Experimental Mathematics

Visualizing Ricci flow of manifolds of revolution

J. Hyam Rubinstein and Robert Sinclair

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 3 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.em/1128371754

Mathematical Reviews number (MathSciNet)
MR2172707

Zentralblatt MATH identifier
1081.53055

Subjects
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)

Keywords
Ricci flow neckpinch mathematical visualization

Citation

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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