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

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: