An orientationpreserving mapping is called quasiconformal or a mapping with bounded distortion if. 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. This volume contains the proceedings of the ams special session on quasiconformal mappings, riemann surfaces, and teichmuller spaces, held in honor of clifford j. Quasiconformal mappings and teichmullers theorem scholars. Quasiconformal mappings, from ptolemys geography to the work of. Quasiconformal mappings and teichmullers theorem scholars choice edition bers, lipman on. This section gives a summary of the basics of teichmull. Quasiconformal mappings and teichmullers theorem paperback august 24, 2017 by lipman bers author see all 7 formats and editions hide other formats and editions. 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. A more algebrogeometric approach to lusztigs total positivity has been developed. What is a good introduction to teichmuller theory, mapping class groups etc. Instead of power series, integral representations, or algebra ic techniques, the theory relies on singular integrals, geometricmeasuretheory,andsobolevspaces.
Jun 04, 2015 the goal of this paper is to develop an arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same. Even if this were proved, it wouldnt a priori be especially helpful if we. Teichmuller, which are the 20thcentury founders of the theory. The geometry of the universal teichmuller space and the euler. Introduction an orientation preserving homeomorphism f from a riemann surface x onto a riemann surface y is given. Quasiconformal maps and teichmuller theoryextremal mappings, overview article pdf available in bulletin classe des sciences mathematiques et natturalles 14538 january 20 with 439 reads. Introduction an orientation preserving homeomorphism ffrom a riemann surface x onto a riemann surface y is given. Athanase papadopoulos teichmuller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different. Interuniversal teichmuller theory ii hodgearakelovtheoretic evaluation shinichi mochizuki april2020 abstract. The fenchelnielsen coordinates so named after werner fenchel and jakob nielsen on the teichmuller space are associated to a pants decomposition of the surface. Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type. Quasiconformal mappings, from ptolemys geography to the work. Quasiconformal mapping encyclopedia of mathematics.
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. 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. Quasiconformal mappings and teichmuller s theorem by bers, lipman. Teichmuller space is, in some sense, the space of all riemann surfaces complex manifolds of dimension. Teichmuller theory and applications to geometry, topology, and dynamics volume 1. Abstract we present a brief exposition of teichmuller s. Quasiconformal maps and teichmuller theory alastair. Proceedings of the year on teichmuller theory hri, allahabad, volume 10, 2009, pages 195228 abstract we present a brief exposition of teichmullers theorem.
Aug 24, 2017 quasiconformal mappings and teichmuller s theorem bers, lipman on. By the theory of \base points at in nity we have a representation g q. The grothendieck teichmul ler conjecture states that g q dgt. Teichmullers work on quasiconformal mappings 43 references 55 1. The mantra is studying objects in families gives important information about the individual objects. Teichmuller theory interpretation of the decomposition of the tensor product of positive representations p p into positive representations p. A more algebrogeometric approach to lusztigs total positivity has been developed by fock and goncharov 17, and applied in the context of higher teichmuller theory. 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.
The geometry of the universal teichmuller space and the. Geometry of teichmuller space with the teichmuller metric. Teichmuller spaces for surfaces with cone points have come into play. Teichmuller theory and applications to geometry, topology. Working on my current research problem, teichmuller spaces for surfaces with cone points have come into play. Teichmuller theory and applications matrix editions. Stack exchange network stack exchange network consists of. Gus schrader uc berkeley quantum higher teichmuller theory. This volume includes a wide range of papers on teichmuller theory and related areas.
Along somewhat di erent lines we describe some recent important work of k. Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000. Quasiconformal teichmuller theory as an analytical foundation for twodimensional conformal field theory. 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. Interaction with analysis, geometry and number theory from 4 to 16 july, 2011. A quasiconformal mapping class is a homotopy equivalence 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. Abstract we present a brief exposition of teichmullers. Teichmuller theory is one of those few wonderful subjects which bring together, at an equally important. For example, it is a special coadjoint orbit of the bottvirasoro group nag and verjovsky1990 and plays. Then there exists a complete hyperbolic structure x with geodesic boundary on the pair of pants s2 \ 3 discs such that lengths of boundary. Kerckhoff receioed 8 december 1978 introduction teichmuller space is the space of conformal structures on a topological. Moduli of riemann surfaces, teichmuller theory see also 14h15, 30fxx secondary.
Teichmuller space encodes the possible number of ways in which you can deform complex structure on a space. 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. 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. Quasiconformal mappings, from ptolemys geography to the. Y 1 2 inf f lnk where k is the dilatation of a quasiconformal map f. Quasiconformal mappings and teichmuller s theorem scholars choice edition bers, lipman on. 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. Given a quasiconformal hyperbolic surface of nite type, s, the teichmull er space modeled on s, denoted t s, is the set of equivalence classes of. Pdf quasiconformal maps and teichmuller theoryextremal. Gardiner and jun hu winter, 2005, hri, allahabad, india. This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\.
Raction on these moduli spaces, the kontsevichzorich cocycle over the teichmuller geodesic. The authors have produced a formidable treatise on the modern theories of quasiconformal mappings, riemann surfaces and teichmuller. This volume uses quasiconformal mapping to give a unified and upto. Ra that gives a combinatorial description of the teichmuller metric. Quasiconformal mappings, riemann surfaces, and teichmuller spaces. Quasiconformal mappings, riemann surfaces, and teichmuller. Quasiconformal maps and teichmuller theoryextremal mappings, overview. The general problem of constructing a quasiconformal mapping from one simplyconnected domain onto another satisfying 2 was posed and solved by m. Teichmuller theory mathematisches institut universitat bonn. Quasiconformal teichmuller theory as an analytical foundation for. The theorem generalizes the riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows.
Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w. The discussion on quasiconformal mappings in chapter 4 is very important but might be one of the. The goal of this paper is to develop an arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the. The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Interuniversal teichmuller theory i construction of hodge theaters shinichi mochizuki april2020 abstract. Basically, this is not due to the history of the topic but rather since. Given a quasiconformal hyperbolic surface of nite type, s, the teichmull er space modeled on s, denoted t s, is the set of equivalence classes of pairs x. 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.
Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable riemann mapping theorem, proved by lars ahlfors and lipman. Another important subject is the study of the action of the action of the mapping class group on teichmuller space. Earle, from october 23, 2010, in syracuse, new york. Use 3 disjoint mutually nonseparating geodesics in h2 such that hyperbolic distances between them are the numbers. Introduction the roots of quasiconformal theory lie in geography, more precisely in the study of mappings from subsets of the. In addition to being of intrinsic interest, teichmuller theory has found wideranging.
The universal teichmuller space appears in many areas of mathematics and mathematical physics. Quasiconformal mappings and teichmuller s theorem scholars choice edition. Lavrentev, one of the founders of the theory of quasiconformal mappings. Introduction to riemann surfaces and teichmuller theory mpi mis. Complex structures, jacobians and the weil petersson form 31 lecture 4. Teichmuller s work on quasiconformal mappings 43 references 55 1. Inthepresentpaper,whichisthesecondinaseriesoffourpa. The teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. Teichmuller theory and applications to geometry, topology, and dynamics, vol. 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.
Quasiconformal mappings and teichmullers theorem by bers, lipman. D is the interior of the unit disk, and denotes extremal. 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. Xis a quasiconformal homeomorphism, with two such pairs deemed to be equivalent if.
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