There is a theorem by Jean Taylor that says that an almost minimal surface in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times a line and all the faces you can make from the center of a tetrahedron and its vertices.

This question has already been answered (in some sense) by Otis Chodosh as negative: Can we obtain those minimal cones by a deformation of minimal surfaces ( i.e. mean curvature = 0)?

Thanks, Mario