Gradient Ambient Obstruction Solitons on Homogeneous Manifolds

TL;DR

This paper investigates homogeneous solitons related to the ambient obstruction flow, revealing conditions under which they are trivial or possess specific geometric structures, especially in four dimensions.

Contribution

It characterizes homogeneous gradient Bach solitons in dimension four, identifying when they are Bach flat or specific product metrics, and constructs new examples of expanding solitons.

Findings

Compact ambient obstruction solitons with constant scalar curvature are trivial.

Steady homogeneous gradient Bach solitons in dimension four are Bach flat.

Non-Bach-flat shrinking solitons are specific product metrics on  imes S^2 and  imes H^2.

Abstract

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on $\mathbb{R}^2\times S^2$ and $\mathbb{R}^2 \times H^2$. We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor $q$.