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

TL;DR

This paper classifies complete four-dimensional manifolds with finite quadratic curvature energy that are critical points of the curvature functional, showing they are either Einstein or specific product manifolds under certain curvature conditions.

Contribution

It extends the classification of critical metrics from compact to complete four-dimensional manifolds with finite energy, under a natural curvature inequality.

Findings

Critical metrics are either Einstein or Riemannian products of constant curvature surfaces.

The classification extends previous compact results to the complete non-compact setting.

The results rely on a natural inequality condition on the curvature operator of the second kind.

Abstract

We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. 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 $-c$ $(c\ne 0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting.