Search

A method for identifying parameters of the conductivity coefficient of objects is generalized for the case of reconstructing an image of a part of a soil massif from the tomography data of the applied quasipotentials. In this case, without diminishing the generality, the reconstruction of the image is carried out in a fragment of a rectangular medium with local bursts of homogeneity present in it. The general idea of the corresponding algorithm consists in the sequential iterative solution of problems on quasiconformal mappings and identification of the parameters of the conductivity coefficient, with an insufficient amount of data on the values of the flow functions on the “inaccessible” part of the boundary. The image was reconstructed according to the data obtained using a full-range gradient array. The developed approach, in comparison with the existing ones, has a number of advantages that make it possible to increase the accuracy of identification of the conductivity coefficient. Namely, it provides an increase, in a qualitative sense, in the amount of input data, allows avoiding the use of Dirac delta functions when modeling areas of application of potentials and sufficiently flexibly take into account the mathematical aspects of the implementation of a quasiconformal mapping of a finite fragment of a half-plane onto a parametric polygon (domain of a complex quasipotential). The solution of the corresponding problem, in particular, occurs not in a single (fixed) investigated fragment of a rectangular soil massif, but in a number of smaller subdomains of the same shape, in the proposed optimal sequence. This saves machine time significantly. The prospects for further practical implementation of the proposed method follow from its ability to give an approximate result with relatively low costs (financial, time), Обобщен метод идентификации параметров коэффициента проводимости объектов на случай реконструкции изображения части грунтового массива по данным томографии приложенных квазипотенциалов. При этом, не уменьшая общности, реконструкция изображения осуществляется в фрагменте среды прямоугольной формы с имеющимися в ней локальными всплесками однородностей. Общая идея соответствующего алгоритма заключается в поочередном итерационном решении задач на квазиконформные отображения и идентификацию параметров коэффициента проводимости, при недостаточном количестве данных о значениях функций течения на «недоступной» части границы. Реконструкция изображения проводилась по данным, полученным с помощью полнодиапазонной градиентной установки. Разработанный подход, по сравнению с существующими, обладает рядом преимуществ, которые позволяют повысить точность идентификации коэффициента проводимости. А именно: обеспечивает увеличение, в качественном смысле, количества входных данных, позволяет избегать применения дельта функций Дирака при моделировании участков приложения потенциалов и достаточно гибко учитывать математические аспекты осуществления квазиконформного отображения конечного фрагмента полуплоскости на параметрический многоугольник (область комплексного квазипотенциала). Решение соответствующей задачи, в частности, происходит не в единственном (фиксированном) исследуемом фрагменте грунтового массива прямоугольной формы, а в ряде меньших подобластей такой же формы, в предложенной оптимальной последовательности. Это позволяет существенно экономить машинное время. Перспективность дальнейшего практического внедрения предложенного метода следует из его способности давать приближенный результат при сравнительно невысоких затратах (финансовых, временных), Узагальнено метод ідентифікації параметрів коефіцієнта провідності об’єктів на випадок реконструкції зображення частини ґрунтового масиву за даними томографії прикладених квазіпотенціалів. При цьому, не зменшуючи загальності, реконструкція зображення здійснюється у фрагменті середовища прямокутної форми із наявними у ньому локальними сплесками однорідностей. Загальна ідея відповідного алгоритму полягає у почерговому ітераційному розв’язанні задач на квазіконформні відображення та ідентифікацію параметрів коефіцієнта провідності, при недостатній кількості даних про значення функцій течії на «недоступній» частині границі. Реконструкція зображення проводилась за даними, отриманими за допомогою повнодіапазонної градієнтної установки. Розроблений підхід, в порівнянні з існуючими, володіє низкою переваг, які дозволяють підвищити точність ідентифікації коефіцієнта провідності. А саме: забезпечує збільшення, у якісному сенсі, кількості вхідних даних, дозволяє уникати застосування дельта функцій Дірака при моделюванні ділянок прикладання потенціалів та досить гнучко враховувати математичні аспекти здійснення квазіконформного відображення скінченного фрагмента пів-площини на параметричний многокутник (область комплексного квазіпотенціалу). Розв’язання відповідної задачі, зокрема, відбувається не в єдиному (фіксованому) досліджуваному фрагменті ґрунтового масиву прямокутної форми, а в низці менших підобластях такої ж форми, у запропонованій оптимальній послідовності. Це дозволяє суттєво економити машинний час. Перспективність подальшого практичного впровадження запропонованого методу слідує із його здатності давати наближений результат при порівняно невисоких затратах (фінансових, часових)

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $${\mathbb {H}}^{1}$$. Several auxiliary properties of quasiconformal mappings between subdomains of $${\mathbb {H}}^{1}$$ are proven, including BMO estimates for the logarithm of the Jacobian. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $${\mathbb {H}}^{1}$$. The theorems are discussed for the sub-Riemannian and the Koranyi distances. This extends results due to Astala–Gehring, Astala–Koskela, Koskela and Bonk–Koskela–Rohde.

The asymptotic Teichmüller space A T ( E ) AT(E) of a closed subset E E of the Riemann sphere C ^ \hat {\mathbb {C}} with at least 4 4 points and the natural asymptotic Teichmüller metric are introduced. It is proved that A T ( E ) AT(E) is isometrically isomorphic to the product space of the asymptotic Teichmüller spaces of the connected components of C ^ ∖ E \hat {\mathbb {C}}\setminus E and the Banach space of the Beltrami coefficients defined on E E . Furthermore, it is proved that there is a complex Banach manifold structure on A T ( E ) AT(E) .

For a self mapping f: D→D of the unit disk in C which has finite distortion, we give a separation condition on the components of the set where the distortion is very large - say greater than a given constant - which implies that f still extends homeomorphically and quasisymmetrically to the boundary S = ∂D. Thus f shares its boundary values with a quasiconformal mapping whose distortion we explicitly estimate in terms of the data. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate on the entire boundary S, but it does not allow a component to run out to the boundary - a necessary restriction. The lift of a Jordan domain in a Riemann surface to its universal cover D is always uniformly separated in modulus, and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map between surfaces and their geometry - again with explicit estimates. As part of our investigations, we study mappings ϕ: S → S which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of ϕ to the disk D. We then extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.

In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S 1 belongs to the Weil-Petersson class WP ( S 1 ) , namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincare metric if and only if g is absolutely continuous such that log ⁡ g ′ belongs to the Sobolev class H 1 2 . In this sequel to [23] , we show that the smooth Hilbert manifold structure on WP ( S 1 ) inherited from H 1 2 by the pullback g ↦ log ⁡ | g ′ | is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27] . This enables us to give a fast approach to some results in our previous papers [23] and [24] .

We study the relation between the geometric properties of a quasicircle~$\Gamma$ and the complex dilatation~$\mu$ of a quasiconformal mapping that maps the real line onto~$\Gamma$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-\mu S)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves., Comment: 27 pages

We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping f in the unit disk is small enough, then f is, indeed, globally univalent in the unit disk and can be extended to a quasiconformal mapping in the extended complex plane.

We consider the infinite dimensional Teichmuller space of a Riemann surface of general type. On the basis of the fact that the action of the quasiconformal mapping class group on the Teichmuller space is not discontinuous, in general, we divide the Teichmuller space into two disjoint subsets, the limit set and the region of discontinuity, according to the discreteness of the orbit by a subgroup of the quasiconformal mapping class group. The asymptotic Teichmuller space is a certain quotient space of the Teichmuller space and there is a natural projection from the Teichmuller space to the asymptotic Teichmuller space. We consider the fibers of the projection over any point in the asymptotic Teichmuller space, and show a coherence of the discreteness on each fiber in the Teichmuller space.

Suppose that $$D$$ is a simply connected domain in the extended complex plane $$\widehat{\mathbb C}$$ with the following homogeneity property: for each pair of points $$a$$ and $$b$$ in $$D$$ there exists a $$K$$ -quasiconformal self-mapping $$f$$ of $$\widehat{\mathbb C}$$ such that $$f(D)=D$$ and $$f(a)=b$$ . This paper classifies the simply connected plane domains $$D$$ with locally connected boundaries that exhibit this property for some $$K$$ . Any such domain $$D$$ falls into one of five (non-empty) categories, each specified by the character of the boundary $$\partial D$$ of $$D$$ , namely $$\partial D$$ is the empty set, a singleton, a quasicircle, a dendrite, or a Swiss cheese.

In this present work, we continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in the regions with piecewise smooth boundary. 1. Introduction and Main Results Let G C be a …nite region, with 0 2 G, bounded by a Jordan curve L := @G; (t; R) := fw : jw tj > Rg, := (0; 1); := extG (with respect to C). Let w = (z) be the univalent conformal mapping of onto the normalized by (1) =1; (1) > 0, and := : Let }n denote the class of arbitrary algebraic polynomials Pn(z), degPn n; n 2 N: Let h(z) be a weight function de…ned in GR: Denote by A(G) the class of functions f which are analytic in G . For any p > 0 we de…ne Ap(h;G) := f : f 2 A(G); kfkpAp(h;G) := ZZ G h(z) jf(z)j d z 0; let Lp(L) := 1, let us set LR := fz : j (z)j = Rg ; GR := intLR; R := extLR. Then, well known Bernstein -Walsh Lemma [13] says that: (1.1) kPnkC(GR) R n kPnkC(G) : Date: January 21, 2013. 2010 Mathematics Subject Classi…cation. Primary 30A10,30C10; Secondary 41A17. Key words and phrases. Algebraic polynomials, Conformal mapping, Smooth curve, Quasiconformal curve. 1 2 F. G. ABDULLAYEV AND C. D. GUN Hence, setting R = 1+ 1 n , we see that the C norm of polynomials Pn(z) in GR and G is identical, i.e. the norm kPnkC(G) increases up to multiplication by a constant in GR. Similar estimation to (1.1) in space Lp(L) was obtained in [9] as following: (1.2) kPnkLp(LR) R n+ 1 p kPnkLp(L) ; p > 0: To give a similar estimation to (1.2) for the Ap(G) norm, …rst of all we will give some de…nitions and notations. De…nition 1.1. [10, p.97], [11]The Jordan arc (or curve) L is calledK quasiconformal (K 1), if there is a K quasiconformal mapping f of the region D L such that f(L) is a line segment (or circle). Let F (L) denote the set of all sense preserving plane homeomorphisms f of the region D L such that f(L) is a line segment (or circle) and let KL := inf fK(f) : f 2 F (L)g ; where K(f) is the maximal dilatation of f: Then L is a quasiconformal curve, if KL 0: Corollary 1.4. If L is an analytic curve or arc, then L is 1 quasiconformal. It has been known that there exist quasiconformal curves which are not recti…able [10, p.104]. Let fzjgmj=1 be a …xed system of distinct points which are ordered in the positive direction on curve L . Consider a so-called generalized Jacobi ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIALS 3 weight function h (z) de…ned as follows

We show that a quasiconformal mapping of Carnot-Caratheodory spaces is absolutely continuous not only on integral curves of horizontal vector fields but also on integral curves of vector fields whose degree differs from one.

We establish the strong asymptotics for Bergman orthogonal polynomials defined over Jordan domains with corners. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains bounded by analytic curves, and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Sigma of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments., 40 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:0910.1788

We provide several estimates which involve the distance to L ∞ in some function spaces, the composition operator induced by a quasiconformal mapping and the logarithm of the Jacobian of a quasiconformal mapping. Our results are sharp in the two dimensional case.

The book is dedicated to the memory of Alexander Grothendieck, and it contains several chapters on his work related to the so-called Grothendieck-Teichmuller theory: the Grothendieck-Teichmuller group and the Teichmuller tower. There are also chapters on dessins d'enfants and Grothendieck’s construction of the complex structure of Teichmuller space. The other chapters concern compactifications of Teichmuller spaces, Mirzakhani’s recursion formula on Weil–Petersson volumes, rigidity phenomena in mapping class groups, harmonic volume and its applications, universal families of Riemann surfaces, representation theory and generalized structures, cubic differentials in the differential geometry of surfaces, two-generator groups acting on the complex hyperbolic plane, configuration spaces of planar linkages, an overview of quasiconformal mappings on the Heisenberg group, hypergeometric Galois actions, a panaroma of the fundamental group of the modular orbifold, a survey of Grothendieck’s tame topology, commentaries on Teichmuller’s paper ``Extremale quasikonforme Abbildungen und quadratische Differentiale", and the English translations and commmentaries of the following three papers by Teichmuller: " Complete solution of an extremal problem of the quasiconformal mapping", " On extremal problems of conformal geometry" and " A displacement theorem of quasiconformal mapping." The authors of the various chapters are: Athanase Papadopoulos, Valentin Poenaru, Norbert A’Campo, Lizhen Ji, Vincent Alberge, Hideki Miyachi, Ken’ichi Ohshika, Yi Huang, Javier Aramayona, Juan Souto, Yuuki Tadokoro, Robin de Jong, John Loftin, Ian McIntosh, Pierre Will, Alexey Sossinsky, Ioannis D. Platis, Pierre Guillot, A. Muhammed Uludag, Ismail Saglam, Ayberk Zeytin, Reiner Kuhnau, Manfred Karbe and Oswald Teichmuller (in English translation).

We provide a commentary on Teichmuller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichmuller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.

This is a mathematical commentary on Teichmuller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flachen" (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichmuller wrote on the theory of moduli. It contains the proof of the so-called Teichmuller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmuller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincare and Klein's use of the so-called ``continuity principle" in their attempts to prove the uniformization theorem.

We show that the Teichmuller space of a surface without boundary and with punctures, equipped with the Thurston metric, is the limit in an appropriate sense of Teichmuller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmuller spaces equipped with the Thurston metric. In this paper, we show that the arc metrics on the Teichmuller space of surfaces with boundary limit to the Thurston metric on the Teichmuller space of a surface without boundary, by making the boundary lengths tend to zero. We use this to prove a result on the translation distances for mapping classes. We introduce some notation before stating precisely the results. In all this paper, S = Sg,p,n is a connected orientable surface of finite type, of genus g with p punctures and n boundary components. We assume that S has negative Euler characteristic, i.e., χ(S) = 2 − 2g − p − n 0, we denote by ∂S the boundary of S. A hyperbolic structure on S is a complete metric of constant curvature −1 such that (i) each puncture has a neighborhood isometric to a cusp, i.e., to the quotient {z = x + iy ∈ H 2 | y > a}/hz 7→z + 1i

Letf:Ω⊂Rn→Rnbe aquasiconformal mappingwhose Jacobian is denoted byJfand letEXP(Ω)be the space of exponentially integrable functions onΩ. We give an explicit bound for the norm of the composition operatorTf:u∈EXP(Ω)↦u∘f-1∈EXP(f(Ω))and, as a related question, we study the behaviour of the norm oflog⁡Jfin the exponential class. TheA∞property ofJfis the counterpart in higher dimensions of the area distortion formula due to Astala in the plane and it is the key tool to prove the sharpness of our results.

This paper is devoted to the study of so-called finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries and quasi-isometries. We obtain several criteria for the homeomorphic extension to the boundary of finitely bi-Lipschitz homeomorphisms f between domains in ℝn, n ≥ 2, whose outer dilatations KO(x, f) satisfy the integral constraints $$\int {\Phi (K_O^{n - 1} (x,f))dm(x) < \infty } $$ with an increasing convex function Φ: [0,∞] → [0,∞]. Note that the integral conditions on the function Φ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of f to the boundary.

Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmuller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and proved an extension theorem for quasiconformal motions over an interval. We prove some new properties of (normalized) quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdorff spaces. As a spin-off, we strengthen the result of Sullivan and Thurston, and show that if a quasiconformal motion of E over an interval has a certain group-equivariance property, then the extended quasiconformal motion can be chosen to have the same group-equivariance property. Our main theorem proves a result on isomorphisms of continuous families of Mobius groups arising from a group-equivariant quasiconformal motion of E over a path-connected Hausdorff space. Our techniques connect the Teichmuller space of the closed set E with quasiconformal motions of E.

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.

For a subgroup of the quasiconformal mapping class group of a Riemann surface in general, we give an algebraic condition which guarantees its discreteness in the compact-open topology. Then we apply this result to its action on the Teichmuller space.

We prove that a harmonic diffeomorphism between two Jordan domains with C2 boundaries is a (K, K′) quasiconformal mapping for some constants K ≥ 1 and K′ ≥ 0 if and only if it is Lipschitz continuous. In this setting, if the domain is the unit disk and the mapping is normalized by three boundary points condition we give an explicit Lipschitz constant in terms of simple geometric quantities of the Jordan curve which surrounds the codomain and (K, K′). The results in this paper generalize and extend several recently obtained results.

The stable quasiconformal mapping class group is a group of quasiconformal mapping classes of a Riemann surface that are homotopic to the identity outside some topologically finite subsurface. Its analytic counterpart is a group of mapping classes that act on the asymptotic Teich\-m\"ul\-ler space trivially. We prove that the stable quasiconformal mapping class group is coincident with the asymptotically trivial mapping class group for every Riemann surface satisfying a certain geometric condition. Consequently, the intermediate Teich\-m\"ul\-ler space, which is the quotient space of the Teich\-m\"ul\-ler space by the asymptotically trivial mapping class group, has a complex manifold structure, and its automorphism group is geometrically isomorphic to the asymptotic Teich\-m\"ul\-ler modular group. The proof utilizes a condition for an asymptotic Teich\-m\"ul\-ler modular transformation to be of finite order, and this is given by the consideration of hyperbolic geometry of topologically infinite surfaces and its deformation under quasiconformal homeomorphisms. Also these arguments enable us to show that every asymptotic Teich\-m\"ul\-ler modular transformation of finite order has a fixed point on the asymptotic Teich\-m\"ul\-ler space, which can be regarded as an asymptotic version of the Nielsen theorem.

We study the maximal number $0\le h\le+\infty$ for a given plane domain $\Omega$ such that $f\in H^p$ whenever $p, Comment: 11 pages

We review recent results on planar mappings of finite distortion. This class of mappings contains all analytic functions and quasiconformal mappings.

For a Riemann surface of analytically infinite type, the action of the quasiconformal mapping class group on the Teichmüller space is not discontinuous in general. In this paper, we consider pure mapping classes that fix all topological ends of a Riemann surface and prove that the pure mapping class group acts on the Teichmüller space discontinuously under a certain geometric condition of a Riemann surface. We also consider the action of the quasiconformal mapping class group on the asymptotic Teichmüller space. Non-trivial mapping classes can act on the asymptotic Teichmüller space trivially. We prove that all such mapping classes are contained in the pure mapping class group.

In this note we show that a harmonic quasiconformal mapping f = u + i v with respect to the Poincare metric of the upper half plane R + 2 onto itself such that v ( x , y ) = v ( y ) or u ( x , y ) = u ( x ) is a conformal mapping.

We prove versions of the Ahlfors–Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric.

For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmuller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmuller space AT(R). We prove that if MCG(R) has a common fixed point α(p) ∈ AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

For an analytically infinite Riemann surface R, the quasiconformal mapping class group MCG(R) always acts faithfully on the ordinary Teichmuller space T(R). However in this paper, an example of R is constructed for which MCG(R) acts trivially on its asymptotic Teichmuller space AT(R).

In this note we prove that the so-called Sierpi\’nski holes in the parameter plane 0 < ¦λ¦ < ∞ of the McMullen family Fλ(z) = z m + λ/z l, m ≥ 2 and l ≥ 1 fixed, are simply connected, and determine the total number of these domains.

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.

We use circle packing techniques to construct approximate solutions to the generalized Beltrami equations with simply and multiply connected regions in the plane. We show convergence of the approximate solutions. This gives a constructive proof for the existence of quasiconformal mappings with a given pair of complex dilations.

In this paper, we consider quasiconformal homeomorphisms ϕn ;S 0→S n (n = 1, 2, ...) of a bordered Riemann surfaceS 0 and discuss how the Dirichlet solutions $$H_{fo\varphi n^{ - 1} }^{S_n } $$ for a continuous functionf on ϖS 0 vary when the maximal dilatations of ϕn converge to one. Furthermore, we consider the smoothness of Dirichlet solutions for parameters of the quasiconformal deformation.

We generalize Martio's paper (14). Indeed the problem studied in this paper is under which conditions on a homeomorphism f between the unit circle S1 := fz : jzj = 1g and a fix convex Jordan curve ∞ the harmonic extension of f is a quasiconformal mapping. In addition, we give some results for some classes of harmonic dieomorphisms. Further, we give some results concerning harmonic quasiconformal mappings (which follow by the results obtained in (10)). Finally, we give some examples which explain that the classes defined in (14) are not big enough to enclose all harmonic quasiconformal mappings of the disc onto itself.

In this paper a condition is obtained in terms of Dirichlet's integral, for a sense-preserving homeomorphism between the unit circumferences to be prolonged into the interior of disk quasiconformally or as extremal Teichmuller mapping, which sharpens and simplifies the widely known theorems by Teichmuller [ Abh. Preuss. Akad. Wiss. Math. Naturw. Kl. 22 (1939) 1-197], Ahlfors [ J. d'Anal. Math ., 3 (1953/54) 1-98], Hamilton [ Trans. Amer. Math. Soc ., 138 (1969) 399-406], Reich [ Ann. Acad. Sci. Fenn. Ser. A. I. Math . 10 (1985) 469-475], Strebel [ Comment. Math. Helv. , 39 (1964) 77-89], Beurling and Ahlfors [ Acta Math ., 96 (1956) 125-142].

This paper has arisen from an effort to provide a comprehensive and unifying development of the L p L^{p} -theory of quasiconformal mappings in R n \mathbb {R}^{n} . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by D − \mathcal {D}^{-} and D + \mathcal {D}^{+} , which act on weakly differentiable deformations f : Ω → R n f:\Omega \to \mathbb {R}^{n} of a domain Ω ⊂ R n \Omega \subset \mathbb {R}^{n} . Solutions to the so-called Cauchy-Riemann equations D − f = 0 \mathcal {D}^{-}f=0 and D + f = 0 \mathcal {D}^{+}f=0 are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental L p L^{p} -estimate ‖ D + f ‖ p ≤ A p ( n ) ‖ D − f ‖ p . \begin{equation*}\|\mathcal {D}^{+}f\|_{p} \le A_{p}(n)\|\mathcal {D}^{-}f\|_{p}. \end{equation*} In quest of the best constant A p ( n ) A_{p}(n) , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

The theory of quasiconformal mappings originated at the beginning of the twentieth century. At that time, it arose out of geometric reasons in the works of Grotzsch on the one hand and as solutions to special types of elliptic systems of differential equations in the works of Lavrentiev on the other hand. Important applications in various fields of mathematics such as discrete-group theory, mathematical physics, and complex differential geometry led to the great development of the theory of quasiconformal mappings, which at present is a branch of complex analysis. Great contributions to this theory were made by M. A. Lavrentiev, H. Grotzsch, L. Ahlfors (who became one of the first Fields laureates (1936)), L. Bers, O. Teichmuller, P. Belinskii, and L. Volkovyskii in the past and C. Earle, I. Kra, V. Zorich, S. Krushkal, Yu. Reshetnyak, A. Sychev, F. Gehring, O. Lehto, C. Andreyan-Cazacu, R. Kuhnau, V. Aseev, V. Gutlyanskii, and V. Sheretov more recently. Many methods, such as the area method, the variational method, the method of parametric representations, the length–area principle, and the extremal-length method, were developed for solving problems on quasiconformal mappings. Many of them became more or less transformations of the ideas of classical methods of conformal analysis. However, not all of them give us a similar effect in solution of specific problems. Therefore, we can compare the Lowner–Kufarev method [2,54] for univalent functions in the conformal case and the parametric method of Shah Dao-Shing [67] and Gehring–Reich [28] in the quasiconformal case. Only a few problems were solved by the latter method, but the main extremal problems such as the Bieberbach conjecture in the conformal case involved the Lowner method. Thus, the problems in the quasiconformal case are much more difficult. At midcentury, it was established that the classical methods of geometric function theory could be extended to complex hyperbolic manifolds. Teichmuller spaces became the most important of them. In 1939, O. Teichmuller [81] proposed and partially realized an adventurous program of investigations in a moduli problem for Riemann surfaces. In the present paper, we often use the word “modulus” in a different way. Therefore, we have to clarify that the classical moduli problem starts from the works of B. Riemann [61] and is based on the fact that the conformal structure of a compact Riemann surface of genus g > 1 depends on 3g − 3 complex parameters (moduli). The problem is to investigate the nature of these parameters for general Riemann surfaces and to induce a real or a complex structure into the corresponding space of Riemann surfaces. O. Teichmuller connected the moduli problem with extremal quasiconformal mappings and with relevant quadratic differentials. This led him to the wellknown theory of Teichmuller spaces. Teichmuller’s ideas were strictly substantiated then in works by L. Ahlfors and L. Bers [6] and other authors. Investigations in Teichmuller spaces were carried out in different directions: the topological one connected with homotopy classes of diffeomorphisms of surfaces of finite conformal type (W. Abikoff, R. Fricke, J. Nielsen, and W. Thurston), mathematical physics connected with applications in conformal-gauge string theory (L. Takhtadzhyan and P. Zograf), the theory of discontinuous groups (H. Zieschang, E. Fogt, H.-D. Coldewey, I. Kra, S. Krushkal, B. Apanasov, and N. Gusevskii), the theory of dynamical systems (R. Devaney, A. Douady, J. Hubbard, L. Keen

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmiiller differential associated with the Teichmilller mapping between compact or finitely punctured Riemann surfaces.

This paper deals with the extremals of some functionals defined on a given homotopy class of quasiconformal mappings between Riemann surfaces, clarifying and extending some results in the literature.

In this paper we shall show that the boundary 9Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincare type inequalities. Then by using Heinonen-Koskela's work, we shall prove Loewner capacity estimates for some families of curves of WIp,q and the fact that every quasiconformal homeomorphism f : 0Ip,q d9Ip,q is quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirty-three YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1-12) is NO.

These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L ∞ norm of an energy density rather than the more familiar L p norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.