Communications in Analysis and Geometry

Volume 21 (2013)

Number 2

Second-order renormalization group flow of three-dimensional homogeneous geometries

Pages: 435 – 467



Karsten Gimre (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Christine Guenther (Department of Mathematics and Computer Science, Pacific University, Forest Grove, Oregon, U.S.A.)

James Isenberg (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)


We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb{R}^3$ and $\text{SO}(3)\times \mathbb{R}$, the flow is qualitatively the same as the Ricci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \mathbb{R}$, if the curvature is small, then the flow expands as in the Ricci flow case, while if the curvature is large, then the flow contracts and forms a singularity in finite time. The main focus of the paper is the flow on the $\text{SU}(2)$, $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\mathbb{R})$ 3-geometries, with two of the three principal directions set equal. The configuration spaces for these geometries are two dimensional, and we can consequently apply phase plane techniques to the study. For the $\text{SU}(2)$ case, the flow is everywhere qualitatively the same as Ricci flow. For the $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\mathbb{R})$ cases, we show that the configuration space is partitioned into two regions which are delineated by a solution curve of the flow that depends on the coupling parameter: in one of the regions, the flow develops cigar or pancake singularities characteristic of the Ricci flow, while in the other both directions shrink. In the $\text{Nil}$ case we obtain a characterization of the full three-dimensional flow.

Published 9 April 2013