Loewner chains and evolution families on parallel slit half-planes
arXiv:2002.03359 · doi:10.1016/j.jmaa.2023.127180
Abstract
In this paper, we define and study Loewner chains and evolution families on finitely multiply-connected domains in the complex plane. These chains and families consist of conformal mappings on parallel slit half-planes and have one and two "time" parameters, respectively. By analogy with the case of simply connected domains, we develop a general theory of Loewner chains and evolution families on multiply connected domains and, in particular, prove that they obey the chordal Komatu-Loewner differential equations driven by measure-valued processes. Our method involves Brownian motion with darning, as do some recent studies.
64 pages; (v3)Theorems 2.2 and 2.5 are improved, and Appendix D is added for the explanation on weak and vague convergences of measures (v4)minor modification (v5)typos are collected in the proof of Proposition C.6; to appear in J. Math. Anal. Appl