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

Abstract

Let $N\ge3$ be an integer and $B$ be a smooth, compact, oriented, $(N-2)$-dimensional boundary in ${\Bbb R}^{N}$. In 1960, H. Federer and W. Fleming proved that there is an $(N-1)$-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. In 1970 H. Federer proved the definitive regularity result for such a codimension one minimizing surface. 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. Specifically, given a neighborhood $U$ around $B$ in ${\Bbb R}^{N}$ 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) 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 $T$.