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.
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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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