# Sliding almost minimal sets and the Plateau problem

## TL;DR

This paper reviews regularity results for minimal and almost minimal sets, focusing on a sliding variant relevant to Plateau problems, and explores boundary regularity and geometric properties using Federer-Fleming projections.

## Contribution

It introduces a sliding minimality concept suited for Plateau problems and studies regularity properties near boundaries, extending classical results with new techniques.

## Key findings

- Establishment of local Ahlfors regularity and rectifiability for almost minimal sets.
- Analysis of density monotonicity and limit behaviors of these sets.
- Application of Federer-Fleming projections to study regularity near boundaries.

## 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.