Harmonic morphisms defined by polynomials

Abstract

AbstractThis chapter establishes the Liouville-type theorem which asserts that any entire (i.e., globally defined) harmonic morphism from a Euclidean space of arbitrary dimension to a Euclidean space of dimension more than two is necessarily polynomial. In fact, the harmonic morphism need only be defined off a closed polar set. It is shown that any horizontally conformal polynomial mapping is necessarily harmonic and, hence, a harmonic morphism. At a critical point of finite order, a horizontally conformal mapping is approximated, at least at the level of the first non-constant term in its Taylor expansion, by a harmonic morphism. Important classes of polynomial harmonic morphisms are classified. This provides information on the local behaviour of a harmonic morphism near a critical point, and leads to global topological restrictions on its domain.