## Harmonic maps via ordinary differential equations

Emma Carberry

Sydney University

Abstract

Harmonic maps are by definition the solutions to the Laplace-Beltrami equation, a second order partial differential equation. However there is a subclass of harmonic maps from a surface to a Lie group or symmetric space which can be described by vastly easier means. These maps of \emph{finite-type} are obtained simply by integrating a pair of commuting vector fields on a finite dimensional space and hence by solving ordinary differential equations. This naturally prompts one to find conditions under which a map is of finite type, for which there are quite general results known when the target manifold is compact.