Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 311-327

REMARKS ON LOEWNER CHAINS DRIVEN BY FINITE VARIATION FUNCTIONS

Atul Shekhar, Huy Tran and Yilin Wang

KTH, Department of Mathematics
Lindstedtsvägen 25, SE-100 44 Stockholm, Sweden; atuls 'at' kth.se

Technische Univesität Berlin, Institut für Mathematik
Strasse des 17. Juni 136, 10623 Berlin, Germany; tranvohuy 'at' gmail.com

ETH Zurich, Department of Mathematics
Rämistrasse 101, 8092 Zurich, Switzerland; yilin.wang 'at' math.ethz.ch

Abstract. To explore the relation between properties of Loewner chains and properties of their driving functions, we study Loewner chains driven by functions U of finite total variation. Under a slow point condition, we show the existence of a simple trace γ and establish the continuity of the map from U to γ with respect to the uniform topology on γ and to the total variation topology on U. In the spirit of the work of Wong [19] and Lind–Tran [10], we also obtain conditions on the driving function that ensures the trace to be continuously differentiable.

2010 Mathematics Subject Classification: Primary 30C55, 34M99.

Key words: Loewner differential equation, finite variation drivers, trace of Loewner chains, continuity of Loewner map.

Reference to this article: A. Shekhar, H. Tran and Y. Wang: Remarks on Loewner chains driven by finite variation functions. Ann. Acad. Sci. Fenn. Math. 44 (2019), 311-327.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4421