Geometry, supports, regulator

Indices are moved with $\eta$ unless $q$ 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; $⟨v,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f⟩=⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}v,f⟩$. Write $n_{\mu }={\partial }_{\mu }t$, $n^{\mu }={{\delta }^{\mu }}_0$.

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 $0<\epsilon <\delta$ once and for all. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is X-D's, flat at both ends. ${\mathcal{G}}_U$ is the group generated formally by $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$; every $F\in {\mathcal{G}}_U$ equals the identity outside a compact subset of $U$.