The triple $\bm{\omega}^{\mathcal{M}}$ is a closed definite triple on $\mathcal{M}$. Furthermore, for any $k\in\mathbb{N}$, there is some constant $C_{k}>0$ independent of the gluing parameter $\beta>0$ such that

(6.56)

$\displaystyle\|Q_{\bm{\omega}}-\Id\|_{C^{k}(\mathcal{M})}\leq C_{k}e^{-\delta_{q}\beta},$

where $\delta_{q}>0$ is a uniform constant independent of $\beta$ and $Q_{\bm{\omega}}=(Q_{ij})$ is defined by

(6.57)

$$\frac{1}{2}\omega_{i}\wedge\omega_{j}=Q_{ij}\dvol_{\bm{\omega}^{\mathcal{M}}}.$$

Here the norm is measured with respect to $g_{\beta}$, the Riemannian metric associated to $\bm{\omega}^{\mathcal{M}}$.