DeTurck background gauges for Yang–Mills heat flow: energy identities, Faddeev–Popov coercivity, and Gribov margin barriers

Background DeTurck gauges provide a strictly parabolic formulation of Yang–Mills heat flowrelative to a fixed reference connection. Three quantitative ingredients are developed for theanalysis of this gauge fixed evolution. Energy identities for the forced DeTurck–Yang–Mills heatsystem are derived, including covariant monotonicity relations adapted to background gauges. Aperturbative coercivity estimate for the corresponding Faddeev–Popov operator is then provedin a critical regime. Finally, a deterministic barrier mechanism is established for the accretivitymargin of a mixed Faddeev–Popov operator along the flow, yielding an explicit lower bound onthe time interval over which the gauge fixing map stays invertible.