Explicit Determination in R-N of (N-1)-Dimensional Area Minimizing Surfaces with Arbitrary Boundaries
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2019, Mathematica Josephina, Inc. Let N 3 be an integer and B be a smooth, compact, oriented, (N- 2) -dimensional boundary in RN. In 1960, Federer and Fleming (Ann Math 72:458520, 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 problemwith 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 neighborhoodU around B inRN and a tolerance> 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 (T) U.(2)B and T are within distance in the Hausdorff distance.(3)B and T are within distance in the flat norm distance.(4)M(T) < + inf { M(S) : S= B}.(5)Every area-minimizing current R with R= T is within flat norm distance ofT.
