Ambient Obstruction Flow

TL;DR

This paper studies the ambient obstruction flow, establishing smoothing estimates, singularity formation, and compactness results for solutions, advancing understanding of its geometric and analytic properties.

Contribution

It provides the first local smoothing estimates and a compactness theorem for the ambient obstruction flow, revealing singularity formation and long-time behavior.

Findings

Curvature blows up at finite time singularities.

Established local $L^2$ and pointwise smoothing estimates.

Proved a compactness theorem for solutions with bounded curvature.

Abstract

We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing estimates for the curvature tensor 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. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions with bounded $C^0$ curvature norm and injectivity radius bounded from below at one point. Finally, we use the compactness theorem to obtain a singularity model from a finite time singular solution and to characterize the behavior at infinity of a nonsingular solution.