Geometric structure of the limiting current

Characterize the limiting current L_\infty beyond the homological information encoded by its associated \Lambdaweighted gradient cycle T, including whether it is an integral current and whether its fibrewise slices are holomorphic curves or translated flat subtori with the prescribed homology classes.

Background

The main compactness theorem extracts from rescaled special Lagrangian or associative cycles a limiting current L_\infty whose currently established information is captured by a \Lambdaweighted gradient cycle T on the base. The paper asks whether L_\infty itself has stronger geometric structure.

In the associative case, the desired description would make L_\infty an integral current whose fibrewise slices are I_vholomorphic curves in K3 fibres, with homology class \delta. In the special Lagrangian case, the analogous slices would be sums of translated flat subtori in the relevant torus-fibre homology class. The unresolved difficulty is controlling oscillation of the slices in the fibre directions over macroscopic base scales; uncontrolled oscillation could yield only closed normal currents or real-weighted averages over moduli spaces rather than integral fibrewise cycles.