Lin’s interior mass-bound problem for confined area-minimizing currents

Determine whether sufficiently small Hausdorff distance between the support of an indecomposable, n-dimensional area-minimizing integral current T in Euclidean space and the unit disk D₁, together with the projection condition p₍#₎T = Q⟦D₁⟧ and vanishing boundary in the cylinder C₁, implies a uniform bound (T⟂C₁/₂) ≤ C(Q,n,k); equivalently, establish whether the interior mass is controlled solely by the projection multiplicity and dimensions.

Background

The paper formulates Lin’s Problem 1 for an indecomposable, n-dimensional area-minimizing integral current T in Euclidean space with orthogonal projection p onto an n-dimensional factor. The hypotheses require p₍#₎T = Q⟦D₁⟧, vanishing boundary inside the cylinder C₁ = D₁ × ℝᵏ, and Hausdorff distance at most δ between supp T and D₁, where δ is sufficiently small. The unresolved issue is whether these confinement and projection assumptions force a bound for the mass of T in the smaller cylinder C₁/₂ depending only on Q, n, and k.

The question is important because the projection multiplicity Q is algebraic and may conceal oppositely oriented sheets that cancel under projection, whereas the mass counts all sheets positively. The paper subsequently proves an affirmative answer with a codimension-independent bound C(n)Q, so the problem is explicitly open in the cited prior context but resolved by the present work.