Lemma 14.18.11ECE72025D8E

Proof. The map $\Gamma \mapsto \eta $ with $\eta ^{n-1}=\Gamma $ is smooth on positive forms, since in a frame it is the inverse of the smooth map $\eta \mapsto \eta ^{n-1}$, whose differential is invertible. The openness argument in the proof of theorem 14.10.1, with the $\mathrm{d}V_0$-mean in place of the $\gamma ^ n$-mean, gives a bounded inverse for the derivative with respect to the potential and the scalar between the relevant Hölder spaces. The implicit function theorem gives local smooth solution maps, and uniqueness shows that they agree on overlaps. Applying this for every order of Hölder regularity gives the assertion for smooth data. $\square$