Quantitative stability of the total $Q$-curvature near minimizing metrics

TL;DR

This paper establishes quantitative stability estimates for the total $Q$-curvature functional near minimizing metrics on smooth closed manifolds, extending previous results and constructing examples with higher-order control of the distance to minimizers.

Contribution

It provides new stability estimates for $Q$-curvature functionals for all $k < n/2$, including quadratic control on generic manifolds and higher power control through constructed examples.

Findings

Quadratic stability estimates near minimizers on generic manifolds.

Existence of manifolds with higher power control of the distance to minimizers.

Extension of previous work from the case $k=1$ to all $k < n/2$.

Abstract

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 $1 \leq k < \frac{n}{2}$. 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 $1 \leq k< \frac{n}{2}$, 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.