Experimental Mathematics

Visualizing Ricci flow of manifolds of revolution

J. Hyam Rubinstein and Robert Sinclair

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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.

Article information

Experiment. Math., Volume 14, Issue 3 (2005), 285-298.

First available in Project Euclid: 3 October 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53-04: Explicit machine computation and programs (not the theory of computation or programming)

Ricci flow neckpinch mathematical visualization


Rubinstein, J. Hyam; Sinclair, Robert. Visualizing Ricci flow of manifolds of revolution. Experiment. Math. 14 (2005), no. 3, 285--298. https://projecteuclid.org/euclid.em/1128371754

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