Constructions and Rigidity of Nondegenerate $\mathbb{Z}_2$ Harmonic Functions with Quadric Branching Sets

Abstract

We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index \(n-1\) and positive critical value. In \(\mathbb R^3\), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.