A nonlocal approximation of the area in codimension two

TL;DR

This paper introduces a fractional $s$-mass concept for codimension two surfaces in $ ^n$, proving its convergence to classical area as $s$ approaches 1, with implications for geometric measure theory.

Contribution

It defines a new fractional $s$-mass for codimension two surfaces and proves its $ ext{Gamma}$-convergence to classical area, extending nonlocal geometric analysis.

Findings

Fractional $s$-mass is well-defined for codimension two surfaces.

The fractional $s$-mass $ ext{Gamma}$-converges to classical area.

Pointwise convergence of fractional $s$-mass to classical area as $s o 1$.

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 $\Gamma$-convergence, with respect to the flat topology, and pointwise convergence to the $(n-2)$-dimensional area.