A Construction of non-degenerate ${\mathbb{Z}}_2$-harmonic functions on ${\mathbb{R}}^n$

We discover an explicit construction of non-degenerate ${\mathbb{Z}}_2$-harmonic functions on ${\mathbb{R}}^n,n\ge 3$, using a variant of ellipsoidal coordinates on ${\mathbb{R}}^n$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate ${\mathbb{Z}}_2$-harmonic $1$-forms on ${\mathbb{R}}^n$ with compact branching sets. Moreover, the graph of the related ${\mathbb{Z}}_2$-harmonic one form in $T^{\ast }{\mathbb{R}}^n$ can be obtained as a certain limit of a specific sequence of Lawlor’s necks in ${\mathbb{C}}^n=T^{\ast }{\mathbb{R}}^n$.