Global hyperbolic branching solutions

Determine whether a global nondegenerate \(\mathbb Z_2\) harmonic function on \(\mathbb R^3\) can have a planar hyperbola as its branching set, equivalently, prove or disprove the conjecture that such functions exist only locally.

Background

The paper constructs a nondegenerate $\mathbb Z_2$ harmonic function near every planar hyperbola in $\mathbb R^3$, but the construction is confined to a proper region because the modified ellipsoidal coordinates degenerate away from that region. The authors conjecture that this local character is intrinsic and that no global nondegenerate solution with a planar hyperbolic branching set exists.

The paper proves a related obstruction: no global critical solution with a planar hyperbola as branching set can have finite Almgren frequency, or finite order, at infinity. This leaves open the possibility of global solutions with sufficiently rapid or otherwise uncontrolled growth.