Introduction much of the theory of riemann surfaces boils down to the following theorem, the twodimensional equivalent of thurstons. Regarding teichmullers work, ahlfors, in his 1953 paper on the development of the theory of conformal mapping and riemann surfaces through. This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. Quasiconformal maps and teichmuller theory alastair. The geometry of the universal teichmuller space and the euler. Download quasiconformal teichmuller theory by frederick p. Pdf quasiconformal maps and teichmuller theoryextremal. Quasiconformal homeomorphisms and dynamics iii harvard. Fletcher, alastair is the author of quasiconformal maps and teichmuller theory, published 2006 under isbn 9780198569268 and isbn 0198569262.
A theme of much of teichmuller theory is to compare the complex analytic theory where points are given by riemann surfaces and the hyperbolic geometry. Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type. Quasiconformal teichmuller theory and nag, the complex analytic theory of teichmuller spaces. Suppose that d is a simply connected domain in c that is. The few papers of teichmuller himself that i read are also exciting, but my poor knowledge of german does not allow me to read all. The theorem generalizes the riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. This newinpaperback creation to topology emphasizes a geometrical procedure with a spotlight on surfaces. If youre more analytically minded, i recommend gardiner and lakic, quasiconformal teichmuller theory and nag, the complex analytic theory of teichmuller spaces.
Quasiconformal teichmuller theory as an analytical foundation for. Online pdf quasiconformal maps and teichmuller theory. Hubbard teichmuller theory pdf vector calculus, linear algebra, and differential forms, a unified approach with barbara burke hubbard. Interaction with analysis, geometry and number theory from 4 to 16 july, 2011. Apr 29, 2020 hubbard teichmuller theory pdf vector calculus, linear algebra, and differential forms, a unified approach with barbara burke hubbard.
It is illustrated with diagrams, examples, and exercises. Geometry of teichmuller space with the teichmuller metric. An orientationpreserving mapping is called quasiconformal or a mapping with bounded distortion if. Beginning with the classical riemann mapping theorem, there is a lot of existence theorems for. Geometric function theory is that part of complex analysis which covers the theory of conformal and quasiconformal mappings. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal teichmuller theory and geometric function theory.
Use 3 disjoint mutually nonseparating geodesics in h2 such that hyperbolic distances between them are the numbers. This book offers a modern, uptodate introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past. The underlying geometric category is based on the moduli space of riemann surfaces with parametrized boundary components and the sewing operation. Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000. The possibility arising from this of extending a quasiconformal automorphism of lobachevskii space to a quasiconformal mapping of the absolute of this space, in accordance with the fact that a quasiconformal mapping of the sphere is conformal and is the trace of a conformal automorphism of the ball, lies at the basis of the proof of the. By clicking post your answer, you acknowledge that. The classical parts of the theory include teichmuller s theorem on the existence and uniqueness of an extremal quasiconformal mapping in a given homotopy class of mappings between riemann surfaces, the theorems of bers and ahlfors on the completeness of poincare theta series for general fuchsian groups and the approximation of integrable. A panoramic overview of interuniversal teichmuller theory. Quasiconformal maps and teichmuller theory 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. Quasiconformal teichmuller theory as an analytical foundation for twodimensional conformal field theory. Teichmuller space encodes the possible number of ways in which you can deform complex structure on a space.
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. Teichmullers work on quasiconformal mappings 43 references 55 1. The patterson and bersgreenberg isomorphism theorems 29 2. What is a good introduction to teichmuller theory, mapping class groups etc. Quasiconformal maps and teichmuller theory book, 2007. Assuming some prior familiarity with riemann surfaces and hyperbolic geometry, the text is illustrated throughout by examples and exercises. Let me try to list briefly teichmullers own contribution to teichmuller theory. Note that teichmullers theory is related to complex analysis, hyperbolic geometry, the.
The mantra is studying objects in families gives important information about the individual objects. An introduction to teichmuller theory notes from a course given by dick canary, fall 2015 november 23, 2015 1 an overview of hyperbolic geometry 1. The fenchelnielsen coordinates so named after werner fenchel and jakob nielsen on the teichmuller space are associated to a pants decomposition of the surface. Interuniversal teichmuller theory i construction of hodge theaters shinichi mochizuki may2020 abstract. If is differentiable at the point, then the linear mapping transforms a ball of the tangent space into an ellipsoid the ratio of the major to. An introduction to the theory of higher dimensional.
Teichmuller space, which give a very nice description of the space for certain. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal. 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. Teichmuller theory extends in a natural way to onedimensional complex orbifolds, lo cally modeled on riemann surfaces modulo finite groups. This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\. The universal teichmuller space appears in many areas of mathematics and mathematical physics. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal teichmuller. Aimed at graduates with a grounding in complex analysis, this book provides an accessible introduction to the theory of quasiconformal maps and teichmuller theory.
The classical parts of the theory include teichmullers theorem on the existence and uniqueness of an extremal quasiconformal mapping in a given homotopy class of mappings between riemann surfaces, the theorems of bers and ahlfors on the completeness of poincare theta series for general fuchsian groups and the approximation of integrable. This thesis contains work on analytical aspects of teichmuller. 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. Raction on these moduli spaces, the kontsevichzorich cocycle over the teichmuller geodesic. On the inclusion of the quasiconformal teichmuller space into. Pdf quasiconformal teichmuller theory as an analytical. The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. Complex structures, jacobians and the weil petersson form 31 lecture 4. Chapters 1 and 2 contain standard results about quasiconformal mappings and teichmuller. Then there exists a complete hyperbolic structure x with geodesic boundary on the pair of pants s2 \ 3 discs such that lengths of boundary curves are l1,l2,l3. The geometry of the universal teichmuller space and the. 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. Athanase papadopoulos teichmuller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different.
Sheretov 1 mathematical notes of the academy of sciences of the ussr volume 16, pages 714 718 1974 cite this article. Quasiconformal mappings and their applications download. Courant institute of mathematical sciences, new york university. Much of the theory of riemann surfaces boils down to the following theorem.
By david radnell, eric schippers and wolfgang staubach. Quasiconformal maps and teichmuller theoryextremal mappings, overview article pdf available in bulletin classe des sciences mathematiques et natturalles 14538 january 20 with 439 reads. A survey of the very important weilpetersson metric can be found in w. Teichmuller theory mathematisches institut universitat bonn. May 02, 2016 the functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. By the theory of quasiconformal mappings, any complex structure on a sur. 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.
Since the correspondence is given by the uniformization theorem. Quasiconformal mappings and teichmullers theorem by bers, lipman. The functorial mathematical definition of conformal field theory was. Quasiconformal teichmuller theory as an analytical foundation. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w. Instead of power series, integral representations, or algebra ic techniques, the theory relies on singular integrals, geometricmeasuretheory,andsobolevspaces. In particular, the obstruction to deforming a quasiconformal conjugacy between two rational maps to a conformal conjugacy is measured by. Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable riemann mapping theorem, proved by lars ahlfors and lipman bers. Quasiconformal teichmuller theory as an analytical.
Chapter 3 presents some results on the distribution of zeros 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. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of riemann surfaces with socalled weilpetersson class parametrizations. 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. The teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. Quasiconformal teichmuller theory as an analytical foundation for twodimensional conformal field theory chapter pdf available january 2017 with 40 reads how we measure reads. Quasiconformal mapping encyclopedia of mathematics. Quasiconformal maps and teichmuller theory 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. Quasiconformal teichmuller theory thoery nag, the complex analytic theory of teichmuller spaces. This relationship yielded a natural metric on the parameter space of teichmuller his last paper in this area was devoted to the complex structure of this space. Quasiconformal mappings, from ptolemys geography to the work of.
Gardiner and jun hu winter, 2005, hri, allahabad, india. Bers embedding and the bers boundary of teichmuller. Abstract we present a brief exposition of teichmullers. Teichmuller theory and quadratic differentials pure. Note that a 1quasiconformal map is conformal by this definition.
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