Geometric replacement for the algebraic proof of three-dimensional rigidity

Determine whether the algebraic proof of rigidity for three-dimensional Pham–Brieskorn rings can be replaced by a geometric argument, or whether the geometric proof can be replaced by an argument using locally nilpotent derivations.

Background

The paper explains that the proof of the n=3 case of the Kaliman–Zaidenberg rigidity conjecture combines locally nilpotent derivations with birational geometry of surfaces. The algebraic component uses reductions involving derivations, while the geometric component uses the relationship between nonzero locally nilpotent derivations and polar cylinders on weighted projective hypersurfaces.

The author explicitly identifies the absence of a known way to replace either part of the proof with the other type of argument. This is a methodological unresolved question rather than a new classification conjecture.