UC Irvine

We establish several results for the ambient obstruction flow (AOF), a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell which is based on the Fefferman-Graham ambient obstruction tensor. The flow may be regarded as a higher order analogue of Ricci flow, and the critical metrics for this flow may be regarded as generalizations of Einstein metrics. First, we obtain local L2 smoothing estimates for the curvature tensor along AOF and use them to prove pointwise smoothing estimates for the curvature tensor. We use the pointwise smoothing estimates to show that the curvature must blow up for a finite time singular solution to AOF. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions to AOF with bounded C0 curvature norm and injectivity radius bounded from below at one point. The compactness theorem allows us to obtain a singularity model from a finite time singular solution to AOF and to characterize the behavior at infinity of a nonsingular solution to AOF. Our final result is a rigidity theorem, which states that under suitable conditions a metric that is critical for AOF and has small scale-invariant integral energy has vanishing Riemann curvature tensor.