Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds

We study critical metrics of the curvature functional $\text{A}(g)={\int }_M\mathrm{\mid }R{\mathrm{\mid }}^2\text{vol}$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\text{A}(g)<\mathrm{\infty }$. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that $(M,g)$ is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$ $(c\mathrm{\ne }0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting.