A local description of 2-dimensional almost minimal sets bounded by a curve

TL;DR

This paper investigates the local regularity of 2-dimensional almost minimal sets bounded by a smooth curve in Euclidean space, providing detailed descriptions near the boundary under certain proximity conditions to specific minimal cones.

Contribution

It offers a local description of such sets near the boundary when they are close to basic minimal cones, extending understanding of boundary regularity for these geometric objects.

Findings

Full description when close to a half plane or plane

Description when near a union of two half planes or a Y/T cone

Development of adapted monotonicity formulae for density

Abstract

We study the local regularity of sliding almost minimal sets of dimension 2 in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren's. We aim for a local description, in particular near L and modulo $C^{1+$\epsilon$}$ diffeomorphisms, of such sets $E$, but in the present paper we only obtain a full description when $E$ is close enough to a half plane, a plane or a union of two half planes bounded by the same line, or a transverse minimal cone of type $Y$ or $T$. The main tools are adapted near monotonicity formulae for the density, including for balls that are not centered on L, and the same sort of construction of competitors as for the generalization of J. Taylor's regularity result far from the boundary.