Explicit Multi-slit Loewner Flows and Their Geometry

In this paper we present explicit solutions to the radial and chordal Loewner PDEs 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=1nbj delta{zeta jeiat}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{t}:=\sum _{j=1}<^>{n}b_j \delta _{\{\zeta _j e<^>{iat}\}}$$\end{document} in the radial case and nu t:=& sum;j=1nbj delta{kj1-t}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _t:=\sum _{j=1}<^>{n}b_j\delta _{\{k_j\sqrt{1-t}\}}$$\end{document} 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.