João Henrique Santos de Andrade

Universidade de São Paulo, Brazil

Title: Quantitative stability of the total Q-curvature near minimizing metrics

Abstract: We discuss new quantitative stability estimates for the total \(Q\)-curvature functional of order \(k\in \mathbb{N}\) near minimizing metrics on any smooth, closed (compact and without boundary) \(n\)-dimensional Riemannian manifold, where \(k \in (1, \frac{n}{2}) \cap\mathbb{Z}\) and \(n\in \mathbb{N}\). More precisely, under suitable positivity assumptions, we show that (generically) the distance to the set of minimizing metrics is quadratically controlled by the energy deficit of the \(Q\)-curvature, extending recent results for the scalar curvature case \ (k = 1\). In the degenerate setting, we further prove the existence of Riemannian manifolds where this quadratic control can be improved: to cubic when \(k \in \{2, 3\}\), and to quartic for general \(k\in (1, \frac{n}{2}) \cap \mathbb{Z}\). These degenerate examples appear to be of independent interest and provide a framework for constructing solutions to the total \(Q\)-curvature flows that converge at slow (polynomial) rates.