From now on, we will fix weight parameters in the definition of weight spaces, which allows us to prove the uniform injectivity estimate in Proposition 7.15 and apply the implicit function theorem to complete the proof the main theorem in Section 7.4. The parameters $\delta$, $\mu$, $\nu$ are fixed as follows (similar to the specification of the parameters in Section 6.1):

1. (GP1)

   (Fix $\nu$) The parameter $\nu\in\mathbb{R}$ is chosen such that

   (7.74)

   $\displaystyle\nu\in(-1,0).$

2. (GP2)

   (Fix $\alpha$) The Hölder constant $\alpha\in(0,1)$ is chosen such that

   (7.75)

   $\displaystyle\nu+\alpha<0.$

3. (GP3)

   (Fix $\delta$) The constant $\delta>0$ is chosen such that

   (7.76)

   $$0<\delta<\delta_{G}\equiv\frac{1}{n\cdot(|k_{-}|+|k_{+}|)}\cdot\min\{\delta_{e},\delta_{Z_{1}},\delta_{Z_{2}},\epsilon_{Z_{1}},\epsilon_{Z_{2}},\sqrt{\lambda_{D}}\},$$

   where $\sqrt{\lambda_{D}}$ is in Lemma 6.7 (Liouville theorem on $Q$), $\delta_{e}>0$ is in Proposition 4.23, $\delta_{Z_{1}},\delta_{Z_{2}}$ are the constants in Proposition 7.4 applied to $Z_{1},Z_{2}$, and $\epsilon_{Z_{1}},\epsilon_{Z_{2}}$ are the constants in Theorem 5.2 applied to $Z_{1},Z_{2}$.

4. (GP4)

   (Fix $\mu$) The parameter $\mu>0$ is chosen as

   (7.77)

   $$\mu=(1-\frac{1}{n})(\nu+2+\alpha).$$

As a comparison, on the neck region $\mathcal{M}_{T}$, the corresponding choice of parameters are given in (6.10), (6.11), (6.12) and (6.13).