Reduction techniques

Abstract

AbstractWhen a map has a certain symmetry, the equations for a harmonic morphism reduce to equations in a smaller number of variables. Here, the appropriate symmetry is equivariant with respect to isoparametric mappings. The latter is discussed and the concept of eigen-harmonic morphism — reduction theorem — is given. The modification of reduction equations by suitable conformal changes of metric is discussed, which allows us to find equivariant harmonic morphisms by first finding an equivariant map which is horizontally weakly conformal, and then rendering it harmonic by a suitable conformal change of metric. This way, solving the second-order system for a harmonic morphism is reduced to solving two first-order systems in turn. This technique allows the construction of many harmonic morphisms by reduction to an ordinary or partial differential equation (ODE or PDE).