A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$

Abstract: We discover an explicit construction of non-degenerate $\mathbb{Z}{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid. Moreover, the graph of the related $\mathbb{Z}{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$, can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$.