Near flat singularities, two-dimensional area-minimizing mod(q) currents in codimension one are C^{1,alpha}-perturbations of graphs of radially homogeneous harmonic multiple-valued functions, and top-density flat singularities are isolated in all codimensions.
An upper Minkowski bound for the interior singular set of area minimizing currents
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We show that for an area minimizing $m$-dimensional integral current $T$ of codimension at least 2 inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $m-2$. This provides a strengthening of the existing $(m-2)$-dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate $T$ along blow-up scales.
citation-role summary
background 1
citation-polarity summary
fields
math.AP 1years
2025 1verdicts
CONDITIONAL 1roles
background 1polarities
unclear 1representative citing papers
citing papers explorer
-
Structure of two-dimensional mod$(q)$ area-minimizing currents near flat singularities: the codimension one case
Near flat singularities, two-dimensional area-minimizing mod(q) currents in codimension one are C^{1,alpha}-perturbations of graphs of radially homogeneous harmonic multiple-valued functions, and top-density flat singularities are isolated in all codimensions.