Harmonic morphisms and minimal submanifolds

Abstract

Harmonic morphisms, maps which preserve Laplace's equation, are intimately connected to the topic of minimal submanifolds. In this article we first characterise harmonic morphisms between Riemannian manifolds as the weakly horizontally conformal maps that preserve the equation for minimal submanifolds of co-dimension $2$. We further derive additional reduction properties of harmonic morphisms for minimal submanifolds of other co-dimensions. These theorems are then applied in an example case, yielding a novel family of degree $4$ area-minimising hypercones in ${\mathbb{R}}^m$, $m\ge 32$.