Removing the technical assumption in associative-cycle regularity

Determine whether the technical assumption in the regularity theorem for associative weighted gradient cyclesthat any two holomorphic curve classes in the same hyperkahler complex structure with areas bounded by the density are proportionalcan be removed.

Background

The paper proves local embedded-graph regularity for associative weighted gradient cycles only under a technical hypothesis on classes in H_2(K3,Z). Specifically, whenever two classes admit holomorphic representatives for a common complex structure and both have area at most the local density, the classes must be proportional. This ensures that all branches sharing a tangent direction correspond to proportional homology classes and therefore combine into gradient flowlines with controlled local structure.

Without the assumption, distinct homology classes may determine approximately parallel gradient flowlines, potentially producing infinitely many intersections or accumulation points. The paper notes that first-order flow equations and the finiteness of bounded-area holomorphic curve classes may prevent such behavior, but does not establish this.