Characterisation of some multivalued harmonic functions on ${\mathbb{R}}^n$

Using different approaches, Simon Donaldson and Dashen Yan recently constructed ${\mathbb{Z}}_2$-harmonic functions on ${\mathbb{R}}^3$ and on ${\mathbb{R}}^n$ for $n⩾3$, respectively. Their examples have quadratic growth at infinity and $\mathcal{O}(r^{3\mathrm{/}2})$ local growth near a smooth codimension-two branching set $\mathrm{\Sigma }$. We show that these properties uniquely characterise such ${\mathbb{Z}}_2$-harmonic functions up to rigid motions of ${\mathbb{R}}^n$.