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We study the behavior of the second order Renormalization Group flow onlocally homogeneous metrics on closed three-manifolds. In the cases $\mathbbR^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as theRicci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \R$, if thecurvature is small, then the flow expands as in the Ricci flow case, while ifthe curvature is large, then the flow contracts and forms a singularity infinite time. The main focus of the paper is the flow on the $\text{SU}(2)$,$\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ 3-geometries, with two of thethree principal directions set equal. The configuration spaces for thesegeometries are two dimensional, and we can consequently apply phase planetechniques to the study. For the $\text{SU}(2)$ case, the flow is everywherequalitatively the same as Ricci flow. For the $\text{Nil}$, $\text{Sol}$, and$\text{SL}(2,\R)$ cases, we show that the configuration space is partitionedinto two regions which are delineated by a solution curve of the flow thatdepends on the coupling parameter: in one of the regions, the flow developscigar or pancake singularities characteristic of the Ricci flow, while in theother both directions shrink. In the $\text{Nil}$ case we obtain acharacterization of the full 3-dimensional flow.

Second-Order Renormalization Group Flow of Three-Dimensional Homogeneous Geometries