# The Case for a General $q$-flow: An Investigation of Ambient Obstruction Solitons

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We will discuss a new program of studying solitons using a geometric flow for a general tensor $q$. We begin by establishing a number of results for solitons to the geometric flow for a general tensor, $q$, examining both the compact and non-compact cases. From there, we will apply these results to the ambient obstruction flow, the Bach flow ($n\geq 5$), and the Cotton flow to see the utility of this approach. We juxtapose this approach with a more hands-on approach used to show that any homogeneous gradient Bach soliton ($n=4$) that is steady must be Bach flat; that the only homogeneous, non-Bach-flat, shrinking gradient solitons are product metrics on $\mathbb{R}^2 \times S^2$ and $\mathbb{R}^2 \times H^2$; and there is a homogeneous, non-Bach-flat, expanding gradient Bach soliton.