Lemma: Exact plateau invariance of the total Lagrangian

Let $F\in {\mathcal{G}}_U$, and let $\mathcal{L}[q,\varphi ]=(1-g){\mathfrak{L}}_{\eta }[\varphi ]+g{\mathfrak{L}}_q[q,\varphi ]-(2{\kappa }^2{)}^{-1}\sqrt{-q}R[q]$. For all smooth $(q,\varphi )$, the density $\mathcal{L}[F^{\ast }q,F^{\ast }\varphi ]-\mathcal{L}[q,\varphi ]$ is supported in a compact subset of $U$ and integrates to zero.

Proof. $F$ is the identity outside a compact $L\subset U$, and $F(L)=L$. Outside $L$ all pulled-back fields and their derivatives coincide with the originals, which gives the support statement.

$\sqrt{-q}R$ and ${\mathfrak{L}}_q$ are natural: $\omega [F^{\ast }q,F^{\ast }\varphi ]=F^{\ast }(\omega [q,\varphi ])$ as densities. Hence $\int (F^{\ast }\omega -\omega )=0$ for $\omega =\sqrt{-q}R$, since $F$ is a diffeomorphism of $M$ onto itself, and $\int g{F}^{\ast }\omega =\int (g\circ F^{-1})\omega =\int g\omega$ for $\omega ={\mathfrak{L}}_q$, since $g\circ F^{-1}=g$: on $L$ both sides equal $1$, and off $L$$F^{-1}=\mathrm{id}$. Finally $\mathrm{supp}(1-g)\subset M\setminus P_g\subset M\setminus L$, where $F^{\ast }\varphi =\varphi$ with all derivatives; so the $(1-g){\mathfrak{L}}_{\eta }$ term is unchanged pointwise.