Soap Film Solutions to Plateau's Problem

TL;DR

This paper introduces a new mathematical approach using differential chains to solve the general Plateau's problem, encompassing a wide variety of surfaces including nonorientable and junctioned ones, across all dimensions.

Contribution

It develops a novel method employing differential chains to prove the existence of area-minimizing surfaces for the general Plateau's problem, surpassing previous limitations.

Findings

First solution for a broad class of surfaces including nonorientable and junctioned surfaces

Applicable to all dimensions and codimensions in Euclidean space

Includes all previously solved special cases and more

Abstract

Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator to solve Plateau's problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Moebius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem because of a lack of a boundary operator. We use the chain complex of differential chains with its continuous boundary operator to solve a general version of Plateau's problem. We find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, both orientable and nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimensions in (\R^n).