Visualizing Ricci Flow of Manifolds of Revolution

Abstract

We present numerical visualizations of Ricci Flow of surfaces and 3-dimensional manifolds of revolution. Ricci_rot is an educational tool which visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in R3 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 3-dimensional manifolds like the 3-sphere, with a metric which is invariant under the action of 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 R3 and R4 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.