Relative metric estimates near reducible Yang–Mills connections

Abstract

We study the natural $L^2$ metric near a central Yang–Mills connection on a closed surface. In the classical quadratic symplectic model, we prove a relative metric estimate with error quadratic in the harmonic deformation parameter, uniformly across all stabilizer types. We also compute the leading metric correction using the Green operators of the reference connection. The proof first eliminates the gauge directions with a fixed positive spectral gap and then minimizes over the remaining finite-dimensional orbit; it requires no lower bound on a positive orbit eigenvalue. The same argument compares every small eigenvalue of the gauge Laplacian with the infinitesimal stabilizer-action operator, including at changes of kernel dimension. As a curved finite-dimensional comparison, quaternionic projective and hyperbolic reductions admit exact orbit-metric formulas on their flat moment-map zero cone. A four-cycle is realized in both settings with independently vanishing weights. The resulting estimates distinguish uniform control of the quotient metric from estimates that depend on the smallest orbit scale.