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, a Poincaré Lemma, and good compactness properties to solve Plateau’s problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Möbius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincaré Lemma, and good compactness properties of the complex to solve Plateau’s problem in such generality as to 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, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝn.
Plateau’s problem Differential chains Differential forms Chainlets Dipole chains Dirac chains Poincare Lemma Extrusion Retraction Prederivative Pushforward Volume functional Soap films Triple branches Moebius strips Compactness Minimal sets
Mathematics Subject Classification
49Q15 49J52 49J99
Almgren, F.J.: Plateau’s Problem, an Invitation to Varifold Geometry. Benjamin, Elmsford (1966)
Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Bull. Am. Math. Soc. 81(1), 151–154 (1975)