Quantitative stability of the total Q-curvature near minimizing metrics

Under appropriate positivity hypotheses, we prove quantitative estimates for the total k-th order Q-curvature functional near minimizing metrics on any smooth, closed n-dimensional Riemannian manifold for every integer 1 k < n2. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the Q-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case k=1. Next we prove, for any integer 1 k< n2, the existence of an n-dimensional Riemannian manifold such that the k-th order Q-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field.