Lemma: Plateau covariance of the switched Euler operator

For $F\in {\mathcal{G}}_U$ and every formal scalar $\psi$,

Here the condition $\overline{U}\subset P_g$ is used.

Proof. Identical to Lemma lem:plateaucoord with ${\sigma }_{\epsilon }$ replaced by $g$, for both ${\mathfrak{A}}_g[q]=(1-g){\eta }^{-1}+g{\mathfrak{A}}_q$ and ${\varrho }_g[q]=(1-g)\cdot 1+g\sqrt{-q}$; the density pullback of the constant density $1$ is $\det DF=1$ outside $U$. The identity for ${\mathfrak{L}}_g$ follows from (eq:Lg) and (eq:nat1) in the same way.