Uniqueness of the finite-energy boundary point for prescribed minimal surfaces

Let a suitably compactified moduli space parametrize degenerating domains and let a given set of minimal surfaces span the prescribed boundary components. A boundary point has a finite-energy limit when the corresponding map energies converge to a finite value.

Uniqueness conjecture. It may be possible that there is only one boundary point of the compactified moduli space having a finite-energy limit of map energies for the given set of minimal surfaces.

The claim concerns the possible uniqueness of the limiting degeneration associated with prescribed minimal surfaces. The surrounding discussion notes that several minimal surfaces can span the same boundary components, so the precise set of surfaces must first be specified; the paper does not establish this uniqueness.