Gradient ambient obstruction solitons on homogeneous manifolds

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 R-2 x S-2 and R-2 x 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.

Automated summary

This article explores the properties of homogeneous solitons of the ambient obstruction flow, proving that compact ambient obstruction solitons with constant scalar curvature are trivial. In the context of dimension 4, it is shown that homogeneous gradient Bach solitons must be Bach flat if they are steady, with non-Bach-flat shrinking gradient solitons being limited to specific product metrics. Additionally, a non-Bach-flat expanding homogeneous gradient Bach soliton is constructed, along with the establishment of various results for solitons in geometric flow by a general tensor q.