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An increasing homeomorphism \(h\) on the real line \(\mathbb{R}\) is said to be strongly symmetric if it can be extended to a quasiconformal homeomorphism of the upper half plane \(\mathbb{U}\) onto itself whose Beltrami coefficient \(\mu\) induces a vanishing Carleson measure \(|\mu(z)|^2/y\,dx\,dy\) on \(\mathbb{U}\). We will deal with the class of strongly symmetric homeomorphisms on the real line and its Teichmüller space, which we call the VMO-Teichmüller space. In particular, we will show that if \(h\) is strongly symmetric on the real line, then it is strongly quasisymmetric such that \(\log h'\) is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Becker-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Beltrami coefficient of a quasiconformal extension. We will also discuss various models of the VMO-Teichmüller space and endow it with a complex Banach manifold structure via the standard Bers embedding.

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains., Comment: 6 pages, 1 figure; slightly extended introduction in version 2

Kompleksna dinamika je področje matematike, ki se ukvarja z dinamičnimi sistemi določenimi z iteracijo kompleksnih preslikav. Cilj diplomske naloge je konstruirati racionalno preslikavo, katere dinamika je na 2-povezanem območju biholomorfno ekvivalentna iracionalni rotaciji. Tak fenomen imenujemo Hermanov kolobar. Najprej se seznanimo z osnovami kompleksne dinamike in opredelimo pojem Hermanovega kolobarja. Nato razvijemo teorijo kvazikonformnih preslikav in s pomočjo kvazikonformne kirurgije kostruiramo racionalno preslikavo, ki premore Hermanov kolobar. Complex dynamics is the study of dynamical systems defined by the iteration of complex valued functions. The goal of this thesis is to construct a rational function with dynamics on a 2-connected region biholomorphically equivalent to an irrational rotation. This phenomenon is called a Herman ring. First we familiarize ourselves with the basics of complex dynamics and define the term Herman ring. Then we develop the theory of quasiconformal maps and with the help of quasiconformal surgery construct a rational function with a Herman ring.

An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.

In this paper, we provide partial solutions to a problem raised by Vaisala on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in $\mathbb {R}^{n}$ and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

An approach to solving the problem of image reconstruction based on applied quasipotential tomographic data in the three-dimensional case is developed. It is based on the synthesis of spatial analogues of numerical quasiconformal mapping methods and algorithm for identifying the parameters of local bursts of homogeneous materials using similar methods on the plane. The peculiarity of the corresponding algorithm is taking into account (for each of the appropriate injections) the presence of only equipotential lines (with given values of the flow function or distributions of local velocities on them) and flow lines (with known potential distributions on them) at the domain boundary. Numerical experiments of simulative restoration of the environment structure are carried out.

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.

In terms of pre-Schwarzian and Schwarzian derivatives, we obtain some sufficient conditions to ensure that a sense-preserving locally univalent harmonic mapping f in the unit disk can be extended to a quasiconformal mapping in the complex plane so that its complex dilatation satisfies some Carleson measure conditions.

The note is devoted to biography and achievements of Ukrainian mathematician Ivan Pesin (1930--1993).

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 study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the corresponding minimal discrepancy density with the asymptotic variance for Bloch functions of the form "Bergman projection of bounded functions" and obtain a corresponding identity. Together with related work of Ivrii, this gives the asymptotic behavior of the universal quasiconformal integral means spectrum for small values of quasiconformality k and small exponents t. In particular, the conjectured behavior is shown to be smaller than conjectured by Prause and Smirnov, which also shows that there are no quasidisks with dimension 1+k^2, at least for small k., Comment: 37 pages

We establish a weighted version of the $H^p$-theory of quasiconformal mappings., Comment: 18pages

In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernandez and Martin [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmuller mappings.

In this paper, we introduce a class of mappings related to the distance ratio metric and study its connection to the freely quasiconformal mapping in Banach spaces. Moreover, we investigate the invariance of Gromov hyperbolic, natural and uniform domains under this class of mappings.

We prove that a measurable mapping of domains in a complete Riemannian manifold induces an isomorphism of Sobolev spaces with the first generalized derivatives whose summability exponent equals the (Hausdorff) dimension of the manifold if and only if the mapping coincides with some quasiconformal mapping almost everywhere.

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.

A numerical method of quasiconformal mappings for solving the coefficient problems of finding eigenvalues of the conductivity tensor having information about its directions in an anisotropic medium using applied quasipotential tomographic data is generalized. The corresponding algorithm is based on the alternate solving of problems on quasiconformal mappings and parameter identification. The results of numerical experiments of imitative restoration of environment structure are presented.

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there exists a quasiconformal mapping $f: X \rightarrow \Omega$ satisfying the modulus inequality $2\pi^{-1}\text{Mod }\Gamma \leq \text{Mod }f\Gamma \leq 4\pi^{-1}\text{Mod }\Gamma$ for all curve families $\Gamma$ in $X$. This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball., Comment: 7 pages, 2 figures, to appear in Indiana Univ. Math. J

This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichm\"uller theory and classical and modern approaches based on Loewner theory., Comment: 25 pages, 3 figs. This paper will be included in the proceedings of the 2nd GSIS-RCPAM International Symposium "Geometric Function Theory and Applications in Sendai" which was held in Tohoku University on September 10th-13th, 2013

The retinotopic map depicts the cortical neurons’ response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmuller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a “Beltrami coefficient map” (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

In this paper, we present quasiconformal mapping kernel machine learning-based intelligent hyperspectral data classification algorithm for the internet-based hyperspectral data retrieval. The contributions include three points: The quasiconformal mapping-based multiple kernels learning network framework is proposed for hyperspectral data classification, and the Mahalanobis distance kernel function is as the network nodes with the higher discriminative ability than Euclidean distance-based kernel function learning, and the objective function of measuring the class discriminative ability is proposed to seek the optimal parameters of the quasiconformal mapping projection. Experiments show that the proposed scheme is effective to the hyperspectral image classification and retrieval.

Z. Zhou et al. proved that in a Teichm\"uller equivalence class, there exists an extremal quasiconformal mapping with a weakly non-decreasable dilatation. In this paper, we prove that in an infinitesimal equivalence class, there exists a weakly non-decreasable extremal Beltrami differential., Comment: 9 pages. arXiv admin note: text overlap with arXiv:1610.09213

In this Letter, we numerically and experimentally demonstrate the carpet cloaking effect by a rectangular lattice two-dimensional photonic crystal (PC) exhibiting a semi-Dirac cone (SDC) dispersion phenomenon. The proposed SDC PC with an anisotropic zero refractive index medium operates as an optical carpet cloak for a perfect electric conductor surface bump. The experimental verification of the cloak is realized at microwave frequencies at around 12.1 GHz via dielectric rods. A good agreement between experimental measurements and numerical calculations is observed.. Finally, features such as rendering larger objects invisible are possible with the proposed idea. (C) 2020 Optical Society of America

Human visual cortex is organized into several functional re-gions/areas. Identifying these visual areas of the human brain (i.e., V1, V2, V4, etc) is an important topic in neurophysiology and vision science. Retinotopic mapping via functional magnetic resonance imaging (fMRI) provides a noninvasive way of defining the boundaries of the visual areas. It is well known from neurophysiology studies that retino-topic mapping is diffeomorphic within each local area (i.e. locally smooth, differentiable, and invertible). However, due to the low signal-noise ratio of fMRI, the retinotopic maps from fMRI are often not diffeomorphic, making it difficult to delineate the boundaries of visual areas. The purpose of this work is to generate diffeomorphic retinotopic maps and improve the accuracy of the retinotopic atlas from fMRI measurements through the development of a specifically designed registration procedure. Although there are sophisticated existing cortical surface registration methods, most of them cannot fully utilize the features of retinotopic mapping. By considering unique retinotopic mapping features, we form a qua-siconformal geometry-based registration model and solve it with efficient numerical methods. We compare our registration with several popular methods on synthetic data. The results demonstrate that the proposed registration is superior to conventional methods for the registration of retinotopic maps. The application of our method to a real retinotopic mapping dataset also results in much smaller registration errors.

International audience; We report on Teichmüller's results on the type problem from his two papers "Eine Anwendung quasikonformer Abbildungen auf das Typenproblem" (An application of quasiconformal mappings to the type problem) (1937) and "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched covers of the sphere. At the same time, we review the theory of line complexes, a combinatorial device used by Teichmüller and others to encode branched coverings of the sphere. In the first paper, Teichmüller proves that any two simply connected Riemann surfaces which are branched coverings of the Riemann sphere with finitely many branch values and which have the same line complex are quasiconformally equivalent. For this purpose, he introduces a technique for piecing together quasiconformal mappings. He also obtains a result on the extension of smooth orientation-preserving diffeomorphisms of the circle to quasiconformal mappings of the disc which are conformal at the boundary. In the second paper, using line complexes, Teichmüller gives a type criterion for a simply-connected surface which is a branched covering of the sphere, in terms of an adequately defined measure of ramification, defined by a limiting process. The result says that if the surface is ``sufficiently ramified" (in a sense which is made precise), then it is hyperbolic. In the same paper, Teichmüller answers by the negative a conjecture made by Nevanlinna which states a criterion for parabolicity in terms of the value of a (different) measure of ramification, defined by a limiting process. Teichmüller's results in his first paper are used in the proof of the results of the second one.

International audience; We report on Lavrentieff's paper ``Sur une classe de repr\'esentations continues'' (On a class of continuous representations), published in 1935, in which he introduced quasiconformal mappings under the name ``almost analytic functions." He set the bases of this theory, obtaining several results that show that these functions constitute natural generalizations of conformal mappings. The results include an existence and uniqueness theorem which constitutes an early version of the measurable Riemann mapping theorem, a result on the existence of isothermal coordinates, a generalization of Picard's big theorem to the setting of almost analytic functions, and several results on the type problem.

Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces. His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Gr{\"o}tzsch, Lavrentieff, Ahlfors and Teichm{\"u}ller. Gr{\"o}tzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented. Teichm{\"u}ller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers. The name Tissot is missing from all the known historical reports on quasiconformal mappings. In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin. The final version of this article will appear in Vol. VII of the Handbook of Teichm{\"u}ller Theory (European Mathematical Society Publishing House, 2020)., Comment: arXiv admin note: text overlap with arXiv:1612.00279

International audience; Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name Tissot indicatrix, and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces. His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Grötzsch, Lavrentieff, Ahlfors and Teichmüller. Grötzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented. Teichm\"uller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers. The name Tissot is missing from all the known historical reports on quasiconformal mappings. In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin.

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 distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. 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 Kor\'anyi distances. This extends results due to Astala--Gehring, Astala--Koskela, Koskela and Bonk--Koskela--Rohde., Comment: 40 pages, updated version, presentation reorganized, major change: quasisymmetry used instead of the modulus argument; to appear in Ann. Mat. Pure Appl

International audience; We survey the life and work of Mikhaïl Lavrentieff, one of the main founders of the theory of quasiconformal mappings, with an emphasis on his training years at the famous Moscow school of theory of functions founded by Luzin. We also mention the major applications of quasiconformal mappings that Lavrentieff developed in the physical sciences. At the same time, we review several connected historical events, including the sad fate of Luzin during Stalin's period, the birth of the scientific Siberian center of Akademgorodok, and the role that Lavrentieff played in the organization of science and research in the Soviet Union.

International audience; Herbert Grötzsch is the main founder of the theory of quasiconformal mappings. In a series of papers written between 1928 and 1932, he introduced these mappings as a natural generalization of conformal mappings and he developed their main properties. He saw that several problems in conformal geometry naturally lead to problems on quasiconformal mappings. In this chapter, we review five of his papers that show the progress of his work from conformal to quasiconformal geometry. The five papers are the following: (1) ``Über einige Extremalprobleme der konformen Abbildung" (On some extremal problems of the conformal mapping), published in 1928; (2) ``Über einige Extremalprobleme der konformen Abbildung. II" (On some extremal problems of the conformal mapping. II), published in 1928; (3) ``Über die Verzerrung bei schlichten nichtkonformen Abbildungen und \"uber eine damit zusammenhängende Erweiterung des Picardschen Satzes." (On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem) \cite{gr3}, published in 1928; (4) ``Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche" (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions), published in 1930; (5) ``Über möglichst konforme Abbildungen von schlichten Bereichen" (On closest-to-conformal mappings of schlicht domains), published in 1932. Translations of these five papers into English are included in Volume VII of the Handbook. In this paper, we present the main results of these papers, providing introductory remarks, establishing the connections between them, and explaining the background. We shall also indicate some relations with works of Lavrentieff and Teichm\"uller.

International audience; This is a commentary on Teichmüller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry, in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a quasiconformal map. The two results find wide applications in value distribution theory, where they allow the efficient use of moduli of doubly connected domains and of quasiconformal mappings. Teichmüller's paper also contains a thorough development of the theory of conformal invariants of doubly connected domains.

For a domain D D in R ¯ n \bar {\mathbb R}^n , the modulus metric is defined by μ D ( x , y ) = inf γ cap ( D , γ ) \mu _D(x,y)=\inf _\gamma \textrm {cap}(D,\gamma ) , where the infimum is taken over all curves γ \gamma in D D joining x x to y y , and “cap" denotes the conformal capacity of the condensers. It has been conjectured by J. Ferrand, G. J. Martin, and M. Vuorinen that isometries in the modulus metric are conformal mappings. We prove the conjecture when n = 2 n=2 . In higher dimensions, we prove that isometries are quasiconformal mappings.

A measurable function $\mu$ on the unit disk $\mathbb{D}$ of the complex plane with $\|\mu\|_\infty, Comment: 7 pages, 1 figure

We study the Dirichlet problem with continuous boundary data in simply connected domains D of the complex plane for the semi-linear partial differential equations whose linear part has the divergent form. We prove that if a Jordan domain D satisfies the so-called quasihyperbolic boundary condition, then the problem has regular (continuous) weak solutions whose first generalized derivatives by Sobolev are integrable in the second degree. We give a suitable example of a Jordan domain with the quasihyperbolic boundary condition that fails to satisfy both the well-known (A)-condition and the outer cone condition. We also extend these results to some non-Jordan domains in terms of the prime ends by Caratheodory. The proofs are based on our factorization theorem established earlier. This theorem allows us to represent solutions of the semilinear equations in the form of composition of solutions of the corresponding quasilinear Poisson equation in the unit disk and quasiconformal mapping of D onto the unit disk generated by the measurable matrix function of coefficients. In the end we give applications to relevant problems of mathematical physics in anisotropic inhomogeneous media.

Herbert Grötzsch is the main founder of the theory of quasiconformal mappings. In a series of papers written between 1928 and 1932, he introduced these mappings as a natural generalization of conformal mappings and he developed their main properties. He saw that several problems in conformal geometry naturally lead to problems on quasiconformal mappings. In this chapter, we review five of his papers that show the progress of his work from conformal to quasiconformal geometry. The five papers are the following: (1) ``Über einige Extremalprobleme der konformen Abbildung" (On some extremal problems of the conformal mapping), published in 1928; (2) ``Über einige Extremalprobleme der konformen Abbildung. II" (On some extremal problems of the conformal mapping. II), published in 1928; (3) ``Über die Verzerrung bei schlichten nichtkonformen Abbildungen und \"uber eine damit zusammenhängende Erweiterung des Picardschen Satzes." (On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem) \cite{gr3}, published in 1928; (4) ``Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche" (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions), published in 1930; (5) ``Über möglichst konforme Abbildungen von schlichten Bereichen" (On closest-to-conformal mappings of schlicht domains), published in 1932. Translations of these five papers into English are included in Volume VII of the Handbook. In this paper, we present the main results of these papers, providing introductory remarks, establishing the connections between them, and explaining the background. We shall also indicate some relations with works of Lavrentieff and Teichmüller. The final version of this paper will appear in Vol. VII of the Handbook of Teichmüller theory.

We introduce a complex Banach manifold structure on the space of normalized symmetric homeomorphisms on the real line.

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) .

Retinotopic mapping, i.e., the mapping between visual inputs on the retina and neuronal activations in cortical visual areas, is one of the central topics in visual neuroscience. For human observers, the mapping is obtained by analyzing functional magnetic resonance imaging (fMRI) signals of cortical responses to slowly moving visual stimuli on the retina. Although it is well known from neurophysiology that the mapping is topological (i.e., the topology of neighborhood connectivity is preserved) within each visual area, retinotopic maps derived from the state-of-the-art methods are often not topological because of the low signal-to-noise ratio and spatial resolution of fMRI. The violation of topological condition is most severe in cortical regions corresponding to the neighborhood of the fovea (e.g., < 1 degree eccentricity in the Human Connectome Project (HCP) dataset), significantly impeding accurate analysis of retinotopic maps. This study aims to directly model the topological condition and generate topology-preserving and smooth retinotopic maps. Specifically, we adopted the Beltrami coefficient, a metric of quasiconformal mapping, to define the topological condition, developed a mathematical model to quantify topological smoothing as a constrained optimization problem, and elaborated an efficient numerical method to solve the problem. The method was then applied to V1, V2, and V3 simultaneously in the HCP dataset. Experiments with both simulated and real retinotopy data demonstrated that the proposed method could generate topological and smooth retinotopic maps., Author summary Retinotopic maps of human observers derived from state-of-the-art methods are often not topological because of the low signal-to-noise ratio and spatial resolution of fMRI. The proposed topological smoothing method can generate topology-preserving and smooth retinotopic maps in V1, V2, and V3 simultaneously from retinotopy fMRI data.

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 this paper we study the hyperbolic and Euclidean area distortion of measurable sets under some classes of K-quasiconformal mappings from the upper half-plane and the unit disk onto themselves, respectively.

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.