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A nonlocal approximation of the area in codimension two

Michele Caselli, Mattia Freguglia, Nicola Picenni

TL;DR

The paper introduces a fractional $s$-mass for codimension-two surfaces in $\mathbb{R}^n$ by pairing a linking constraint with a fractional Sobolev energy, and proves that $(1-s)^2\mathbb{M}_s(\Sigma)$ Γ-converges, under flat convergence of boundaries, to the $(n-2)$-dimensional area with multiplicity $\frac{2\pi\omega_{n-1}}{n}\sum_i d_i\mathcal{H}^{n-2}(\Sigma_i)$. The authors establish both a liminf inequality via slicing and a local positivity argument, and a matching limsup construction using vortex-type configurations and a Ginzburg-Landau reduction to obtain energy upper bounds. The results generalize the codimension-one BBM-type convergence to codimension two, with the correct $(1-s)^2$ scaling, and are local in nature, suggesting robust extensions to ambient Riemannian manifolds and to higher codimensions. The framework is poised for min-max applications to existence and regularity questions for codimension-two minimal-like objects, and connects fractional Sobolev energies, linking theory, and geometric measure theory in a unified relaxation approach to area minimization.

Abstract

For $s\in (0,1)$ we introduce a notion of fractional $s$-mass on $(n-2)$-dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $Γ$-convergence, with respect to the flat topology, and pointwise convergence to the $(n-2)$-dimensional area.

A nonlocal approximation of the area in codimension two

TL;DR

The paper introduces a fractional -mass for codimension-two surfaces in by pairing a linking constraint with a fractional Sobolev energy, and proves that Γ-converges, under flat convergence of boundaries, to the -dimensional area with multiplicity . The authors establish both a liminf inequality via slicing and a local positivity argument, and a matching limsup construction using vortex-type configurations and a Ginzburg-Landau reduction to obtain energy upper bounds. The results generalize the codimension-one BBM-type convergence to codimension two, with the correct scaling, and are local in nature, suggesting robust extensions to ambient Riemannian manifolds and to higher codimensions. The framework is poised for min-max applications to existence and regularity questions for codimension-two minimal-like objects, and connects fractional Sobolev energies, linking theory, and geometric measure theory in a unified relaxation approach to area minimization.

Abstract

For we introduce a notion of fractional -mass on -dimensional closed, orientable surfaces in . Moreover, we prove its -convergence, with respect to the flat topology, and pointwise convergence to the -dimensional area.
Paper Structure (21 sections, 28 theorems, 252 equations)

This paper contains 21 sections, 28 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Motivation and related works
  4. Future directions
  5. Definitions and notations
  6. Linking number between submanifolds
  7. Fractional Sobolev spaces
  8. Slicing with subspaces
  9. Flat convergence of boundaries
  10. Prescribing singularities and weakly linking maps
  11. Existence of a competitor
  12. Existence of a minimizer
  13. Linking with squares
  14. Positivity of the fractional mass
  15. Proof of the main result
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Sigma\subset \mathbb R^n$ be as in defn:Sigma. Then it holds that where the $\Gamma$-limit is intended with respect to the flat convergence of boundaries in $\mathbb R^n$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Proposition 1.2: Lower bound
  • Proposition 1.3: Upper bound
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 53 more