Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interior regularity of area minimizing currents within a $C^{2,α}$-submanifold

Stefano Nardulli, Reinaldo Resende

TL;DR

The work addresses interior regularity for area-minimizing $m$-currents $T$ in a $C^{2,\alpha}$ submanifold $\Sigma$, proving $\dim_{\mathcal{H}}(\operatorname{Sing}_{\mathrm{i}}(T))\le m-2$. Building on the De Lellis–Spadaro framework, the authors introduce an external center manifold $\mathcal{M}^{\ast}$ (potentially outside $\Sigma$) and an internal center manifold $\mathcal{M}$ (inside $\Sigma$), and develop Lipschitz approximations, $Q$-valued normal approximations $N^{\ast}$, and a Whitney decomposition to analyze branching. A PDE-based elliptic system controls the approximations on tilted cylinders, while a frequency function and a blow-up argument are used to derive a contradiction if the interior singular set were too large; this yields the desired bound and extends Allard–Allard–Almgren-type interior regularity to arbitrary codimension. The approach leverages a refined construction of $\mathcal{M}^{\ast}$ with $C^{3,\kappa}$ regularity to obtain the necessary monotonicity for the frequency, circumventing the need for $C^{3,\alpha}$-regularity of $\Sigma$ and enabling a codimension-robust interior regularity theory via Nash embedding considerations. The result sharpens the understanding of interior singularities for area-minimizing currents and provides a foundational tool for extending regularity results to Riemannian manifolds with low regularity.

Abstract

Given an area-minimizing integral $m$-current in $Σ$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $Σ$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}^{m+n}$ of class $C^{2,α}$, where $α>0$. This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class $C^{2,α}$.

Interior regularity of area minimizing currents within a $C^{2,α}$-submanifold

TL;DR

The work addresses interior regularity for area-minimizing -currents in a submanifold , proving . Building on the De Lellis–Spadaro framework, the authors introduce an external center manifold (potentially outside ) and an internal center manifold (inside ), and develop Lipschitz approximations, -valued normal approximations , and a Whitney decomposition to analyze branching. A PDE-based elliptic system controls the approximations on tilted cylinders, while a frequency function and a blow-up argument are used to derive a contradiction if the interior singular set were too large; this yields the desired bound and extends Allard–Allard–Almgren-type interior regularity to arbitrary codimension. The approach leverages a refined construction of with regularity to obtain the necessary monotonicity for the frequency, circumventing the need for -regularity of and enabling a codimension-robust interior regularity theory via Nash embedding considerations. The result sharpens the understanding of interior singularities for area-minimizing currents and provides a foundational tool for extending regularity results to Riemannian manifolds with low regularity.

Abstract

Given an area-minimizing integral -current in , we prove that the Hausdorff dimension of the interior singular set of cannot exceed , provided that is an embedded -submanifold of of class , where . This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class .
Paper Structure (12 sections, 20 theorems, 130 equations, 3 figures)

This paper contains 12 sections, 20 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Whitney decomposition
  3. Whitney decomposition, behaviour of the various parameters, and stopping conditions
  4. Fine properties of the Whitney decomposition
  5. Existence of the external center manifold
  6. Lipschitz approximations on tilted cylinders and interpolating functions
  7. Construction of the external center manifold Lg
  8. Proof of the construction estimates for Lg through the elliptic system
  9. Normal approximation on the external center manifold
  10. Frequency function and blow-up argument
  11. The frequency function
  12. Blow-up argument

Key Result

Theorem 1.1

Let $\alpha>0$, $m \geq 2$, $n \geq 1$, $\bar{n} \geq 0$, and let $\Sigma$ be an embedded $(m+\bar{n})$-submanifold of class $C^{2,\alpha}$ of ${\mathbb R}^{m+n}$. If $T$ is an area minimizing integral $m$-current in $\Sigma$, then

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (56)

  • Definition
  • Definition
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Stopping cubes conditions
  • ...and 46 more