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 $\frac12\ln r$ the singular limiting solution, then for every integer $k\ge0$, there exist positive constants $C_k$ and $c_k$ such that \[ \big\| u_\epsilon - \frac12\ln r \big\|_{C^k([r_0,2r_0])} \le C_k e^{-c_k/\epsilon}. \] Consequently, all results of the preceding paper can be refined.