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. Athanase papadopoulos teichmuller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different. 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. 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. Quasiconformal mappings and teichmuller s theorem scholars choice edition. Quasiconformal mappings, riemann surfaces, and teichmuller spaces. Earle, from october 23, 2010, in syracuse, new york. D is the interior of the unit disk, and denotes extremal. Teichmuller s work on quasiconformal mappings 43 references 55 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. Y 1 2 inf f lnk where k is the dilatation of a quasiconformal map f. Introduction an orientation preserving homeomorphism ffrom a riemann surface x onto a riemann surface y is given. 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. 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. Quasiconformal mappings, from ptolemys geography to the work of. Interuniversal teichmuller theory i construction of hodge theaters shinichi mochizuki april2020 abstract. Kerckhoff receioed 8 december 1978 introduction teichmuller space is the space of conformal structures on a topological. Quasiconformal maps and teichmuller theoryextremal mappings, overview. Teichmuller theory mathematisches institut universitat bonn.
By the theory of \base points at in nity we have a representation g q. 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. Stack exchange network stack exchange network consists of. 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. Instead of power series, integral representations, or algebra ic techniques, the theory relies on singular integrals, geometricmeasuretheory,andsobolevspaces. Quasiconformal mappings, from ptolemys geography to the. Pdf quasiconformal maps and teichmuller theoryextremal.
Even if this were proved, it wouldnt a priori be especially helpful if we. Gus schrader uc berkeley quantum higher teichmuller theory. Quasiconformal mappings and teichmullers theorem scholars. The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Quasiconformal mappings, from ptolemys geography to the work. 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. This volume includes a wide range of papers on teichmuller theory and related areas. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w. Proceedings of the year on teichmuller theory hri, allahabad, volume 10, 2009, pages 195228 abstract we present a brief exposition of teichmullers theorem. Quasiconformal mappings and teichmuller s theorem by bers, lipman. Complex structures, jacobians and the weil petersson form 31 lecture 4.
Quasiconformal mappings and teichmullers theorem scholars choice edition bers, lipman on. 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. This section gives a summary of the basics of teichmull. 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. In addition to being of intrinsic interest, teichmuller theory has found wideranging. 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. Teichmuller theory is one of those few wonderful subjects which bring together, at an equally important. Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable riemann mapping theorem, proved by lars ahlfors and lipman. Quasiconformal mappings and teichmullers theorem paperback august 24, 2017 by lipman bers author see all 7 formats and editions hide other formats and editions. Quasiconformal teichmuller theory as an analytical foundation for twodimensional conformal field theory.
Quasiconformal maps and teichmuller theoryextremal mappings, overview article pdf available in bulletin classe des sciences mathematiques et natturalles 14538 january 20 with 439 reads. The general problem of constructing a quasiconformal mapping from one simplyconnected domain onto another satisfying 2 was posed and solved by m. Introduction an orientation preserving homeomorphism f from a riemann surface x onto a riemann surface y is given. Introduction the roots of quasiconformal theory lie in geography, more precisely in the study of mappings from subsets of the. Then there exists a complete hyperbolic structure x with geodesic boundary on the pair of pants s2 \ 3 discs such that lengths of boundary. Teichmuller theory and applications to geometry, topology. Teichmuller theory and applications to geometry, topology, and dynamics, vol. Raction on these moduli spaces, the kontsevichzorich cocycle over the teichmuller geodesic. Abstract we present a brief exposition of teichmullers. 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. Teichmullers work on quasiconformal mappings 43 references 55 1.
Geometry of teichmuller space with the teichmuller metric. Teichmuller space encodes the possible number of ways in which you can deform complex structure. The discussion on quasiconformal mappings in chapter 4 is very important but might be one of the. Aug 24, 2017 quasiconformal mappings and teichmuller s theorem bers, lipman on. The theorem generalizes the riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Ra that gives a combinatorial description of the teichmuller metric. Teichmuller spaces for surfaces with cone points have come into play. The universal teichmuller space appears in many areas of mathematics and mathematical physics. Moduli of riemann surfaces, teichmuller theory see also 14h15, 30fxx secondary. This volume uses quasiconformal mapping to give a unified and upto. Working on my current research problem, teichmuller spaces for surfaces with cone points have come into play. 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.
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 teichmuller space \tx\ is the space of marked conformal structures on a given quasiconformal surface \x\. The authors have produced a formidable treatise on the modern theories of quasiconformal mappings, riemann surfaces and teichmuller. The geometry of the universal teichmuller space and the. This volume uses quasiconformal mapping to give a unified and uptodate treatment of \tx\. Interaction with analysis, geometry and number theory from 4 to 16 july, 2011. Interuniversal teichmuller theory ii hodgearakelovtheoretic evaluation shinichi mochizuki april2020 abstract. Xis a quasiconformal homeomorphism, with two such pairs deemed to be equivalent if. 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. 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.
Teichmuller theory interpretation of the decomposition of the tensor product of positive representations p p into positive representations p. Quasiconformal teichmuller theory as an analytical foundation for. Quasiconformal maps and teichmuller theory alastair. Use 3 disjoint mutually nonseparating geodesics in h2 such that hyperbolic distances between them are the numbers. Quasiconformal mapping encyclopedia of mathematics. Teichmuller space is, in some sense, the space of all riemann surfaces complex manifolds of dimension. 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. Inthepresentpaper,whichisthesecondinaseriesoffourpa. What is a good introduction to teichmuller theory, mapping class groups etc. Lavrentev, one of the founders of the theory of quasiconformal mappings. The mantra is studying objects in families gives important information about the individual objects. Abstract we present a brief exposition of teichmuller s. Along somewhat di erent lines we describe some recent important work of k. Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type.
Quasiconformal mappings, riemann surfaces, and teichmuller. Gardiner and jun hu winter, 2005, hri, allahabad, india. Quasiconformal mappings and teichmuller s theorem scholars choice edition bers, lipman on. Introduction to riemann surfaces and teichmuller theory mpi mis. 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. Teichmuller theory and applications to geometry, topology, and dynamics volume 1. The grothendieck teichmul ler conjecture states that g q dgt. Quasiconformal mappings and teichmullers theorem by bers, lipman. 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, which are the 20thcentury founders of the theory. The geometry of the universal teichmuller space and the euler.
A quasiconformal mapping class is a homotopy equivalence class. Another important subject is the study of the action of the action of the mapping class group on teichmuller space. Teichmuller theory and applications matrix editions. Teichmuller space encodes the possible number of ways in which you can deform complex structure on a space. For example, it is a special coadjoint orbit of the bottvirasoro group nag and verjovsky1990 and plays. 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.
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