## Experimental Mathematics

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

### Visualizing Ricci flow of manifolds of revolution

J. Hyam Rubinstein and Robert Sinclair

#### 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

#### Supplemental materials

- Supplementary Material: Associated software.