Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 803-835

LOEWNER CHAINS AND HÖLDER GEOMETRY

Kyle Kinneberg

Rice University, Department of Mathematics
MS 136, 6100 Main Street, Houston, TX 77005-1892, U.S.A.; kyle.kinneberg 'at' rice.edu

Abstract. The Loewner equation provides a correspondence between continuous real-valued functions λt and certain increasing families of half-plane hulls Kt. In this paper we study the deterministic relationship between specific analytic properties of λt and geometric properties of Kt. Our motivation comes, however, from the stochastic Loewner equation (SLEκ), where the associated function λt is a scaled Brownian motion and the corresponding domains H \ Kt are Hölder domains. We prove that if the increasing family Kt is generated by a simple curve and the final domain H \ KT is a Hölder domain, then the corresponding driving function has a modulus of continuity similar to that of Brownian motion. Informally, this is a converse to the fact that SLEκ curves are simple and their complementary domains are Hölder, when κ < 4. We also study a similar question outside of the simple curve setting, which informally corresponds to the SLE regime κ > 4. In the process, we establish general geometric criteria that guarantee that Kt has a Lip(1/2) driving function.

2010 Mathematics Subject Classification: Primary 30C20; Secondary 30C45.

Key words: Loewner equation, Hölder domains, John domains.

Reference to this article: K. Kinneberg: Loewner chains and Hölder geometry. Ann. Acad. Sci. Fenn. Math. 40 (2015), 803-835.

Full document as PDF file

doi:10.5186/aasfm.2015.4044

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