Visualizing Ricci Flow of Manifolds of Revolution

Abstract

We present numerical visualizations of Ricci flow of surfaces and three-dimensional manifolds of revolution. Ricci_ rot is an educational tool that 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 three-dimensional manifolds like the 3-sphere, with a metric that 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.

Keywords:

• Primary 53C44, 53-04
• Ricci flow
• neckpinch
• mathematical visualization