Intersecting-line branching sets

Construct, or prove the nonexistence of, critical and nondegenerate \(\mathbb Z_2\) harmonic functions on \(\mathbb R^3\) whose branching set is a pair of transverse intersecting lines, without imposing homogeneity.

Background

The paper discusses the singular branching configuration formed by two transverse intersecting lines through the origin. Results of Chen and He rule out homogeneous critical Z2\mathbb Z_2 harmonic functions for this configuration, but do not address the nonhomogeneous case.

The authors explicitly ask whether relaxing homogeneity permits a critical, or even nondegenerate, solution. They also suggest that such a solution might arise as a limit of the hyperbolic constructions, since a pair of intersecting lines is the asymptotic cone of a planar hyperbola.

References

Nevertheless, it remains an open question whether such a critical \mathbb Z_2 harmonic function can be constructed if the homogeneity constraint is relaxed. This leads to the following question: is it possible to construct critical or even nondegenerate \mathbb Z_2 harmonic functions in \mathbb Rn whose branching sets are non-smooth and exhibit singularities, particularly in the case where \Sigma is a pair of intersecting lines in \mathbb R3?

— Constructions and Rigidity of Nondegenerate $\mathbb{Z}_2$ Harmonic Functions with Quadric Branching Sets  (2608.14040 - Zhou, 14 Aug 2026) in Section 3.2.1, immediately after Theorem 3.2 (the theorem labeled \(\ref{chenhe}\))