Jul 14, 2006 lars ahlforss lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type. Moduli of riemann surfaces, teichmuller theory see also 14h15, 30fxx. Let us consider two riemann surface structures s 0 and s 1 on a xed topological compact surface of genus g 1. Teichmullers work on quasiconformal mappings 43 references 55 1. Ergodic theory of teichmuller flow with respect to masurveech measure. Teichmuller space encodes the possible number of ways in which you can deform complex structure on a space. They started out being used in nevanlinnas value distribution theory but now form a fundamental component of geometric function theory. Denote by compgi c compg the subspace of all metrics g compg such that k 1. A quasiconformal map is the uniquely extremal representative in its teichmuller class if and only if its beltrami coefficient \i is uniquely ex tremal in its infinitesimal class. Quasiconformal mappings and teichmullers theorem scholars choice edition. Quasiconformal mappings, riemann surfaces, and teichmuller. Every proof comes with geometric intuition, aided by many diagrams.
Pdf quasiconformal maps and teichmuller theoryextremal. Quasiconformal maps and the mapping theorem quasiconformal maps form a branch of complex analysis. The intimate relationship between quasiconformal maps and beltrami di erentials is revealed by the following profound theorem of l. This is a decomposition of into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in teichmuller space, and another real parameter called the. I read a little bit on kashaevs or chekhov and focks, but find that i need to fill. Hyperbolic geometry and fenchelnielsen coordinates. Over 10 million scientific documents at your fingertips. Geometry of teichmuller space with the teichmuller metric. Lectures on quasiconformal mappings lars valerian ahlfors. I found the subject dicult to learn, mainly because i had a hard time appreciating how smooth the maps are. What is a good introduction to teichmuller theory, mapping class groups etc. The fenchelnielsen coordinates so named after werner fenchel and jakob nielsen on the teichmuller space are associated to a pants decomposition of the surface. Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000.
Every homeomorphic surface map can be considered as a quasiconformal map, with its local nonconformal distortion given by its beltrami. This volume uses quasiconformal mapping to give a unified and uptodate treatment of tx. The theory of qc maps was designed as an optimal flexibilization of the theory of. The geometry of the universal teichmuller space and the euler. A quasiconformal mapping class is a homotopy equivalence class. Quasiconformal mapping encyclopedia of mathematics. Complex structures, jacobians and the weil petersson form 31 lecture 4. Lars ahlforss lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Quasiconformal mappings and teichmuller s theorem by bers, lipman. Introduction to riemann surfaces and teichmuller theory mpi mis. Quasiconformal harmonic diffeomorphism and the universal teichmuller space. Interuniversal teichmuller theory i construction of hodge theaters shinichi mochizuki april2020 abstract. Quasiconformal mappings, from ptolemys geography to the.
This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\. There is also a version of teichmuller contraction for atr. The moduli space mg of the riemann surfaces of genus g is the quotient space of the teichmuller. Foundations for the theory of quasiconformal mappings on the heisenberg group. They are somehow rather magical, with properties that seem contradictory. Newest quasiconformalmaps questions mathematics stack. The manipulation of surface homeomorphisms is an important aspect in 3d modeling and surface processing. The teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. Quasiconformal maps and teichmuller theoryextremal mappings, overview. Earle, from october 23, 2010, in syracuse, new york. Quasiconformal mappings, riemann surfaces, and teichmuller spaces. These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic.
Quasiconformal mappings, from ptolemys geography to the work of. An introduction to the theory of higherdimensional. The theory continues to be active, with numerous studies of the complex structure of. Geometry of teichmuller space with the teichmuller metric howard masur january 20, 2010 1 introduction the purpose of this chapter is to describe recent progress in the study of teichmuller geometry. On the minimal factorization of the higher dimensional quasiconformal maps3 suppose 0 and let s. Teichmuller theory and applications to geometry, topology.
Quasiconformal maps in metric spaces with controlled. Preprint 1 computing quasiconformal maps on riemann surfaces using discrete curvature flow w. For example, it is a special coadjoint orbit of the bottvirasoro group nag and verjovsky1990 and plays an important role in the theory of riemann surfaces, several complex variables, and. For example, it is a special coadjoint orbit of the bottvirasoro group nag and verjovsky1990 and plays an important role in the theory of riemann surfaces, several complex variables, and quasiconformal maps gardiner. Louis funar, athanase papadopoulos and bob penner abstracts of talks norbert acampo ideal triangulations and variation of the complex structure by monodromy abstract. Emphasis is placed on parts of the theory applicable. Quasiconformal mappings, from ptolemys geography to the work. The geometry of the universal teichmuller space and the. H is an orientation preserving homeomorphism which is absolutely continuous on lines and which satis. The set of 1 quasiconformal maps forms a group under composition. Two quasiconformal maps f from s to fs and g from s to. The regularity of harmonic maps into spheres and applications to bernsteing problems jost, jurgen, xin, yuanlong, and yang, ling, journal of differential geometry, 2012. Ahlfors,lectures on quasiconformal mappings, van nostrand, princeton. Computation of quasiconformal surface maps using discrete beltrami flow abstract.
Teichmuller theory mathematisches institut universitat bonn. In the same way that a quasiconformal mapping turns out to be a mobius transformation even without the a priori assumption that it is a homeomorphism, a quasiconformal mapping is locally homeomorphic as soon as its coefficient of quasiconformality is sufficiently close to 1. We show that there exists a universal constant kc so that every. Quasiconformal mappings and teichmullers theorem by bers, lipman. I would like to ask for some detailed reference for quantum teichmuller theory, better in a mathematical taste. Quasiconformal maps in metric spaces with controlled geometry. The space of k quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact. Quasiconformal mappings and teichmullers theorem scholars. The mantra is studying objects in families gives important information about the individual objects. We aim to discuss this and a more quantitative version of theorem 1. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w.
The inverse of a k quasiconformal homeomorphism is k quasiconformal. Preprint 1 computing quasiconformal maps on riemann surfaces. Introduction the roots of quasiconformal theory lie in geography, more precisely in the study of mappings from subsets of the sphere to the euclidean plane, and the attempts. Teichmuller theory, for general surfaces, there is an one to. Quasiconformal maps and teichmuller theory alastair. Gluing quasiconformal mappings in the complex plane 3 denote by bels the banach space of beltrami di. Teichmuller theory, for general surfaces, there is an one to one correspondence between the set of quasi. Ahlfors used quasiconformal mappings in his geometric approach to nevanlinnas value distribution theory. Teichmuller theory is one of those few wonderful subjects which bring together, at an equally important. This volume includes a wide range of papers on teichmuller theory and related areas.
Earle,teichmuller theory, indiscrete groups and automorphic functions. Liftings of holomorphic maps into teichmuller spaces. Volumes 2 through 4 prove four theorems by william thurston. Now plane quasiconformal maps play important roles in a. And the theory of derivatives in the sense of distributions. The universal teichmuller space appears in many areas of mathematics and mathematical physics. Preprint 1 computing quasiconformal maps on riemann. His approach to quasiconformal maps is more friendly to those who shy away from the hard analysis approach usually taken on the subject. A survey of the very important weilpetersson metric can be found in w. Each point in may be regarded as an isomorphism class of marked riemann surfaces, where a marking is an isotopy class of homeomorphisms from to itself. Bers,riemanns mapping theorem for variable metrics. Teichmuller spaces and holomorphic motions springerlink. Gehring friend, mentor, and inadvertent matchmaker. By the theory of quasiconformal mappings, any complex structure on a sur.
A quasiconformal mapping between two riemann surfaces is a. There is a map from teichmuller space to the moduli space of riemann. The goal of this course is to present a brief proof of the original teichmuller. Quasiconformal mappings and teichmullers theorem scholars choice edition bers, lipman on. Quasiconformal mappings and teichmuller s theorem scholars choice edition bers, lipman on. Introduction to teichmuller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards giovanni forni and carlos matheus abstract. This text is an expanded version of the lecture notes of a minicourse with the same. Nieminen, tomi, illinois journal of mathematics, 2009. Contemporary mathematics modular groups acting on in. Gluing quasiconformal mappings in the plane guowu yao abstract. Athanase papadopoulos teichmuller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different. We will show that it is a complete metric space, and that it is homeomorphic to euclidean space. Gu abstract surface mapping plays an important role in geometric processing.
An introduction to the theory of higherdimensional quasiconformal mappings. Energy quantization for harmonic maps lin, fanghua and riviere, tristan, duke mathematical journal, 2002. Mappings, quasiconformal homogeneity, and nonamenable covers of riemann surfaces petra bonferttaylor, gaven martin, alan w. Teichmuller theory and applications to geometry, topology, and dynamics hardcover june 6, 2006. Grad riemann surface algebraic geometry complex analysis differential geometry ergodic theory geometry. Introduction the roots of quasiconformal theory lie in geography, more precisely in the study of mappings from subsets of the sphere to the euclidean plane, and the attempts to nd the \best such mappings. Advances in teichmuller theory february 4 8, 20 international erwin schr odinger institute for mathematical physics esi university of vienna, vienna organized by. Quasiconformal maps are generalizations of conformal maps. The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. In other words, every quasiconformal map can be fully determined by the beltrami differentials and is unique up to a. Teichmuller theory the teichmuller space of a riemann surface is the space of all deformations of the complex structure, with a topological marking that insures that the teichmuller space is simply connected. In addition to being of intrinsic interest, teichmuller theory has found wideranging. The goal of this paper is to present some of the more recent activity us. Quasiconformal mappings and teichmuller s theorem scholars choice edition.
For example, the mapping theorem is proved without ever mentioning calderon zygmund theory. Preface xvn this result is called teichmuller contraction. Riemann surface is a connected 1dimensional complex manifold. After the introduction of quasiconformal mappings into the subject, the theory can be said to deal with classes consisting of quasiconformal mappings of a riemann surface. Poincares theorem and teichmuller theory for open surfaces 357 return now to m2 of the above topological type. Based on a series of graduate lectures given by vladimir markovic at the university of warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and teichmuller theory. In mathematics, the teichmuller space t s \displaystyle ts \displaystyle ts of a real. So2 the special feature of the complex dimension 1 is that the classes of locally biholomorphic and locally conformal maps in subdomains of c.
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