Celebratio Mathematica — Rubinstein

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Works connected to Robert Michael Sinclair

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J. H. Rubinstein and R. Sinclair: “Visualizing Ricci flow of manifolds of revolution,” Exp. Math. 14 : 3 (2005), pp. 285–298. MR 2172707 Zbl 1081.53055 ArXiv 0406189 article

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 \( \mathbb{R}^3 \) is what makes direct visualization possible. The numerical lessons gained in developing this tool may be applicable to numerical simulation of the 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 \( \operatorname{SO}(3) \) 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 \( \mathbb{R}^3 \) and \( \mathbb{R}^4 \), 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.