On the regularity of area minimizing currents at boundaries with arbitrary multiplicity

TL;DR

The paper resolves Allard’s boundary regularity question for area-minimizing $m$-currents with arbitrary boundary multiplicity by developing a full boundary regularity theory in codimension $n\ge1$. The approach combines one-sided and two-sided analyses, a center-manifold/normal-approximation framework, and a universal frequency function to capture excess decay and tangent-cone structure, enabling ${\mathcal{H}}^{m-3}$-rectifiability of the one-sided boundary singular set and openness/density of the boundary regular set. A linearized (Dirichlet-minimizing) problem is treated to establish rectifiability results that transfer to the nonlinear case, with the Jones $\beta_2$ coefficients playing a central role via the Naber–Valtorta/Azzaam–Tolsa rectifiability machinery. Under a convex barrier, the boundary singular set is shown to be $\mathcal{H}}^{m-3}$-rectifiable, and the paper also provides a detailed decomposition framework and regularity criteria for flat one-sided and two-sided boundary points. Overall, the work delivers sharp, dimensionally optimal regularity results across arbitrary $Q$, $m$, and $n$, advancing the boundary theory for area-minimizing currents in higher codimension.

Abstract

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $Σ$ of $\mathbb{R}^{m+n}$, taking a boundary $Γ$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $Γ$ and $Σ$ are $C^{3, κ}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $Γ$ which is also dimensionally sharp. Moreover, we prove that if $p \in Γ$ admits an open neighborhood in $Γ$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular. Furthermore, we show that if the convex barrier condition is satisfied-namely, if $Γ$ is a closed manifold that lies at the boundary of a uniformly convex set and $Σ= \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $Γ$ that enable us to provide further information about the singular boundary set.

Figures (2)

• Figure 1: If $T$ is given by $\left\llbracket{D_1}\right\rrbracket + \left\llbracket{D_2}\right\rrbracket$, then $p$ is neither a one-sided nor a two-sided point.
• Figure :

Theorems & Definitions (116)

• Definition 1.1
• Definition 1.2
• Definition 1.3: Regular one-sided boundary points
• Definition 1.4: Regular two-sided boundary points
• Remark 1.5
• Definition 1.6: Singular points
• Remark 1.7
• Theorem A
• Theorem 1.9
• Remark 1.10