Phase transition for a family of complex-driven Loewner hulls

Lind, Joan; Utley, Jeffrey

doi:10.2140/involve.2022.15.447

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Phase transition for a family of complex-driven Loewner hulls

Joan Lind and Jeffrey Utley

Vol. 15 (2022), No. 3, 447–474

DOI: 10.2140/involve.2022.15.447

Abstract

Building on H. Tran’s study of Loewner hulls generated by complex-valued driving functions, which showed the existence of a phase transition, we answer the question of whether the phase transition for complex-driven hulls matches the phase transition for real-driven hulls. This is accomplished through a detailed study of the Loewner hulls generated by driving functions $c \sqrt{1 - t}$ and $c \sqrt{τ + t}$ for $c \in ℂ$ and $τ \geq 0$ . This family also provides examples of new geometric behavior that is possible for complex-driven hulls but prohibited for real-driven hulls.

Keywords

Loewner evolution, Loewner hulls

Mathematical Subject Classification

Primary: 30C35

Milestones

Received: 29 June 2021

Revised: 8 November 2021

Accepted: 8 November 2021

Published: 2 December 2022

Communicated by Michael Dorff

Authors

Joan Lind
Department of Mathematics University of Tennessee Knoxville, TN United States

Jeffrey Utley
Department of Mathematics University of Tennessee Knoxville, TN United States

Vol. 15, No. 3, 2022