Twist points of planar domains

Arcozzi, Nicola; Casadio Tarabusi, Enrico; Di Biase, Fausto; Picardello, Massimo

doi:10.1090/S0002-9947-05-03855-9

Twist points of planar domains
HTML articles powered by AMS MathViewer

by Nicola Arcozzi, Enrico Casadio Tarabusi, Fausto Di Biase and Massimo A. Picardello PDF

Trans. Amer. Math. Soc. 358 (2006), 2781-2798 Request permission

Abstract:

We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.

References

N. Arcozzi, E. Casadio Tarabusi, F. Di Biase and M. Picardello, On the McMillan Twist Theorem: a potential theoretic approach, Preprint number 70, Chalmers Institute of Technology, October 2001. http://www.math.chalmers.se/Math/Research/Preprints/.
Albert Baernstein II, Ahlfors and conformal invariants, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 3, 289–312. MR 994466, DOI 10.5186/aasfm.1988.1323
Christopher J. Bishop, How geodesics approach the boundary in a simply connected domain, J. Anal. Math. 64 (1994), 291–325. MR 1303516, DOI 10.1007/BF03008413
K. Burdzy, My favorite open problems, http://www.math.washington.edu/˜burdzy/ open.shtml.
A. Cantón, A. Granados, and Ch. Pommerenke, Conformal images of Borel sets, Bull. London Math. Soc. 35 (2003), no. 4, 479–483. MR 1979001, DOI 10.1112/S0024609303002078
Joan Josep Carmona and Christian Pommerenke, Twisting behaviour of conformal maps, J. London Math. Soc. (2) 56 (1997), no. 1, 16–36. MR 1462823, DOI 10.1112/S0024610797005334
J. J. Carmona and Ch. Pommerenke, On the argument oscillation of conformal maps, Michigan Math. J. 46 (1999), no. 2, 297–312. MR 1704213, DOI 10.1307/mmj/1030132412
J. J. Carmona and C. Pommerenke, On prime ends and plane continua, J. London Math. Soc. (2) 66 (2002), no. 3, 641–650. MR 1934297, DOI 10.1112/S0024610702003587
Henri Cartan, Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces, Hermann, Paris, 1967 (French). MR 0231303
Martin Chuaqui and Christian Pommerenke, An integral representation formula of the Schwarzian derivative, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 155–163. MR 1931608, DOI 10.1007/BF03320982
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360
Philip Davis and Henry Pollak, On the zeros of total sets of polynomials, Trans. Amer. Math. Soc. 72 (1952), 82–103. MR 45245, DOI 10.1090/S0002-9947-1952-0045245-2
Fausto Di Biase, Bert Fischer, and Rüdiger L. Urbanke, Twist points of the von Koch snowflake, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1487–1490. MR 1443822, DOI 10.1090/S0002-9939-98-04226-9
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335–400.
F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR 643101
H. Grunsky, Neue Abschätzungen zur konformen Abbildungen ein - und mehrfach zusammenhängender Berichte, Schr. Math. Seminars Inst. Angew. Math. Univ. Berlin 1 (1932) 95–140.
E. Hille Analytic function theory, Vol. 1, Chelsea, 1982.
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
J. G. Llorente, Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach, Potential Anal. 9 (1998), no. 3, 229–260. MR 1666891, DOI 10.1023/A:1008655312464
A. I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, Second English edition, Chelsea Publishing Co., New York, 1977. Translated and edited by Richard A. Silverman. MR 0444912
J. E. McMillan, Boundary behavior of a conformal mapping, Acta Math. 123 (1969), 43–67. MR 257330, DOI 10.1007/BF02392384
Raghavan Narasimhan and Yves Nievergelt, Complex analysis in one variable, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1803086, DOI 10.1007/978-1-4612-0175-5
A.I. Plessner, Uber das Verhalten analytischer Funktionen am Rande ihres Definitionsbereichs, J. Reine Angew. Math. 158 (1927) 219–227.
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
Ch. Pommerenke, Boundary behaviour of conformal mappings, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 313–331. MR 623475
Ch. Pommerenke, On ergodic properties of inner functions, Math. Ann. 256 (1981), no. 1, 43–50. MR 620121, DOI 10.1007/BF01450942
Christian Pommerenke, Über das Randverhalten schlichter Funktionen, Mathematical papers given on the occasion of Ernst Mohr’s 75th birthday, Tech. Univ. Berlin, Berlin, 1985, pp. 199–209 (German). MR 809004
Ch. Pommerenke, On the boundary continuity of conformal maps, Pacific J. Math. 120 (1985), no. 2, 423–430. MR 810781
Ch. Pommerenke, On the boundary derivative for univalent functions, Math. Ann. 276 (1987), no. 3, 499–504. MR 875343, DOI 10.1007/BF01450844
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Ch. Pommerenke and S. E. Warschawski, On the quantitative boundary behavior of conformal maps, Comment. Math. Helv. 57 (1982), no. 1, 107–129. MR 672848, DOI 10.1007/BF02565849
W. Seidel, Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. 104 (1931), 182–243.
H. von Koch, Une méthode géométrique élémentaire pour l’étude de certains questions de la théorie des courbes planes, Acta Math. 30 (1906), 145–174.
J. L. Walsh, Note on the shape of level curves of Green’s function, Amer. Math. Monthly 60 (1953), 671–674. MR 58789, DOI 10.2307/2307144
N. Wiener, Certain notions in potential theory, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 24–51.