Quasiconformal mappings, with applications to differential equations, function theory and topology

The theory of quasiconformal mappings is nearly 50 years old (see [44] for references to the papers by Grötzsch, Ahlfors Lavrent'ev and Morrey from the 20's and 30's) and the interest in them does not seem to wane. These mappings may be studied for their own sake or as a tool for attacking other mathematical problems; they are indeed a powerful and flexible tool. The purpose of this lecture is to state two basic theorems about quasiconformal mappings in two dimensions (the existence theorem, the first version of which is due to Gauss, and Teichmüller's theorem about extremal quasiconformal mappings) and to discuss some applications of these theorems. The discussion will necessarily be sketchy and several important topics will be slighted or not even mentioned. (Some of those are covered in [15], [20], [23].) (At St. Louis I learned about an interesting application, due to J. Sachs and K. Uhlenbeck, to the theory of minimal surfaces. Added in proof.) Lack of time and of competence prevents me from saying anything about the subtle theory of quasiconformal mappings in «-space, initiated in a short note [48] by my late teacher Loewner, and developed by Gehring, Vâisâlâ and others (see the references in [32], [67]), a theory which also has important applications (Mostow [55]). We begin by defining the concept of quasiconformality.

[1]  H. Royden.  Automorphisms and Isometries of Teichmilller Space , 1971 .

[2]  L. Bers,et al.  Holomorphic convexity of Teichmüller spaces , 1964 .

[3]  E. Reich.  Extremal Quasiconformal Mappings with Given Boundary Values , 1974 .

[4]  E. Hille,et al.  Remarks on a paper by Zeev Nehari , 1949 .

[5]  A. Gramain.  Le type d'homotopie du groupe des difféomorphismes d'une surface compacte , 1973 .

[6]  A uniqueness theorem for Beltrami equations , 1962 .

[7]  L. Ahlfors,et al.  On quasiconformal mappings , 1953 .

[8]  Louis Nirenberg,et al.  On nonlinear elliptic partial differential equations and hölder continuity , 1953 .

[9]  F. Gehring.  Univalent functions and the Schwarzian derivative , 1977 .

[10]  I. Kra,et al.  On Holomorphic Mappings between Teichmüller Spaces , 1974 .

[11]  C. Earle,et al.  Teichmüller theory for surfaces with boundary , 1970 .

[12]  Automorphic Forms and General Teichmüller Spaces , 1965 .

[13]  L. Nirenberg.  VI. On a Generalization of Quasi-Conformal Mappings and its Application to Elliptic Partial Differential Equations , 1955 .

[14]  Oswald Teichmüller,et al.  Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen , 1943 .

[15]  Correction to “Spaces of Riemann surfaces as bounded domains” , 1961 .

[16]  P. Griffiths.  Complex-Analytic Properties of Certain Zariski Open Sets on Algebraic Varieties , 1971 .

[17]  H. Masur.  On a class of geodesics in Teichmiuller space , 1975 .

[18]  Mathématiques DE L’I.H.É.S,et al.  Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms , 1968 .

[19]  L. Bers.  Uniformization by Beltrami equations , 1961 .

[20]  James Eells,et al.  A fibre bundle description of Teichmüller theory , 1969 .

[21]  C. Loewner.  On the Conformal Capacity in Space , 1959 .

[22]  EXTREMAL QUASICONFORMAL MAPPINGS WITH PRESCRIBED BOUNDARY VALUES , 1969 .

[23]  C. Earle,et al.  Reduced Teichmüller spaces , 1967 .

[24]  Charles B. Morrey,et al.  On the solutions of quasi-linear elliptic partial differential equations , 1938 .

[25]  A. Marden.  On Homotopic Mappings of Riemann Surfaces , 1969 .

[26]  S. Krushkal.  Teichmuller's theorem on extremal quasi-conformal liappings , 1967 .

[27]  L. Ahlfors,et al.  Lectures on quasiconformal mappings , 1966 .

[28]  L. Bers.  An extremal problem for quasiconformal mappings and a theorem by Thurston , 1978 .

[29]  Lipman Bers,et al.  Uniformization, Moduli, and Kleinian Groups , 1972 .

[30]  R. Kühnau.  Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhängiger Dilatationsbeschränkung , 1969 .

[31]  Lipman Bers,et al.  Existence and uniqueness of a subsonic flow past a given profile , 1954 .

[32]  Zeev Nehari,et al.  The Schwarzian derivative and schlicht functions , 1949 .

[33]  L. Ahlfors,et al.  RIEMANN'S MAPPING THEOREM FOR VARIABLE METRICS* , 1960 .

[34]  Vorlesungen über die Theorie der automorphen Funktionen , 1914 .

[35]  Oswald Teichmüller,et al.  Extremale quasikonforme Abbildungen und quadratische Differentiale , 1940 .

[36]  Stephen Smale,et al.  DIFFEOMORPHISMS OF THE 2-SPHERE , 1959 .

[37]  Jussi Väisälä,et al.  Lectures on n-Dimensional Quasiconformal Mappings , 1971 .

[38]  B. Maskit.  On the classification of Kleinian groups: I—Koebe groups , 1975 .