Minimal surfaces and the new main inequality

We establish the new main inequality as a minimizing criterion for minimal maps to products of $\mathbb{R}$-trees, and the infinitesimal new main inequality as a stability criterion for minimal maps to ${\mathbb{R}}^n$. Along the way, we develop a new perspective on destabilizing minimal surfaces in ${\mathbb{R}}^n$, and as a consequence we reprove the instability of some classical minimal surfaces; for example, the Enneper surface.