AI 中文总结
该研究分析渐近锥形3-流形上$\mathrm{SU}(2)$单极子的大质量极限，证明其在非集中点指数阿贝尔化，收敛到含狄拉克奇点的可约单极子，确定逸出电荷并描述剩余阿贝尔歧义。
AI 中文摘要
设$(A_i,\Phi_i)$为一端渐近锥形3-流形上电荷$k>0$的有限能量$\mathrm{SU}(2)$单极子，其质量$m_i\to\infty$。取子序列后，质量重正化的能量测度集中于有限多个点$x_a$，浓度权重为$4\pi K_a$，其中$K_a$是位于$x_a$处的质量为1的欧氏单极子完整有限簇的总电荷。我们证明，在这些点的补集$M$上，场以指数方式阿贝尔化。通过沿希格斯场的单位方向平移希格斯场并应用规范变换，平移后的配对在局部光滑收敛到可约单极子$(A_\infty,\Phi_\infty)$，形式为$\Phi_\infty=-u\Psi_\infty$，$F_{A_\infty}=-*du\\,\Psi_\infty$，$u=4\pi\sum_aK_aG(\\,\cdot\\,,x_a)$，其中$\Psi_\infty$是平行单位截面，$G$是最小正格林函数。因此，$x_a$是电荷为$K_a$的狄拉克奇点。剩余极限的奇异部分由浓度点的加权0-循环和总簇电荷决定，不再保留单个欧氏轮廓或其分离层级。我们还证明$k-\sum_aK_a$恰好是通过渐近锥形端点逸出的电荷，并描述了剩余的平坦阿贝尔歧义。
英文摘要
Let $(A_i,Φ_i)$ be finite energy $\mathrm{SU}(2)$ monopoles of charge $k>0$ on an asymptotically conical $3$-manifold with one end, with masses $m_i\to\infty$. After passing to a subsequence, the mass-renormalized energy measures concentrate at finitely many points $x_a$ with concentration weights $4πK_a$, where $K_a$ is the total charge of the complete finite cluster of mass-one Euclidean monopoles lying over $x_a$. We prove that, on the complement $M$ of these points, the fields abelianize exponentially. After translating the Higgs fields by their masses along the unit Higgs directions and applying gauge transformations, the translated pairs converge smoothly locally to a reducible monopole $(A_\infty,Φ_\infty)$ of the form \[ Φ_\infty=-uΨ_\infty, \qquad F_{A_\infty}=-*du\,Ψ_\infty, \qquad u=4π\sum_aK_aG(\,\cdot\,,x_a), \] where $Ψ_\infty$ is a parallel unit section and $G$ is the minimal positive Green function. Consequently, $x_a$ is a Dirac singularity of charge $K_a$. The singular part of the residual limit is determined by the weighted $0$-cycle of concentration points and total cluster charges, and does not retain the individual Euclidean profiles or their separation hierarchy. We also show that $k-\sum_aK_a$ is exactly the charge escaping through the asymptotically conical end, and describe the residual flat abelian ambiguity.
Comments20 pages, no figures. Comments are welcome
