Tetrahedral points and classification of Almgren-minimizing sets in ${\mathbb{R}}^3$

We prove that a two-dimensional Almgren-minimizing set in three dimensions which is sufficiently close in Hausdorff distance to a tetrahedral cone contains an interior tetrahedral point. As a consequence, we obtain quantitative regularity near tetrahedral cones and we complete the classification of entire Almgren-minimizers in ${\mathbb{R}}^3$.