The following uniform curvature estimates hold for every point in $\mathcal{M}$:

1. (1)

   Let $r(\bm{x})$ denote the Euclidean distance to the monopole points, then there exists constants $C>0$ so that such that for each $1\leq m\leq m_{0}$ for every $\bm{x}\in B_{r_{0}}(p_{m})$ with $r_{0}\equiv\frac{1}{2}\InjRad_{g_{0}}(\mathbb{T}^{2})$, the following curvature estimates hold,

   (7.22)

   $\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C\beta,&0\leq r(\bm{x})<\beta^{-1},\\ \frac{C}{\beta^{2}r(\bm{x})^{3}},&\beta^{-1}\leq r(\bm{x})<r_{0}.\end{cases}$

   In terms of the intrinsic distance function with respect to the Riemannian metric $g_{j}$,

   (7.23)

   $\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C\beta,&0\leq r(\bm{x})<\beta^{-1},\\ \frac{C}{\beta^{\frac{1}{2}}d_{m}(\bm{x})^{3}},&\beta^{-1}\leq r(\bm{x})<r_{0}.\end{cases}$

2. (2)

   If $\bm{x}$ is in the neck region but has some definite distance away from the monopoles, the following curvature estimates hold for some uniform constant $C>0$,

   (7.24)

   $\displaystyle|\Rm|(\bm{x})\leq\begin{cases}\frac{C}{\beta^{2}z(\bm{x})},&\frac{r_{0}}{10}<|z(\bm{x})|<\beta,\\ \frac{C}{\beta^{3}},&-T_{1}\leq z(\bm{x})\leq-\beta\text{and}\ \beta\leq z(\bm{x})<T_{2}.\end{cases}$

3. (3)

   For $\bm{x}\in X_{b_{\pm}}(T_{\pm})\subset\mathcal{M}$, there is a constant $C$ so that

   (7.25)

   $\displaystyle|\Rm|(\bm{x})\leq\begin{cases}C&d(\bm{x})<\zeta_{\pm}\\ \frac{C}{d(\bm{x})^{2}}&d(\bm{x})\geq\zeta_{\pm},\end{cases}$

   where $d(\bm{x})$ is the distance to a base point in $X_{b\pm}$.

4. (4)

   For $\bm{x}\in DZ_{\pm}\subset\mathcal{M}$, there is a constant $C>0$ so that

   (7.26)

   $\displaystyle|\Rm|(\bm{x})\leq\frac{C}{\beta^{3}}.$