Backward uniqueness with critical covariant drift and a quantitative background slice theorem

Two analytic ingredients that repeatedly appear in background gauge formulations ofYang–Mills type parabolic flows are isolated and treated in a single place. The first partconcerns bundle valued parabolic operators whose principal part is covariant and whose lowerorder term is a scale critical covariant drift; a Carleman inequality is derived in this regimeand is then used to obtain backward uniqueness and an ancient rigidity principle. Such driftterms are not optional: in DeTurck or background gauge fixed evolutions they arise from thegauge fixing transport and from the covariant linearization.The second part provides the complementary geometric estimate that makes the gaugefixed formulation quantitatively stable. A local slice theorem around a smooth backgroundconnection is proved with explicit control in terms of an accretivity margin for the correspondingFaddeev–Popov operator, yielding in particular a quantitative description of afirst background Gribov region and coercivity on the orthogonal complement of the gaugedirections. Combined, these statements allow one to place the linearized and gauge fixedYang–Mills heat operator into the class covered by the Carleman theory without conditionalassumptions.