Visualizing Ricci flow of manifolds of revolution

Abstract

We present numerical visualizations of Ricci flow of surfaces and three-dimensional manifolds of revolution. {\tt Ricci\_rot} is an educational tool that visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in {\small $\mathbb{R}^3$} is what makes direct visualization possible. The numerical lessons gained in developing this tool may be applicable to numerical simulation of Ricci flow of other surfaces. Similarly for simple three-dimensional manifolds like the 3-sphere, with a metric that is invariant under the action of {\small $SO(3)$} with 2-sphere orbits, the metric can be represented by a 2-sphere of revolution, where the distance to the axis of revolution represents the radius of a 2-sphere orbit. Hence we can also visualize the behaviour of such a metric under Ricci flow. We discuss briefly why surfaces and 3-manifolds of revolution remain embedded in {\small $\mathbb{R}^3$} and {\small $\mathbb{R}^4$}, respectively, under Ricci flow and finally indulge in some speculation about the idea of Ricci flow in the larger space of positive definite and indefinite metrics.