Almgren's type regularity for Semicalibrated Currents

TL;DR

This paper extends Almgren's regularity theorem to semicalibrated currents, showing that their singular set has Hausdorff dimension at most m-2, under certain smoothness conditions.

Contribution

It proves Almgren's type regularity result for semicalibrated currents in manifolds with specific smoothness, generalizing previous results for area minimizing currents.

Findings

Singular set dimension at most m-2

Extension of Almgren's theorem to semicalibrated currents

Regularity result under C^{3,ε} manifold conditions

Abstract

In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an $m$-dimensional semicalibrated current in a $(n+m)$-dimensional $C^{3,\varepsilon_0}$ manifold, semicalibrated by a $C^{2,\varepsilon_0}$ $m$-form, has singular set of Hausdorff dimension at most $m-2$.