Singular points of harmonic maps from 4-dimensional domains into 3-spheres

Abstract

In this article we show that if a stable-stationary harmonic map $u$ from a domain in ${\mathbb{R}}^4$ into ${\mathbb{S}}^3$ has an isolated singular point $\xi$, then the mapping degree $\mathrm{deg}(u,\xi )$ of $u$ at $\xi$ is $+1$ or $-1$. Furthermore, the complete characterization of stable-stationary tangent maps from ${\mathbb{B}}^4$ to ${\mathbb{S}}^3$ is given