(The $C^{2}$ estimate.) There are constants $C$ and $\epsilon_{0}$ depending only on $J$ (possibly smaller than the $\epsilon_{0}$ of Theorem 4.5) such that for all $\epsilon<\epsilon_{0}$,

$$C^{-1}\omega_{\epsilon}\leq\tilde{\omega}_{\epsilon}\leq C\omega_{\epsilon}$$

where $\tilde{\omega}_{\epsilon}=\omega_{\epsilon}+i\partial\bar{\partial}u_{\epsilon}$.