Explicit multi-slit Loewner flows and their geometry

Abstract

In this paper we present explicit solutions to the radial and chordal Loewner PDE and we make an extensive study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses $\mu_{t}:=\sum_{j=1}^{n}b_j \delta_{\{\zeta_j e^{iat}\}}$ in the radial case and $\nu_t:=\sum_{j=1}^{n}b_j\delta_{\{k_j\sqrt{1-t}\}}$ in the chordal case, where all the above parameters are chosen arbitrarily. Furthermore, we investigate their close connection to the semigroup theory of holomorphic functions, which also allows us to map the chordal case to the radial one.