Interactive visualization of curvature flows

After a short motivation, we introduce several different curvature flows: A naive flow on the curvature under the heat equation, the curve-shortening flow, the mean curvature flow for imbedded surfaces, and the Ricci flow on surfaces of revolution and for abstract 2-manifolds. The main focus is on interactive visualizations using animations of curves, surfaces, and in the case of the Ricci also using flow on a metric field similar to Tissot’s indicatrix. We refer to and briefly demonstrate an existing applet for the curve-shortening flow, and present our own code written in SageMath / Python (and MAPLE) for other flows, including for the Ricci flow on surfaces of revolution, thus recreating animations first presented by Rubinstein and Sinclair. Our code is publicly available on GitHub and invites for further experimentation, especially with different initial shapes.