A symplectic functional analytic proof of the conformal welding theorem
HTML articles powered by AMS MathViewer
- by Eric Schippers and Wolfgang Staubach
- Proc. Amer. Math. Soc. 143 (2015), 265-278
- DOI: https://doi.org/10.1090/S0002-9939-2014-12225-8
- Published electronically: September 24, 2014
- PDF | Request permission
Abstract:
We give a new functional-analytic/symplectic geometric proof of the conformal welding theorem. This is accomplished by representing composition by a quasisymmetric map $\phi$ as an operator on a suitable Hilbert space and algebraically solving the conformal welding equation for the unknown maps $f$ and $g$ satisfying $g \circ \phi = f$. The univalence and quasiconformal extendibility of $f$ and $g$ is demonstrated through the use of the Grunsky matrix.References
- N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Conference on partial differential equations, University of Kansas (1954), Studies in eigenvalue problems, Technical report 14.
- P. Ebenfelt, D. Khavinson, and H. S. Shapiro, Two-dimensional shapes and lemniscates, Complex analysis and dynamical systems IV. Part 1, Contemp. Math., vol. 553, Amer. Math. Soc., Providence, RI, 2011, pp. 45–59. MR 2868587, DOI 10.1090/conm/553/10931
- F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR 0198152
- V. M. Gol′dshteĭn and Yu. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Mathematics and its Applications (Soviet Series), vol. 54, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated and revised from the 1983 Russian original; Translated by O. Korneeva. MR 1136035, DOI 10.1007/978-94-009-1922-8
- Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, vol. 148, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1448404
- G. F. Mandžavidze and B. V. Hvedelidze, On the Riemann-Privalov problem with continous coefficients, Dokl. Akad. Nauk SSSR 123 (1958), 791–794 (Russian). MR 0112961
- Zair Ibragimov, Quasi-isometric extensions of quasisymmetric mappings of the real line compatible with composition, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 221–233. MR 2643406, DOI 10.5186/aasfm.2010.3513
- Eri Jabotinsky, Representation of functions by matrices. Application to Faber polynomials, Proc. Amer. Math. Soc. 4 (1953), 546–553. MR 59359, DOI 10.1090/S0002-9939-1953-0059359-0
- A. A. Kirillov, Kähler structure on the $K$-orbits of a group of diffeomorphisms of the circle, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 42–45 (Russian). MR 902292
- A. A. Kirillov and D. V. Yur′ev, Kähler geometry of the infinite-dimensional homogeneous manifold $M=\textrm {Diff}_+(S^1)/\textrm {Rot}(S^1)$, Funktsional. Anal. i Prilozhen. 20 (1986), no. 4, 79–80 (Russian). MR 878052
- A. A. Kirillov and D. V. Yur′ev, Kähler geometry of the infinite-dimensional homogeneous space $M=\textrm {Diff}_+(S^1)/\textrm {Rot}(S^1)$, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35–46, 96 (Russian). MR 925071
- Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463
- Subhashis Nag and Dennis Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle, Osaka J. Math. 32 (1995), no. 1, 1–34. MR 1323099
- Bryan Penfound and Eric Schippers, Power matrices for Faber polynomials and conformal welding, Complex Var. Elliptic Equ. 58 (2013), no. 9, 1247–1259. MR 3170696, DOI 10.1080/17476933.2012.662222
- Albert Pfluger, Ueber die Konstruktion Riemannscher Flächen durch Verheftung, J. Indian Math. Soc. (N.S.) 24 (1960), 401–412 (1961) (German). MR 132827
- Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
- C. L. Siegel, Topics in complex function theory. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Automorphic functions and abelian integrals; Translated from the German by A. Shenitzer and M. Tretkoff; With a preface by Wilhelm Magnus; Reprint of the 1971 edition; A Wiley-Interscience Publication. MR 1008931
- S. L. Sobolev, Some applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, vol. 90, American Mathematical Society, Providence, RI, 1991. Translated from the third Russian edition by Harold H. McFaden; With comments by V. P. Palamodov. MR 1125990, DOI 10.1090/mmono/090
- Leon A. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc. 183 (2006), no. 861, viii+119. MR 2251887, DOI 10.1090/memo/0861
- Izu Vaisman, Symplectic geometry and secondary characteristic classes, Progress in Mathematics, vol. 72, Birkhäuser Boston, Inc., Boston, MA, 1987. MR 932470, DOI 10.1007/978-1-4757-1960-4
Bibliographic Information
- Eric Schippers
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
- MR Author ID: 651639
- Email: eric_schippers@umanitoba.ca
- Wolfgang Staubach
- Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
- MR Author ID: 675031
- Email: wulf@math.uu.se
- Received by editor(s): February 25, 2013
- Received by editor(s) in revised form: March 27, 2013
- Published electronically: September 24, 2014
- Additional Notes: The first author was partially supported by the National Sciences and Engineering Research Council.
- Communicated by: Jeremy Tyson
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 265-278
- MSC (2010): Primary 30C35, 30C62, 30F60; Secondary 53D30
- DOI: https://doi.org/10.1090/S0002-9939-2014-12225-8
- MathSciNet review: 3272752