Extended Solitons of the Ambient Obstruction Flow

TL;DR

This paper investigates ambient obstruction solitons, proving they are flat with constant scalar curvature, and extends these results to a broader class of solitons, revealing implications for the structure of the ambient obstruction flow.

Contribution

It introduces a generalized extended soliton equation allowing arbitrary conformal factors and establishes new integral inequalities and identities related to ambient obstruction solitons.

Findings

Any closed ambient obstruction soliton is flat with constant scalar curvature.

On compact manifolds, the flow has no fixed points other than flat metrics.

The results generalize the Bourguignon-Ezin identity and analyze special cases like homogeneous metrics.

Abstract

In this paper we expand on the work of the first author on ambient obstruction solitons, which are self-similar solutions to the ambient obstruction flow. Our main result is to show that any closed ambient obstruction soliton is ambient obstruction flat and has constant scalar curvature. We show, in fact, that the first part of this result is true for a more general extended soliton equation where we allow an arbitrary conformal factor to be added to the equation. We discuss how this implies that, on a compact manifold, the ambient obstruction flow has no fixed points up to conformal diffeomorphism other than ambient obstruction flat metrics. These results are the consequence of a general integral inequality that can be applied to the solitons to any geometric flow. Additionally, we use these results to obtain a generalization of the Bourguignon-Ezin identity on a closed Riemannian manifold and study the converse problem on a closed extended $q$-soliton. We also study this extended equation further in the case of homogeneous and product metrics.