreover, this holds for any Teichmüller space); see (, pp. 75–82, ).
1.6 The Grunsky-Milin inequalities revised
Denote by the subclass of formed by univalent -holomorphic functions in with expansions near admitting quasiconformal extensions to . It is dense in in the weak topology of locally uniform convergence on .
Each Grunsky coefficient is a polynomial of a finite number of the initial coefficients of ; hence it depends holomorphically on Beltrami coefficients of extensions of as well as on the Schwarzian derivatives .
Consider the set
consisting of the integrable holomorphic quadratic differentials on D having only zeros of even order and put
Theorem 4. For all,
where is an extremal Beltrami coefficient in the equivalence class . The last equality is equivalent to .
If and the equivalence class of (the collection of maps equal to on ) is a Strebel point, then the extremal in this class is necessarily of the form
Note that geometrically (16) means the equality of the Carathéodory and Teichmüller distances on the geodesic disk in the universal Teichmüller space and that the mentioned above the strict inequality is valid on the (open) dense subset of in both strong and weak topologies (i.e., in the Teichmüller distance and in locally uniform convergence on ).
An important property of the Grunsky coefficients is that these coefficients are holomorphic functions of the Schwarzians on the universal Teichmüller space . Therefore, for every and each , the series
defines a holomorphic map of the space into the unit disk , and .
The convergence and holomorphy of the series (18) simply follow from the inequalities
Using Parseval’s equality, one obtains that the elements of the distinguished set are represented in the form
1.7 The first Fredholm eigenvalue and Grunsky norm
One of the basic tools in quantitative estimating the Freholm eigenvalues of quasicircles is given by the classical Kühnau-Schiffer theorem mentioned above. This theorem states that the valueis reciprocal to the Grunsky normof the Riemann mapping function of the exterior domain of(see. [53, 83]).
Another important tool is the following Kühnau’s jump inequality :
If a closed curve contains two analytic arcs with the interior intersection angle , then
This implies the lower estimate for and . By approximation, this inequality extends to smooth arcs.
One of the standard ways of establishing the reflection coefficients (respectively, the Fredholm eigenvalues ) consists of verifying wether the equality in (5) or the equality hold for a given curve (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 whose moduli vary in the interval ; moreover, in this case . The method exploited relied on an explicit construction of an extremal reflection. The complete answer was given in .
The relation between the basic curvelinear functionals intrinsically connected with conformal and quasiconformal maps is described in Kühau’s paper .
1.8 Holomorphic motions
Let be a subset of containing at least three points.
A holomorphic motion of is a function such that:
for every fixed the function is holomorphic in ;
for every fixed , the map is injective;
for all .
The remarkable lambda-lemma of Mañé, Sad, and Sullivan  yields that such holomorphic dependence on the time parameter provides quasiconformality of in the space parameter . Moreover: (i)extends to a holomorphic motion of the closureof;
(ii) eachis quasiconformal; (iii)is jointly continuous in.
Quasiconformality here means, in the general case, the boundedness of the distortion of the circles centered at the points or of the cross-ratios of the ordered quadruples of points of .
Extended lambda-lemma: Any holomorphic motioncan be extended to a holomorphic motion, with.
The corresponding Beltrami differentials are holomorphic in via elements of , and Schwarz’s lemma yields
or equivalently, the maximal dilatations . 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 . McMullen and Sullivan introduced in  the Teichmüller spaces for arbitrary holomorphic dynamical systems, and this approach is now one of the basic in complex dynamics.
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 and 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 and the Finsler metrics on holomorphic disks in 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 domainwith piecewise-smooth boundary(and all its fractional linear images), we have the equalities
where and denote the appropriately normalized conformal maps and , respectively, and are the minimal dilatations of their quasiconformal extensions to and stand for their Grunsky norms, and is the opening of the least interior angle between the boundary arcs . Here if the corresponding vertex is finite and 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, whereis the opening of the largest interior angle of.
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.
Let be a closed unbounded curve with the convex interior, which is smooth at all finite points and has at infinity the asymptotes approaching the interior angle . For any such curve, Theorem 4 yields the equalitiesE22
More generally, assume that also has a finite angle point with the angle opening . 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 , because any such lune is a Moebius image of the exterior domain for the above curve .
Other quantitative examples illustrating Theorem 5 are presented in .
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 be a rectilinear polygon with the finite vertices and with vertex , and let the interior angle at the vertex be equal to and at be equal to , where and all , so that . Let be the conformal map of the upper half-plane onto , which without loss of generality, can be normalized by as (assuming that contains the origin ).
An important geometric characteristic of polynomials is the quantity
it valuates the local boundary quasiconformal dilatation of .
Using this quantity, we first prove that an assertion similar to Theorem 4 fails for the generic rectilinear polygons.
Theorem 6.There exist rectilinear polygonswhose conformal mapping functionssatisfy
whereis defined via (23).
Proof. We shall use the rectangles ; in this case all . It is known that the mapping function of any rectangle has equal Grunsky and Teichmüller norms,
Using the Moebius map , we transform the rectangle into a (nonconvex) circular quadrilateral with angles and mutually orthogonal edges so that two unbounded edges from these are rectilinear and two bounded are circular, and note that for sufficiently long rectangles must be
where denotes the conformal map .
Indeed, as was established by Kühnau , the quadrilaterals with the side ratios (conformal module) greater than have the reflection coefficient (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 ); this proves (25).
Any circular quadrilateral satisfying (25) can be approximated by appropriate rectilinear polygons . 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 and are lower continuous functionals under locally uniform convergence of quasiconformal maps (i.e., as in the indicated topology, and similarly for the reflection coefficient). This contradiction proves the theorem.
3.3 The main result
The main result of this section is
Theorem 7.  Letbe a unbounded rectilinear polygon, neither convex nor concave, and hence contain the verticeswhose inner angleshave openingswith. Assume that all suchsatisfy
whereis given by (23) (which means that the maximal value in (22) is attained at some vertexwith).
For any such polygon, taking appropriate Moebius map , we have the equalities
Proof. Let be an unbounded rectilinear polygon. Its conformal mapping function fixing the infinite point and with is represented by the Schwarz-Christoffel integral
where all and are the corresponding complex constants. The logarithmic derivative of this map has the form
Letting , we construct for an ambient complex isotopy (holomorpic motion)
(containing as a fiber map), which is injective in the space coordinate for any fixed , holomorphic in for a fixed and .
First observe that for real the solution to the equation with the initial conditions satisfies
If the interior angles of the initial polygon satisfy the assumption (26), then all the functions are represented by an integral of type (27) (replacing by , and with suitable constants ).
Geometrically this means that the exterior angle at any finite vertex decreases with (but the value increases if ). Under the assumption (26), the admissible bounds for the possible values of angles ensure the univalence of this integral on for every indicated . This yields that every also is a polygon with the interior angles for , while .
Now we pass to the conformal map of the unit disk onto , using the function . This map is represented similar to (28) by
where the points are the preimages of vertices on the unit circle . Pick to have . For this function, we have a natural complex isotopy
Following (31), we set for ,
The relations (32) yield that this function also is univalent in .
The corresponding Schwarzians fill a real analytic line in the universal Teichmüller space (modeled as a bounded domain in the complex Banach space of hyperbolically bounded holomorphic functions on ). This line is located in the holomorphic disk , where denotes the map and is a simply connected planar domain.
This generates the holomorphic motions (complex isotopies) and with .
The basic lambda-lemma for holomorphic motions implies that every fiber map is the restriction to of a quasiconformal automorphism of the whole sphere , and the Beltrami coefficients
in the lower half-plane depend holomorphically on t as elements of the space .
So we have a holomorphic map from the disk into the unit ball of Beltrami coefficients supported on ,
and the classical Schwarz lemma implies the estimate
It follows that the extremal dilatation of the initial map satisfies
Hence, also and by the inequality (10), .
On the other hand, Kühnau’s lower bound (20) implies
Together with (5), this yields that the polygon admits all equalities (27) completing the proof of the theorem.
3.4 Some applications
Theorem 6 widens the collections of curves with explicitly given Fredholm eigenvalues and reflection coefficients.
For example, let 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 quasicircleof the indicated form, its quasireflection coefficientand Fredholm eigenvalueare given by
whereis defined similar to (22) by angles between the subintervals of. The same is valid for imagesunder the Moebius maps.
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 :
For an arbitrary finitely or infinitely generated Fuchsian group is the Bers embedding of its Teichmüller space starlike?
Recall that in this embedding is represented as a bounded domain formed by the Schwarzian derivatives of holomorphic univalent functions in the lower half-plane (or in the disk) admitting quasiconformal extensions to the Riemann sphere compatible with the group acting on .
It was shown in  that universal Teichmüller space 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  (see also ).
This implies, in particular, that universal Teichmüller space is not starlike with respect to any of its points, and there exist points for which the line interval contains the points from , where is the Banach space of hyperbolically bounded holomorphic functions in the half-plane with norm
and denotes the set of all Schwarzian derivatives of univalent functions on . These points correspond to holomorphic functions on which are only locally univalent.
Toki  extended the result on the nonstarlikeness of the space 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  is the same as in .
On the other hand, it was established in  that also all finite dimensional Teichmüller spaces 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 with finite vertices and . Denote the exterior angles at by so that . Then, similar to (22), the conformal map of the lower half-plane onto the complementary polygon is represented by the Schwarz-Christoffel integral
with and complex constants ; here . Its Schwarzian derivative is given by
It defines a point of the universal Teichmüller space modeled as a bounded domain in the space of hyperbolically bounded holomorphic functions on with norm .
Denote by the positive root of the equation
and put . Then for appropriate , we have.
Theorem 8.  For any convex polygon, the Schwarziansdefine for anya univalent functionwhose harmonic Beltrami coefficientinis extremal in its equivalence class, and
By the Ahlfors-Weill theorem , every with is the Schwarzian derivative of a univalent function in , and this function has quasiconformal extension onto the upper half-plane with Beltrami coefficient of the form
called harmonic. Theorem 7 yields that any with does not admit extremal quasiconformal extensions of Teichmüller type, and in view of extremality of harmonic coefficients the Schwarzians for some between and must lie outside of the space ; so this space is not a starlike domain in .
4.2 There are unbounded convex polygons for which the equalities (33) are valid in the strengthened form
for all , completing the bounds (21).
We illustrate this on the case of triangles. Let be a triangle with vertices and and exterior angles . The logarithmic derivative of conformal map has the form
with , and similar to (34),
If the angles of satisfy , where is the angle at , the arguments from  yield that the harmonic Beltrami coefficient satisfies (35).
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 whose conformal mapping functions have different Grunsky and Teichmüller norms, and therefore, .
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 .
One of the interesting questions is whether the equality of Teichmüller and Grunsky norms is preserved under the affine deformations
with (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 .
More precisely, we consider the maps , which are conformal in the complementary domain and have in a constant quasiconformal dilatation , regarding such maps as the affine deformations and the collection of domains as the affine class of .
If is a quasiconformal automorphism of conformal in mapping the disk onto a domain , then for a fixed the maps and differ by a conformal map and hence have in the disk the same Beltrami coefficient.
Note that the inequality equivalent to 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 by zero to the complementary domain).
The following theorem solves the problem positively.
Theorem 9.For any functionwithmapping the diskonto the complement of a bounded domain (quasidisk)and any affine deformationof this domain (with), we have the equality
Theorems 9 essentially increases the set of quasicircles for which 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 [40, 53, 54, 55] and for rectilinear or circular quadrilaterals having a common tangent circle ).
5.2 Scheme of the proof of Theorem 9
The proof follows the lines of Theorem 1.1 in  and is divided into several lemmas.
First, we establish some auxiliary results characterizing the homotopy disk of a map with .
Take the generic homotopy function
Then and this point-wise map determines a holomorphic map so that the homotopy disks foliate the space . Note also that
and if maps the unit disk onto a convex domain, then all level lines for are starlike.
Lemma 1.If the homotopy functionofsatisfyfor some, then the equalityholds for alland the homotopy diskhas no critical pointswith.
Take the univalent extension of to a maximal disk and define
Its Beltrami coefficient in is defined by holomorphic quadratic differentials of the form (19), and we have the holomorphic map, for a fixed ,
of the disk into . In view of our assumption on , the series (37) is convergent in some wider disk .
Using the map (37), we pull back the hyperbolic metric to the disk (parametrized by ) and define on this disk the conformal metric with
of Gaussian curvature at noncritical points. In fact, this is the supporting metric at for the upper envelope of metrics (38) followed by its upper semicontinuous regularization
(supporting means that and in a neighborhood of ).
The metric is logarithmically subharmonic on and its generalized Laplacian
(while for we have at its noncritical points ).
As was mentioned above, the Grunsky coefficients define on the tangent bundle a new Finsler structure dominated by the infinitesimal Teichmüller metric . This structure generates on any embedded holomorphic disk the corresponding Finsler metric and reconstructs the Grunsky norm by integration along the Teichmüller disks:
Lemma 2.  On any extremal Teichmüller disk(and its isometric images in), we have the equality
Taking into account that the disk touches at the point the Teichmüller disk centered at the origin of and passing through this point and that the metric does not depend on the tangent unit vectors whose initial points are the points of , one obtains from Lemma 3 and the equality that also
The following lemma is a needed reformulation of Theorem 3.
Lemma 3.  The infinitesimal formsandof both Kobayashi and Teichmüller metrics on the tangent bundleofare continuous logarithmically plurisubharmonic inand have constant holomorphic sectional curvature.
We compare the metric with using Lemmas 2, 3, and Minda’s maximum principle given by.
Lemma 4.  If a functionis upper semicontinuous in a domainand its (generalized) Laplacian satisfies the inequalitywith some positive constantat any point, where, and if
then eitherfor allor elsefor all.
Lemma 4 and the equality (39) imply that the metrics must be equal in the entire disk , which yields by Lemma 2 the equality
for all (with depending on ) and that for any with its homotopy diskhas only a singularity at the origin of.
We may now investigate the action of affine deformations on the set of functions with equal Grunsky and Teichmüller norms.
Lemma 5.For any affine deformationof a convex domainwith expansionnear, we have
and for sufficiently small all composite maps
Finally, we use the following important result of Kühnau .
Lemma 6.For any functionwith, the extremal quasiconformal extensions of the homotopy functionstoare defined for sufficiently smallby nonvanishing holomorphic quadratic differentials, and therefore,.
Using these lemmas, one establishes the equalities on the disk and
The final step of the proof is to extend the last equality to all c with .
Applying again the chain rule for Beltrami coefficients from the unit ball in ,
and (so for fixed, depends holomorphically on in norm) and defining the corresponding functions (37), one gets now the holomorphic functions of . Then, constructing in a similar way the corresponding Finsler metrics
and taking their upper envelope and its upper semicontinuous regularization, one obtains a subharmonic metric of Gaussian curvature on the nonsingular disk . One can repeat for this metric all the above arguments using the already established equality (40) for small .
The arguments in the proof of Theorem 9 are extended almost straightforwardly to more general case:
Theorem 10. Letand. Letbe a holomorphic mapwithout critical points inand. Denote bythe univalent solution of the Schwarzian equation
where, on the domain. Then, for any, the compositionalso satisfies.
Note that by the lambda lemma for holomorphic motions, the map determines a holomorphic disk in the ball of Beltrami coefficients on , which yields, together with assumptions of the theorem, that for small ,
with . 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 polygonswith sufficiently large number of sides such that
It follows simply from Theorem 8 that if a polygon , whose edges are quasiconformal arcs, satisfies 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 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 holds in particular for quadrilaterals 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, we have
where F is the appropriately normalized conformal map ofonto.
The proof of this theorem essentially relies on Theorem 8 and on result of  that the equalities (41) are valid for all rectangles, and hence for their affine transformations.
Fix such a quadrilateral and consider the collection of quadrilaterals with the same first three vertices and variable ; the corresponding runs over a subset of the trice punctured sphere .
The collection contains the trapezoids, for which we have the equalities (41) by Theorem 8 (and consequently, the infinitesimal equality (39) at the corresponding points ).
Similar to the proof of Theorem 6, one obtains in the universal Teichmüller space a holomorphic disk extending the real analytic curve filled by the Schwarzians, which correspond to the values on . On this disk, one can construct, similar to (38), the corresponding metric . Lemmas 4–6 again imply that this metric must coincide at all points of with the dominant infinitesimal Teichmüller-Kobayashi metric of . Together with Lemma 2, this provides the global equalities (41) for all points of the disk (and hence for the prescribed quadrilateral ).
5.5 An open problem here is the following question of Kühnau (personal communication)
Question: Does the reflection coefficient of a rectanglebe a monotone nondecreasing function of its conformal module(the ratio of the vertical and horizontal side lengths)?
The results of Kühnau and Werner for the rectangles state that if the module satisfies , then
On the other hand, the reflection coefficients of long rectangles are close to , 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 with 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 , he established in  the reflection coefficient of the set , which consists of the interval and a separate point . All these results are presented in .
Theorems 4 and 6 open a new way in solving this problem following the lines of the first example after Theorem 4.
6.2 Reflections across the finite collections of quasiintervals
Theorems 5 and 7 open a new way in solving this problem following the lines of the first example after Theorem 5. Namely, given a finite union
of smooth curvelinear quasiintervals (possibly mutually separated) such that can be extended without adding new vertices (angular points) to a quasicircle containing and bounding a convex polygon that satisfies the assumptions of Theorem 4 or a polygon considered in Theorem 7, then by these theorems, the reflection coefficient of the setequals
whereis defined forsimilar 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 can be smoothly extended and then rounded off.
Note also that adding to L a finite number of appropriately located isolated points does not change the reflection coefficient (42).