Convergence under topological changes of branching sets

Develop a convergence theory for \(\mathbb Z_2\) harmonic functions when the branching sets undergo more general topological changes than mutually diffeomorphic deformations.

Background

The proof of the hyperbola nonexistence result compares rescaled functions whose branching sets remain mutually diffeomorphic. To do so, the authors pull the associated flat real line bundles back to a common reference bundle using diffeomorphisms of the branching configurations.

The authors note that this method does not address situations in which the topology of the branching set changes. A general compactness and convergence framework for such degenerations remains unresolved.