Negative regular sublevels force two-homogeneous projection escape

Since $G_{\nu }=0$ $\nu$-almost everywhere but $\nu$ is not a minimizer, [F2] gives a point ${\theta }_0$ with $G_{\nu }({\theta }_0)<0$. Choose $\eta >0$ so that $G_{\nu }({\theta }_0)<-\eta <0$ is a regular value; if $G_{\nu }<0$ on the whole sphere, take $K={\mathbb{S}}^{d-1}$ and omit its boundary. Otherwise compactness and regularity make $K=\{G_{\nu }\le -\eta \}$ compact with nonempty interior and give a positive lower bound $\beta$ for $\mathrm{\mid }{\mathrm{\nabla }}_S{G}_{\nu }\mathrm{\mid }$ on its boundary. With the outward normal, $-{\mathrm{\nabla }}_S{G}_{\nu }\cdot n_{out}\le -\beta$.

For projected measures $\zeta$ in a sufficiently small $BL$ neighborhood of $\nu$, [F1] and duality give $\mathrm{\parallel }{m}_{\zeta }-m_{\nu }\mathrm{\parallel }\le C_{\mathrm{\Phi }}\mathrm{\parallel }{h}^2(\zeta )-\nu {\mathrm{\parallel }}_{BL}$, where $C_{\mathrm{\Phi }}={\sup }_{\mathrm{\parallel }z\mathrm{\parallel }\le 1}(\mathrm{\parallel }⟨z,\mathrm{\Phi }⟩{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{Lip}⟨z,\mathrm{\Phi }⟩)$. The Lipschitz bound in [F4] then yields $\mathrm{\parallel }{G}_{\zeta }-G_{\nu }{\mathrm{\parallel }}_{C^1}\le C_R{C}_{\mathrm{\Phi }}\mathrm{\parallel }{h}^2(\zeta )-\nu {\mathrm{\parallel }}_{BL}$ for a finite $C_R$. Choose $ϵ$ small enough that, whenever the projection remains in that ball, $G_{h^2({\mu }_t)}\le -\eta \mathrm{/}2$ on $K$ and the outward normal velocity is at most $-\beta \mathrm{/}2$ on $\mathrm{\partial }K$.

While the projection remains in the chosen ball, a characteristic with direction in $K$ cannot leave $K$. At its boundary the potential is differentiable, its angular velocity is $-{\mathrm{\nabla }}_S{G}_{h^2({\mu }_t)}$, and the strict negative outward normal component from step 2.1 rules out a first outward crossing: at a first exit the signed distance to $K$ would have nonnegative right derivative, contradicting the strictly negative derivative. For $u=r\theta$ in this cone, [F5] gives $r^{\mathrm{\prime }}=-2r{G}_{h^2({\mu }_t)}(\theta )\ge \eta r$, so $r(t)\ge r(s)e^{\eta (t-s)}$.

The initial cone has positive mass and is preserved by step 3.1. Since it is the increasing union of subsets with $r\ge 1\mathrm{/}n$, one such subset has mass $m_{\ast }>0$ for some $r_{\ast }=1\mathrm{/}n$. For every particle in that subset, its radius grows at least as $r_{\ast }{e}^{\eta (t-t_0)}$ until the projection exits the ball. Therefore $h^2({\mu }_t)({\mathbb{S}}^{d-1})\ge m_{\ast }{r}_{\ast }^2{e}^{2\eta (t-t_0)}$ throughout that interval.

The constant test function $1$ has bounded-Lipschitz norm $1$. Step 4.1 makes $\int 1d{h}^2({\mu }_t)$ grow without bound while $\nu ({\mathbb{S}}^{d-1})$ stays fixed and finite. Hence $\mathrm{\parallel }{h}^2({\mu }_t)-\nu {\mathrm{\parallel }}_{BL}\ge \mathrm{\mid }{h}^2({\mu }_t)({\mathbb{S}}^{d-1})-\nu ({\mathbb{S}}^{d-1})\mathrm{\mid }$ exceeds $ϵ$ after finite time if the flow stayed in the ball. It must exit by that time, proving the assertion.