Large mass limits of $\mathrm{G}_2$ and Calabi--Yau monopoles: calibrated concentration, Higgs zeros, and abelianization

Abstract

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical $\mathrm{G}_2$-manifolds and Calabi--Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise--Pigati--Stern, and Li's singular abelian compactness theory in a common $Θ$-monopole framework, we prove that the mass-renormalized Yang--Mills--Higgs and intermediate energy measures converge to $8π\|T\|$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal S$ is the calibrated support, $\mathcal Z$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal C$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal S\subset\mathcal Z\subset\mathcal C=\mathcal S\cup\mathcal O$, where $\mathcal O$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant--Salamon $\mathrm{G}_2$-manifolds and the Stenzel Calabi--Yau $3$-fold, we prove that $\mathcal O=\varnothing$. On $X\setminus\mathcal C$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.