Hermitian–Yang–Mills flow at and beyond a stability wall

Abstract

We prove all-time confinement and two-sided decay estimates for Hermitian–Yang–Mills flow whose initial metric is a sufficiently small transverse perturbation of a metric on a fixed finite-dimensional invariant graph. The bundles are small integrable deformations of a specified sum of four line bundles on the product of three projective lines. At the stability wall the directed extension graph is acyclic and connected; beyond it we treat a connected triangular class with locally free Harder–Narasimhan line quotients. In a moving diagonal presentation, the harmonic coefficients remain small although the diagonal metric factors become unbounded. Their energy decays reciprocally at the wall and crosses to exponential decay on the unstable side, uniformly in the ratio of wall distance to initial energy. This confinement yields an asymptotic phase relative to a chosen invariant graph: each such initial metric selects a unique reduced trajectory with exponentially decaying relative metric error. The phase is continuously differentiable in the initial metric at fixed bundle and polarization. At each fixed nearby unstable polarization, subtracting the linear diagonal drift leaves a limiting residual that is locally invertible in the three real diagonal initial coordinates.