(Exponential decay to semiflat metric away from $\mathfrak{D}$) Assume the setup of Theorem 3.33. In the subregion $\{\tilde{\ell}\gtrsim 1\}\subset M^{+}_{\nu}$, the deviation of $\omega_{+}$ from its zeroth Fourier mode $\bar{\omega}_{+}$ decays exponentially:

(3.13)

$$|\omega_{+}-\bar{\omega}_{+}|\leq CA^{-3/4}\nu e^{-\kappa\tilde{\ell}}.$$

The constant depends only on $\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{ij}$. The decay rate $0<\kappa<1$ can be chosen arbitrarily close to 1.