Proposition: Support of the plateau remainder

Let $A\gamma =0$ on $U$ and $\xi \in {\mathfrak{g}}_U$. The remainder $R_{\xi }:={\mathcal{A}}_1[K{\zeta }_{\mathrm{gf}}[s_{\xi }];{\varphi }_0]$ in (eq:plateausum) is given by (eq:switch) with $\zeta ={\zeta }_{\mathrm{gf}}[s_{\xi }]$. As a functional of $\zeta$ its kernel is supported in

There is no ramp term: ${\zeta }_{\mathrm{gf}}[s_{\xi }]$ vanishes on $\{t\le \epsilon \}$ and ${\mathsf{X}}_1[K{\zeta }_{\mathrm{gf}}]={\zeta }_{\mathrm{gf}}$ exactly. With the observation-plateau hypothesis $\mathrm{supp}f\subset P_g$ the second set is empty and the kernel is supported in $\mathrm{supp}\mathrm{d}g\cap J^{+}(\mathrm{supp}\xi )\cap {\mathcal{D}}_f$. Without it, $\mathrm{supp}((1-g)f)$ remains, as in XI-E's Route 2.

Proof. Lemma lem:switch and Proposition prop:gfprops(b). The first term of (eq:switch) is supported in $\mathrm{supp}f\cap \{g\ne 1\}\cap \mathrm{supp}\zeta$; the second, after transposition onto ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$, in $\mathrm{supp}\partial g\cap \mathrm{supp}\zeta \cap {\mathcal{D}}_f$.