一类埃尔米特-杨-米尔斯度量的极限行为，II：指数逼近
Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation
AI总结：
本文是相关研究续篇，针对两个一维复环面乘积上稳定秩二全纯向量丛的近似埃尔米特-杨-米尔斯度量，证明大凯勒极限下该近似度量对精确度量具有任意阶指数精度的渐近描述，克服了全局C^0估计等解析难点。
AI中文摘要：
本文是文献[9]的续篇，在该文献中，第一作者在由两个一维复环面乘积上的双重谱覆盖导出的稳定秩二全纯向量丛上，构造了一族近似埃尔米特-杨-米尔斯度量$H_{0,ε}$。\n 我们证明，在大凯勒极限下，这些近似度量给出了精确埃尔米特-杨-米尔斯度量的任意阶、指数精度的渐近描述。更确切地说，$H_{0,ε}$的平均曲率在每个$C^k$范数下都呈指数衰减。此外，设$H_{1,ε}$为精确埃尔米特-杨-米尔斯度量，且有$H_ε=H_{0,ε}^{-1}H_{1,ε}$，则在归一化后，对每个非负整数$k$，存在正常数$C_k$和$c_k$，使得$\|H_ε- Id\
英文摘要：
This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics $H_{0,ε}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large Kähler limit. More precisely, the mean curvature of $H_{0,ε}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,ε}$ denotes the exact Hermitian Yang--Mills metric and \[ H_ε=H_{0,ε}^{-1}H_{1,ε}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_ε- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}ε}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.
