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Visualizing Ricci flow of manifolds of revolution. (English) Zbl 1081.53055

Summary: 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 \(\text{SO}(3)\) 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.
On the web page access to Supplementary Material: Associated software is offered.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53-04 Software, source code, etc. for problems pertaining to differential geometry
65-05 Experimental papers (numerical analysis) (MSC2010)
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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