The theorem generalizes the riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. In addition to the references we mentioned above, there is a partial list of references 3, 11, 12, 20, 21 about holomorphic motions and quasiconformal mapping theory. Quasiconformal maps in metric spaces with controlled geometry juha heinonen university of michigan ann arbor, mi, u. An improved bound for sullivans convex hull theorem, proceedings of the london math. Quasiconformal teichmuller theory as an analytical foundation for. Teichmullers work on quasiconformal mappings 43 references 55 1. Ahlfors used quasiconformal mappings in his geometric approach to nevanlinnas value distribution theory. Preprint 1 computing quasiconformal maps on riemann. Quasiconformal homeomorphisms and dynamics iii citeseerx. Quasiconformal maps can be viewed as a generalization of conformal. Quasiconformal mapping encyclopedia of mathematics. This volume uses quasiconformal mapping to give a unified and uptodate treatment of tx.
An introduction to the theory of higher dimensional. A survey of the very important weilpetersson metric can be found in w. Free worlds best collection of great games pdf download. Download for offline reading, highlight, bookmark or take notes while you read an introduction to mathematical analysis for economic theory and econometrics. This book offers a modern, uptodate introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in. Teichmuller theory and quasiconformal deformations of geometrically finite groups maskits combination theorems mcmullens proof of kras theta conjecture compactness of ahm for acylindrical m. Teichmuller spaces are central objects in geometry and complex analysis today, with deep connections to quasiconformal mappings in particular. Convergence of an iterative algorithm for teichm\ uller maps via. The inverse of a kquasiconformal homeomorphism is kquasiconformal. Quasiconformal maps and teichmuller theoryextremal mappings, overview article pdf available in bulletin classe des sciences mathematiques et natturalles 14538 january 20 with 439 reads. This equation motivates our definition of quasiconformal mapping. He also coined the term quasiconformal inhis1935workonuberlagerungs. In order to study the geometry of the quasiconformal conjugacy class of f in ratd and ratd, mcmullen and sullivan introduced in ms98 the dynamical teichmuller space of a rational map f, as a dynamical analogue of the teichmuller theory of surfaces see gl00 and hub06 for an introduction to teichmuller theory. Click download or read online button to get an introduction to the theory of higher dimensional quasiconformal mappings book.
Jul 17, 2019 we define a universal teichmuller space for locally quasiconformal mappings whose dilatation grows not faster than a certain rate. The reader may refer to 5,18,23,27 for a description of the requisite gibbs measuretheory, to14forthescalingfunctiontheory, andto1,2,4,7,8,11,12,14,19,26 for the quasiconformal mapping theory and teichmuller. Martin bridgeman morrissey college of arts and sciences. Dirichlet spaces of domains bounded by quasicircles. Transactions of the american mathematical society volume 365, number 12, december 20, pages 65176541 s 000299472058928 article electronically published on may 30, 20. Numerical analysis of quasiconformal mappings 95 numerical computation of quasiconformal mapping is an important aspect to its applica tions.
Oct 20, 20 a numerical method for computing the extremal teichmuller map between multiplyconnected domains is presented. The quasiconformal maps enjoy a variety of important properties see the recent book for a modern treatment from the point of view of pde theory or the classic treatise and are often a useful tool for situations, where conformal maps turn out to be too rigid. The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. Download for offline reading, highlight, bookmark or take notes while you read analysis ii. There is a map from teichmuller space to the moduli space of riemann. Quasiconformal homeomorphisms and dynamics iii harvard.
We show that there exists a universal constant kc so that every. String theory and quasiconformal maps springerlink. Quasiconformal maps are generalizations of conformal maps. The best possibility of gluing quasiconformal mappings is investigated. 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. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w.
Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000. Read or download quasiconformal teichmuller theory pdf. It is illustrated with diagrams, examples, and exercises for the reader. Conformal invariants, quasiconformal maps, and special functions. Quasiconformal mappings, riemann surfaces, and teichmuller. His approach to quasiconformal maps is more friendly to those who shy away from the hard analysis approach usually taken on the subject. In this paper, we introduce a special class of quasiconformal maps called the teichmuller maps for planar domain parameterization.
The patterson and bersgreenberg isomorphism theorems 29 2. Emphasis is placed on parts of the theory applicable. The fenchelnielsen coordinates so named after werner fenchel and jakob nielsen on the teichmuller space are associated to a pants decomposition of the surface. Quasiconformal maps and teichmuller theory banjotooie primas official strategy guide irons in the fire the business history of the tayloe family and virginias gentry, 17001860. 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. There are more than 1 million books that have been enjoyed by people from all over the world. First we touch very briefly on the foundations of deformation theory, which is given by the theory of quasiconformal maps. Quasiconformal mappings, riemann surfaces, and teichmuller spaces. 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. Riemann surface is a connected 1dimensional complex manifold.
Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type. 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. This newinpaperback advent to topology emphasizes a geometrical strategy with a spotlight on surfaces. Quasiconformal maps and the mapping theorem quasiconformal maps form a branch of complex analysis.
This project is supported in part by nsf of china60575004, 10231040, nsf of guangdong, grants from the ministry of education of chinancet040791 and grants from sun yatsen university. Given two multiplyconnected domains, there exists a unique teichmuller map tmap between them minimizing the conformality distortion. Markovic quasiconformal maps and teichmuller theory. Quasiconformal maps and teichmuller theory alastair. A quasiconformal mapping between two riemann surfaces is a. This volume includes a wide range of papers on teichmuller theory and related areas. We present an outline of the theory of universal teichmuller space, viewed as part of the theory of qs, the space of quasisymmetric homeomorphisms of a circle. Quasiconformal maps and teichmuller theory oxford graduate texts in mathematics book 11. The teichmuller spaces of riemann surfaces are defined as quotient spaces of the spaces of beltrami differentials, and the teichmuller distances are defined to measure quasiconformal deformations between the riemann surfaces representing points in the teichmuller spaces. For example, the mapping theorem is proved without ever mentioning calderon zygmund theory. It was conjectured by schoen that any quasisymmetric homeomorphism of the circle has a unique quasiconformal harmonic extension to the hyperbolic plane, see richard m. Quasiconformal mappings, from ptolemys geography to the. Teichmuller theory and applications to geometry, topology, and dynamics hardcover june 6, 2006. Teichmuller theory and applications to geometry, topology.
This reduces the problem of quantization of string theory to the quantization of the space of smooth loops. Earle, from october 23, 2010, in syracuse, new york. Mappings, quasiconformal homogeneity, and nonamenable covers of riemann surfaces petra bonferttaylor, gaven martin, alan w. In mathematics, the teichmuller space t s \displaystyle ts \displaystyle t s of a real. Quasiconformal maps and teichmuller theoryextremal mappings, overview. Pdf quasiconformal maps and teichmuller theoryextremal. If is differentiable at the point, then the linear mapping transforms a ball of the tangent space into an ellipsoid the ratio of the.
Paralleling the classical teichmuller theory, we prove results of existence and uniqueness for extremal mappings in the generalized teichmuller class. 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. The phase space of the closed string theory may be identified with the space of smooth loops. A homeomorphism is 1quasiconformal if and only if it is conformal. The universal teichmuller space t, introduced by ahlfors and bers, plays a key role in the theory of quasiconformal maps and riemann surfaces. Accepted version quasiconformal maps have become an important class of homeomorphisms, for they arise in many elds of mathematics, such as pdes, teichmuller theory, hyperbolic geometry, complex dynamics etc. They are somehow rather magical, with properties that seem contradictory. E is a compact subset of s, where kfs\e is the maximal dilatation of fs\e. Gehring friend, mentor, and inadvertent matchmaker. The teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. 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. An ample information on the quadratic differentials and quasiconformal mappings can be found in. Harmonic maps have also played a role in compactifications and parametrizations of teichmuller space, via classical results of mike wolf, who gave a compactification in terms of hyperbolic harmonic maps.
Jan 22, 2016 the theory of quasiconformal mappings plays an important role in teichmuller theory. The proof depends on the teichmuller theory developed. In mathematics, the theory of quasiconformal maps and teichmuller maps has a long history and provides a nice introduction about the subject. These notes were written for some expository talks on teichmuller theory, given at. The connection between extremal and harmonic maps will be explained in section 2. Teichmuller s work on quasiconformal mappings 43 references 55 1. But the symplectic form of string theory is correctly defined not only on the. We will show that it is a complete metric space, and that it is homeomorphic to euclidean space.
This handbook is an essential reference for graduate students and researchers interested in teichmuller theory and its ramifications, in particular for mathematicians working in topology, geometry. Computing extremal teichmuller map of multiplyconnected. Identities on hyperbolic manifolds, the handbook of teichmuller theory, vol. However, the original ideas of teichmuller extended much further to the theory of quasiconformal maps and teichmuller spaces where quadratic di erentials were used for construction of extremal quasiconformal mappings and the teichmuller metric on the moduli space of riemann surfaces. Schatz, teichmuller theory for surfaces with boundary, to appear. Quasiconformal maps and teichmuller theory bookask. Feb 17, 2009 an introduction to mathematical analysis for economic theory and econometrics ebook written by dean corbae, maxwell b. Gluing quasiconformal mappings in the plane guowu yao abstract in this paper, several versions of gluing theorems for quasiconformal mappings in the plane are obtained. Gluing quasiconformal mappings in the complex plane 3 denote by bels the banach space of beltrami di. The grothendieck teichmuller theory dessins denfants, grothendiecks reconstruction principle, and the teichmuller theory of the solenoid. To describe the rational maps g quasiconformally conjugate to a given. This newinpaperback creation to topology emphasizes a geometrical procedure with a spotlight on surfaces.
Harmonic measure and polynomial julia sets binder, i. Download quasiconformal teichmuller theory by frederick p. They started out being used in nevanlinnas value distribution theory but now form a fundamental component of geometric function theory. Always update books hourly, if not looking, search in the book search column. For n 3, there are no known expressions and so we rely on qualitative properties and inequalities. Handbook of teichm ller theory ebook download free pdf. Bers embedding and the bers boundary of teichmuller.
This volume contains the proceedings of the ams special session on quasiconformal mappings, riemann surfaces, and teichmuller spaces, held in honor of clifford j. Get ebooks handbook of teichm ller theory on pdf, epub, tuebl, mobi and audiobook for free. Schoen, the role of harmonic mappings in rigidity and deformation problems, complex geometry osaka, 1990, lecture notes in pure and appl. An orientationpreserving mapping is called quasiconformal or a mapping with bounded distortion if. Complex structures, jacobians and the weil petersson form 31 lecture 4. The theory continues to be active, with numerous studies of the complex structure of. Staubach, quasiconformal teichmuller theory as an analytic foundation for twodimensional conformal field theory, in lie algebras, vertex operator algebras, and related topics, contemporary mathematics, vol.
Planar domain parameterization for isogeometric analysis. This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\. What is a good introduction to teichmuller theory, mapping class groups etc. Although elements of qs act in one dimension, most results depend on a twodimensional pr. Every proof comes with geometric intuition, aided by many diagrams. I found the subject dicult to learn, mainly because i had a hard time appreciating how smooth the maps are. Discrete and continuous dynamical systems, number, july. As an application, we provide a new short proof of the gluing theorem obtained by jiang and qi. Variational principles in the theory of quasiconformal maps. The reader who is interested in holomorphic motions may refer to those papers and books. Newest quasiconformalmaps questions mathematics stack. The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. An introduction to mathematical analysis for economic theory.
1049 719 438 992 140 1030 619 1496 958 1314 480 140 1326 1240 1182 467 1345 1060 1102 1249 1027 536 771 955 346 820 1182 1315 1357 835 1396 1237 368 1269 1261 322 392 491 616 928 842 1055