Limiting behavior of a class of Hermitian Yang-Mills metrics, II: exponential decay

In this note, the geometric set-up, the rank two bundle, the local HYM ansatz, and the global gluing construction are the same as in the preceding work \cite{Fu}. The new point is an exponential estimate for the radial ordinary differential equation obtained near each branch point. If $u_{\epsilon }$ denotes the local radial solution and $\frac{1}{2}\ln r$ the singular limiting solution, then for every integer $k\ge 0$, there exist positive constants $C_k$ and $c_k$ such that $$\parallel u_{\epsilon }-\frac{1}{2}\ln r{\parallel }_{C^k([r_0,2r_0])}\le C_k{e}^{-c_k\mathrm{/}\epsilon }\mathrm{.}$$ Consequently, all results of the preceding paper can be refined.