These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic. His approach to quasiconformal maps is more friendly to those who shy away from the hard analysis approach usually taken on the subject. Nieminen, tomi, illinois journal of mathematics, 2009. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w. 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. 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. An introduction to the theory of higherdimensional. 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. We show that there exists a universal constant kc so that every. I would like to ask for some detailed reference for quantum teichmuller theory, better in a mathematical taste. He also coined the term quasiconformalinhis1935workonuberlagerungs. 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. Denote by compgi c compg the subspace of all metrics g compg such that k 1. This volume contains the proceedings of the ams special session on quasiconformal mappings, riemann surfaces, and teichmuller spaces, held in honor of clifford j.
Interuniversal teichmuller theory i construction of hodge theaters shinichi mochizuki april2020 abstract. Ergodic theory of teichmuller flow with respect to masurveech measure. The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. Athanase papadopoulos teichmuller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different. Preface xvn this result is called teichmuller contraction. Geometry of teichmuller space with the teichmuller metric. Preprint 1 computing quasiconformal maps on riemann surfaces using discrete curvature flow w. An introduction to the theory of higherdimensional quasiconformal mappings. The theory of qc maps was designed as an optimal flexibilization of the theory of. The inverse of a k quasiconformal homeomorphism is k quasiconformal. Teichmuller theory, for general surfaces, there is an one to one correspondence between the set of quasi. 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. Quasiconformal maps and teichmuller theoryextremal mappings, overview. Riemann surface is a connected 1dimensional complex manifold.
The fenchelnielsen coordinates so named after werner fenchel and jakob nielsen on the teichmuller space are associated to a pants decomposition of the surface. Teichmuller theory mathematisches institut universitat bonn. Computation of quasiconformal surface maps using discrete beltrami flow abstract. Two quasiconformal maps f from s to fs and g from s to. Teichmuller theory and applications to geometry, topology, and dynamics hardcover june 6, 2006. Earle, from october 23, 2010, in syracuse, new york. 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. Poincares theorem and teichmuller theory for open surfaces 357 return now to m2 of the above topological type. The intimate relationship between quasiconformal maps and beltrami di erentials is revealed by the following profound theorem of l. Louis funar, athanase papadopoulos and bob penner abstracts of talks norbert acampo ideal triangulations and variation of the complex structure by monodromy abstract.
Quasiconformal mappings and teichmuller s theorem scholars choice edition bers, lipman on. On the minimal factorization of the higher dimensional quasiconformal maps3 suppose 0 and let s. 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. Quasiconformal mappings, riemann surfaces, and teichmuller.
Bers,riemanns mapping theorem for variable metrics. They are somehow rather magical, with properties that seem contradictory. In other words, every quasiconformal map can be fully determined by the beltrami differentials and is unique up to a. Quasiconformal mappings, riemann surfaces, and teichmuller spaces.
The regularity of harmonic maps into spheres and applications to bernsteing problems jost, jurgen, xin, yuanlong, and yang, ling, journal of differential geometry, 2012. A quasiconformal mapping class is a homotopy equivalence class. The theory continues to be active, with numerous studies of the complex structure of. Earle,teichmuller theory, indiscrete groups and automorphic functions. Quasiconformal mappings and teichmullers theorem scholars choice edition bers, lipman on. The set of 1 quasiconformal maps forms a group under composition. Teichmuller theory is one of those few wonderful subjects which bring together, at an equally important. Emphasis is placed on parts of the theory applicable. The manipulation of surface homeomorphisms is an important aspect in 3d modeling and surface processing. Quasiconformal mappings, from ptolemys geography to the work of. Let us consider two riemann surface structures s 0 and s 1 on a xed topological compact surface of genus g 1.
And the theory of derivatives in the sense of distributions. This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\. Moduli of riemann surfaces, teichmuller theory see also 14h15, 30fxx. Quasiconformal harmonic diffeomorphism and the universal teichmuller space. Now plane quasiconformal maps play important roles in a. They started out being used in nevanlinnas value distribution theory but now form a fundamental component of geometric function theory. The mantra is studying objects in families gives important information about the individual objects.
What is a good introduction to teichmuller theory, mapping class groups etc. I read a little bit on kashaevs or chekhov and focks, but find that i need to fill. 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. A quasiconformal mapping between two riemann surfaces is a. In addition to being of intrinsic interest, teichmuller theory has found wideranging.
Gehring friend, mentor, and inadvertent matchmaker. Every homeomorphic surface map can be considered as a quasiconformal map, with its local nonconformal distortion given by its beltrami. Quasiconformal mappings and teichmuller s theorem scholars choice edition. Quasiconformal mappings and teichmullers theorem scholars choice edition. Ahlfors used quasiconformal mappings in his geometric approach to nevanlinnas value distribution theory. 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 teichmuller s theorem by bers, lipman. Newest quasiconformalmaps questions mathematics stack. Over 10 million scientific documents at your fingertips. Teichmullers work on quasiconformal mappings 43 references 55 1.
Grad riemann surface algebraic geometry complex analysis differential geometry ergodic theory geometry. Pdf quasiconformal maps and teichmuller theoryextremal. Complex structures, jacobians and the weil petersson form 31 lecture 4. Gluing quasiconformal mappings in the complex plane 3 denote by bels the banach space of beltrami di. Quasiconformal mappings and teichmullers theorem scholars.
Quasiconformal maps in metric spaces with controlled geometry. 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. Teichmuller space encodes the possible number of ways in which you can deform complex structure on a space. For example, the mapping theorem is proved without ever mentioning calderon zygmund theory. We will show that it is a complete metric space, and that it is homeomorphic to euclidean space. I found the subject dicult to learn, mainly because i had a hard time appreciating how smooth the maps are. H is an orientation preserving homeomorphism which is absolutely continuous on lines and which satis. Quasiconformal mapping encyclopedia of mathematics. The geometry of the universal teichmuller space and the euler. This volume uses quasiconformal mapping to give a unified and uptodate treatment of tx. Quasiconformal mappings and teichmullers theorem by bers, lipman.
Gu abstract surface mapping plays an important role in geometric processing. Quasiconformal maps and the mapping theorem quasiconformal maps form a branch of complex analysis. Foundations for the theory of quasiconformal mappings on the heisenberg group. A survey of the very important weilpetersson metric can be found in w. The geometry of the universal teichmuller space and the. Introduction to riemann surfaces and teichmuller theory mpi mis. This text is an expanded version of the lecture notes of a minicourse with the same. The universal teichmuller space appears in many areas of mathematics and mathematical physics. Quasiconformal mappings, from ptolemys geography to the work. Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type. In mathematics, the teichmuller space t s \displaystyle ts \displaystyle ts of a real. Energy quantization for harmonic maps lin, fanghua and riviere, tristan, duke mathematical journal, 2002. The moduli space mg of the riemann surfaces of genus g is the quotient space of the teichmuller. Mappings, quasiconformal homogeneity, and nonamenable covers of riemann surfaces petra bonferttaylor, gaven martin, alan w.
Teichmuller theory and applications to geometry, topology. There is also a version of teichmuller contraction for atr. 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. Contemporary mathematics modular groups acting on in.
Volumes 2 through 4 prove four theorems by william thurston. Gluing quasiconformal mappings in the plane guowu yao abstract. Teichmuller theory, for general surfaces, there is an one to. Preprint 1 computing quasiconformal maps on riemann. We aim to discuss this and a more quantitative version of theorem 1. Every proof comes with geometric intuition, aided by many diagrams. Advances in teichmuller theory february 4 8, 20 international erwin schr odinger institute for mathematical physics esi university of vienna, vienna organized by. Preprint 1 computing quasiconformal maps on riemann surfaces. In some cases they can be used to obtain new properties. By the theory of quasiconformal mappings, any complex structure on a sur.
The goal of this paper is to present some of the more recent activity us. Lectures on quasiconformal mappings lars valerian ahlfors. So2 the special feature of the complex dimension 1 is that the classes of locally biholomorphic and locally conformal maps in subdomains of c. Hyperbolic geometry and fenchelnielsen coordinates. The teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. Quasiconformal mappings, from ptolemys geography to the. The space of k quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact. Quasiconformal maps in metric spaces with controlled.
Quasiconformal maps and teichmuller theory alastair. Quasiconformal maps are generalizations of conformal maps. There is a map from teichmuller space to the moduli space of riemann. 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. Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000. Ahlfors,lectures on quasiconformal mappings, van nostrand, princeton. 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. Introduction to teichmuller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards giovanni forni and carlos matheus abstract. Liftings of holomorphic maps into teichmuller spaces. 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 spaces and holomorphic motions springerlink. This volume includes a wide range of papers on teichmuller theory and related areas. Teichmuller space is, in some sense, the space of all riemann surfaces complex manifolds of. The goal of this course is to present a brief proof of the original teichmuller.
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