# Explicit Determination in \(\mathbb {R} ^{N}\) of \((N-1)\)-Dimensional Area Minimizing Surfaces with Arbitrary Boundaries

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## Abstract

Let \(N\ge 3\) be an integer and

*B*be a smooth, compact, oriented, \((N-2)\)-dimensional boundary in \(\mathbb {R} ^{N}\). In 1960, Federer and Fleming (Ann Math 72:458–520, 1960) proved that there is an \((N-1)\)-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. Thus, it is a question of long standing whether there is a numerical algorithm that will closely approximate the area-minimizing surface. The principal result of this paper is an algorithm that solves this problem—with the proviso that since one cannot guarantee the uniqueness of the area-minimizing surface with a particular given boundary, one must be willing to alter the boundary slightly, but by no more than a small amount that can be limited in advance. Our algorithm is currently theoretical rather than practical. Specifically, given a neighborhood*U*around*B*in \(\mathbb {R} ^{N}\) and a tolerance \(\epsilon >0\), we prove that one can explicitly compute in finite time an \((N-1)\)-dimensional integral current*T*with the following approximation requirements:- (1)
\({\text {spt}}(\partial T)\subset U\).

- (2)
*B*and \(\partial T\) are within distance \(\epsilon \) in the Hausdorff distance. - (3)
*B*and \(\partial T\) are within distance \(\epsilon \) in the flat norm distance. - (4)
\(\mathbb {M} (T)<\epsilon +\inf \{\mathbb {M} (S):\partial S=B\}\).

- (5)
Every area-minimizing current

*R*with \(\partial R=\partial T\) is within flat norm distance \(\epsilon \) of*T*.

## Keywords

Area-minimizing surfaces Mass-minimizing currents Codimension one surfaces Flat norm approximation Multigrid approximation Finite time algorithm## Mathematics Subject Classification

49Q15 49Q20 49Q05## Notes

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