Compare flows side by side — Willmore, inverse MCF, and exotic evolutions
Some flows shrink a shape, some keep its area, some only take out the bends. Here they run side by side, and then you can draw a curve of your own.
Pick a shape and press play to run four flows on it at once. Plain curve shortening shrinks it to a point, while the area-preserving, affine, and length-preserving versions each keep something fixed.
Throughout this module we have studied individual geometric flows in depth. Now we bring them together, comparing and contrasting how different evolution equations act on the same shapes. Each flow has its own character — some shrink, some expand, some preserve volume — and watching them side by side reveals the deep relationships between geometry, analysis, and physics.
This lesson surveys four important flows beyond those we have already covered, then provides an interactive playground where you can draw any curve and evolve it under the flow of your choice.
Curve shortening flow shrinks to a point, area-preserving CSF makes the curve circular without shrinking, affine CSF is invariant under affine transformations, and length-preserving flow keeps the total arc length constant.
Where CSF shrinks surfaces, inverse MCF expands them outward at a rate inversely proportional to the mean curvature. Flat regions expand quickly while highly curved regions barely move.
The Yamabe flow evolves the conformal factor of a Riemannian metric to achieve constant scalar curvature R. It stays within a fixed conformal class, changing only the local scale factor.
| Flow | Order | Shrinks? | Preserves | Key Application |
|---|---|---|---|---|
| Heat Equation | 2nd | N/A (scalar) | Total heat (mass) | Diffusion, smoothing |
| Curve Shortening | 2nd | Yes (to a point) | Nothing | Topology, image processing |
| Mean Curvature | 2nd | Yes (can pinch) | Nothing | Soap films, minimal surfaces |
| Ricci Flow | 2nd | Can shrink/expand | Topology (with surgery) | Poincaré conjecture |
| Willmore | 4th | No | Surface area (approx) | Cell membranes, elasticity |
| Inverse MCF | 2nd | No (expands) | Hawking mass increases | General relativity |
| Surface Diffusion | 4th | No | Volume | Crystal growth, materials |
Press play to run Willmore flow and curve shortening flow on the same shape. Willmore flow takes out the bends without shrinking; watch its energy fall while CSF shrinks the curve away.
The Willmore energy of a surface is the integral of the squared mean curvature: W = ∫ H² dA. Willmore flow is the gradient flow that minimizes this bending energy. For curves, the analogous functional is ∫ κ² ds.
Unlike curve shortening flow, Willmore flow does not shrink curves — it only removes excess bending, driving shapes toward circles (which minimize bending energy for a given enclosed area). This fourth-order flow appears in computational geometry and biology, where it models the shapes of cell membranes and lipid vesicles.
Surface diffusion is a fourth-order flow governed by V = −Δ_s H, where Δ_s is the surface Laplacian. Unlike mean curvature flow, surface diffusion preserves the enclosed volume while minimizing surface area.
This flow models atomic diffusion along crystal surfaces in materials science. It smooths roughness while preserving mass — exactly how real material surfaces evolve at high temperatures.
Click to place points and close the curve by clicking near the start or pressing Enter. Then choose a flow, hit Run, and watch the perimeter, area, and roundness change.
Every geometric flow is a tool for improving geometry. The heat equation smooths functions. Curve shortening and mean curvature flow smooth shapes. Ricci flow smooths metrics. Willmore flow removes bending. Each flow encodes a different notion of “better geometry” through its choice of energy functional.
The interplay between these flows has driven some of the deepest results in modern mathematics — from the Poincaré conjecture to the Penrose inequality in general relativity. Understanding one flow illuminates them all.
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