Sliding almost minimal sets and the Plateau problem

Abstract

We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren's notion of minimality, which is well suited in the context of Plateau problems relative to soap films. We are especially interested in regularity properties near a boundary curve, where we would like to get a local C1 description of 2-dimensional almost minimal sets in the spirit of J. Taylor's theorem, but we first study weaker and more general results (local Ahlfors regularity, rectifiability, limits, monotonicity of density), which we describe far from the boundary for simplicity. There we insist on some simpler techniques, in particular the use of Federer-Fleming projections.