Harmonic morphisms and minimal submanifolds

TL;DR

This paper explores the relationship between harmonic morphisms and minimal submanifolds, providing characterizations and reduction properties, and introduces new examples of area-minimising hypercones in high-dimensional Euclidean spaces.

Contribution

It characterizes harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations and constructs new area-minimising hypercones in high dimensions.

Findings

Harmonic morphisms are characterized as weakly horizontally conformal maps preserving minimal submanifold equations.

Derived reduction properties for harmonic morphisms in various co-dimensions.

Constructed a novel family of degree 4 area-minimising hypercones in rom 

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 $\mathbb R^m$, $m\geq32$.