Sliding minimal cones in the 3-dimensional half-space

TL;DR

This paper classifies minimal cones in a half-space with sliding boundary conditions, providing new two-dimensional examples crucial for understanding boundary regularity in minimal surface problems.

Contribution

It introduces a classification of one-dimensional minimal cones and presents four novel two-dimensional minimal cones in a three-dimensional half-space.

Findings

Classified one-dimensional minimal cones in the half-plane.

Constructed four new two-dimensional minimal cones in the half-space.

These cones cannot be derived from Cartesian products of lower-dimensional cones.

Abstract

Solving the Plateau problem means to find the surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists of giving a suitable definition to the notions of 'surface', 'area' and 'boundary'. The sliding boundary condition has been introduced by David in order to study the boundary regularity of minimal sets. In order to do that an important step is to know the list of minimal boundary cones, that is to say tangent cones on boundary points of minimal surfaces. In this paper we focus on cones contained in an half-space and whose boundary can slide along the bounding hyperplane. After giving a classification of one-dimensional minimal cones in the half-plane we provide four new two-dimensional minimal cones in the three-dimensional half space (which cannot be obtained as the Cartesian product of the real line with one of the previous cones).