Geometry, supports, regulator

As in X-E XE:

Indices are moved with $\eta$ unless $q$ or $\bar{q}$ is written. ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ are the Klein–Gordon Green operators, ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ the massless ones. Fix $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;\mathbb{R})$, $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, a relatively compact plateau $U$ with $\overline{U}\subset P_g=\mathrm{int}\{g=1\}$, ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$, and $\delta >0$ with

Fix $0<\epsilon <\delta$. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is X-D's: smooth, ${\sigma }_{\epsilon }=0$ on $t\le 0$, ${\sigma }_{\epsilon }=1$ on $t\ge \epsilon$. In particular ${\sigma }_{\epsilon }=1$ on a neighbourhood of $\{t\ge 2\delta \}\supset \overline{U}\cup \mathrm{supp}g\cup K_f$. ${\mathcal{G}}_U$ is the group generated formally by $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$. For a vector field $\xi$, $F_s$ denotes its flow and $K\xi :={\mathcal{L}}_{\xi }\eta$, $(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$.