The Loewner differential equation and slit mappings

Journal of the American Mathematical Society

Volumn 18, Issue 4, 2005, Pages 763-778

  Marshall, Donald E ^a   Rohde, Steffen ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Conformal maps; Conformal welding; Harmonic measure; H lder continuous; Lipschitz continuous; Loewner's differential equation; Quasiarc; Quasiconformal maps

Indexed keywords

EID: 26844434094     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-05-00492-3     Document Type: Article

References (21)

1. L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand (1966). MR0200442 (34:336)
2. L. de Dranges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152. MR0772434 (86h:30026)
3. L. Carleson, N. Makarov, Aggregation in the plane and Loewner's equation, Comm. Math. Phys. 216 (2001), 583-607. MR1815718 (2001m:82079)
4. C. Earle, A. Epstein, Quasiconformal variation of slit domains, Proc. Amer. Math. Soc. 129 (2001), 3363-3372. MR1845014 (2002f:30009)
5. P. Duren, Univalent functions, Springer (1983). MR0708494 (85j:30034)
6. J. L. Fernández, J. Heinonen, O. Martio, Quasilines and conformal mappings, J. Analyse Math. 52 (1989), 117-132. MR0981499 (90a:30017)
7. J. Garnett, D. E. Marshall, Harmonic Measure, Cambridge University Press (2005).
8. F. Gehring, Characteristic properties of quasidisks, Presses de l'Université de Montréal, Montreal, 1982. MR0674294 (84a:30036)
9. P. P. Kufarev, A remark on the integrals of the Loewner equation, Dokl. Akad. Nauk SSSR, 57 (1947), 655-656 (in Russian) MR0023907 (9:421d)
10. R. Kühnau, Numerische Realisierung konformer Abbildungen durch "Interpolation" Z. Angew. Math. Mech. 63 (1983), 631-637 (in German). MR0737000 (85b:30012)
11. K. Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I, Math. Ann. 89 (1923), 103-121.
12. J. Lind, A sharp condition for the Loewner equation to generate slits, to appear in Ann. Acad. Sci. Fenn.
13. G. Lawler, O. Schramm, W. Werner, Values of Brownian intersection exponents I: Half-plane exponents, Acta Math. 187 (2001), 237-273. MR1879850 (2002m:60159a)
14. G. Lawler, O. Schramm, W. Werner The dimension of the planar Brownian frontier is 4/3, Math. Res. Lett. 8 (2001), 401-411. MR1849257 (2003a:60127b)
15. O. Lehto, K. Virtanen, Quasiconformal mappings in the plane, Springer 1973. MR0344463 (49:9202)
16. D. E. Marshall, S. Rohde, Convergence of the Zipper algorithm for conformal mapping, preprint available at http://www.math.washington.edu/ ~marshall/preprints/preprints.html
17. Chr. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen (1975). MR0507768 (58:22526)
18. Chr. Pommerenke, Boundary behaviour of conformal maps, Springer (1992). MR1217706 (95b:30008)
19. S. Rohde, O. Schramm, Basic properties of SLE, arXiv:math.PR/0106036, to appear in Ann. Math.
20. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel Jour. Math. 118 (2000), 221-288. MR1776084 (2001m:60227)
21. S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), 239-244. MR1851632 (2002f:60193)