Lemma: Vanishing of the breaking density on the plateau

${\mathcal{K}}_{\nu }[\gamma ,\varphi ](x)=0$ for every $x\in P_g\cap \{t>\epsilon \}$ and every smooth $(\gamma ,\varphi )$. In particular ${\mathcal{K}}_{\nu }=0$ on a neighbourhood of $\overline{W}$.

Proof. On the open set $P_g$, $g\equiv 1$, so $1-g$ and all derivatives of $g$ vanish: ${\mathcal{B}}_{g,\nu }=0$. For $t>\epsilon$, ${\sigma }_{\epsilon }\equiv 1$, so $w=0$ and all derivatives of ${\sigma }_{\epsilon }$ vanish: $w{Q}_{\nu }(A\gamma )=0$, ${{\mathcal{D}}_{\sigma }}_{\epsilon }=0$, ${Z_{\sigma }}_{\epsilon }=w{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }=0$, and every term of ${\mathfrak{R}}_{{\sigma }_{\epsilon },\nu }$ carries $\partial {\sigma }_{\epsilon }$ or ${\partial }^2{\sigma }_{\epsilon }$. By (eq:W), $\overline{W}\subset P_g\cap \{\epsilon <t<T\}$, which is open.