A separating two-homogeneous flow has globally minimizing projection limits

Suppose for contradiction that $\nu$ is not a global minimizer. By [F1] and [F2], $G_{\nu }$ vanishes $\nu$-almost everywhere but is negative at some sphere point. Choose a negative regular sublevel $K$ with that point in its interior, as in [F2].

Weak convergence on the compact sphere implies convergence of total masses by testing the constant function. The bounded-Lipschitz unit ball is equicontinuous and uniformly bounded; Arzelà–Ascoli [F6] gives a finite uniform $\delta$-net for it. Weak convergence controls each member of that finite net, and the convergent masses bound the error from replacing any test by its net representative. Letting $\delta \downarrow 0$ proves bounded-Lipschitz convergence. Thus there is $t_0$ after which $\mathrm{\parallel }{h}^2({\mu }_t)-\nu {\mathrm{\parallel }}_{BL}<ϵ$, where $ϵ$ is the escape radius from [F4]. By [F3], the support at $t_0$ separates some spheres ${\rho }_a{\mathbb{S}}^{d-1}$ and ${\rho }_b{\mathbb{S}}^{d-1}$ with $0<{\rho }_a<{\rho }_b$. For a point ${\theta }_0\in \mathrm{int}K$, the radial path $r\mapsto r{\theta }_0$ between these spheres must meet $\mathrm{spt}{\mu }_{t_0}$. That support point has positive radius and direction in $\mathrm{int}K$; every sufficiently small open neighborhood of it has positive ${\mu }_{t_0}$ mass, so the negative cone in [F4] has positive mass.

Applying [F4] at time $t_0$ gives $t_1>t_0$ with $\mathrm{\parallel }{h}^2({\mu }_{t_1})-\nu {\mathrm{\parallel }}_{BL}\ge ϵ$, contradicting the choice of $t_0$. Thus $\nu$ is a global minimizer.

By [F5], $F({\mu }_t)\to F(\nu )$. Since $\nu$ is a minimizer over all finite nonnegative measures, $F(\nu )={\inf }_{\lambda \ge 0}F(\lambda )$. This proves both conclusions.