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 λ_t and certain increasing families of half-plane hulls K_t. In this paper we study the deterministic relationship between specific analytic properties of λ_t and geometric properties of K_t. Our motivation comes, however, from the stochastic Loewner equation (SLE_κ), where the associated function λ_t is a scaled Brownian motion and the corresponding domains H \ K_t are Hölder domains. We prove that if the increasing family K_t is generated by a simple curve and the final domain H \ K_T 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 SLE_κ 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 K_t 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.

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doi:10.5186/aasfm.2015.4044