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

Abstract

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 \[ \big\| u_ε- \frac12\ln r \big\|_{C^k([r_0,2r_0])} \le C_k e^{-c_k/ε}. \] Consequently, all results of the preceding paper can be refined.