Mean Curvature Flow

Surfaces evolving by mean curvature — soap films and singularity formation

Mean Curvature Flow

Curve shortening, one dimension up: every point of a surface moves at a speed set by how curved it is. Bumps melt, spheres shrink, and thin necks can pinch off.

Surface Evolution under MCF

Pick a surface, press play, and drag to rotate it. Red-hot regions are highly curved and move fastest; cool blue regions barely move.

Step: 0
Area: 0.000
Remaining: 100.0%
Low HH=0High H
Go deeper:Mean Curvature Flow

Mean curvature flow (MCF) evolves a surface by moving each point in the direction of its mean curvature normal: ∂F/∂t = H·n. At every instant, each point of the surface moves inward where it is convex and outward where it is concave, at a speed proportional to how curved it is.

MCF is the gradient flow of surface area — it decreases area as fast as possible. This makes it the natural higher-dimensional analogue of curve shortening flow. Soap films, which minimize area under constraints, sit at the fixed points of MCF where H = 0 everywhere — these are minimal surfaces.

The discrete Laplacian approximates how each vertex should move — vertices in regions of high curvature move faster, colored red-hot, while flat regions stay cool blue.

Go deeper:The Mean Curvature Flow Equation

For a family of immersions F(x, t) of a surface into R³, MCF is the PDE:

∂F/∂t = H · n = ½ΔF

Here H is the mean curvature (average of the two principal curvatures, κ₁ and κ₂), n is the unit normal, and ΔF is the Laplace-Beltrami operator applied to the position. The identity ΔF = 2Hn reveals that MCF is simply the heat equation on the surface itself.

The Dumbbell Neckpinch

Press play and watch the thin neck glow red: it shrinks far faster than the round ends and pinches off before the rest of the surface can collapse.

Step: 0
Neck radius: 0.0000
Neck remaining: 100.0%
Low HHigh H (neck)
Go deeper:Singularities in MCF

Unlike curve shortening flow (where simple closed curves always become round), MCF in higher dimensions can form singularities before the surface vanishes. The curvature blows up at isolated points or along curves:

  • Type I (spherical): The surface shrinks to a round point, like a convex surface collapsing. The blowup rate is |A|² ~ 1/(T - t).
  • Type II (neckpinch): A thin neck pinches off while the rest survives. This is the dumbbell singularity — curvature blows up faster than Type I.
  • Surgery: To continue the flow past singularities, one performs topological surgery — cutting the neck and capping off the resulting holes.

Area Functional Descent

Pick a starting radius and run the flow. The area graph falls in a straight line, A(t) = A(0) − 8πt; compare the numerical curve with the prediction.

Time: 0.000
R=2: A = 50.3
R=1.5: A = 28.3
R=1: A = 12.6
Progress: 0%
Go deeper:Gradient Flow of Area

MCF is the steepest descent for the area functional. For a sphere, area decreases linearly: A(t) = A(0) - 8πt, vanishing at the extinction time T = A(0)/(8π).

dA/dt = -2∫ H² dA ≤ 0
  • Area always decreases under MCF (unless H = 0 everywhere)
  • Rate of decrease equals twice the integral of H² over the surface
  • Fixed points (H = 0) are minimal surfaces — like soap films
Go deeper:Sphere Collapse
R(t) = √(R₀² - 2t)
  • Spheres remain spheres but shrink to a point
  • Cylinders remain cylindrical but collapse
  • Convex surfaces become round before vanishing (Huisken 1984)
Go deeper:Soap Films and Minimal Surfaces

A soap film spanning a wire frame finds the surface of least area — a minimal surface where H = 0 at every point. MCF can be thought of as the process by which a surface “relaxes” toward a minimal surface: it flows downhill on the area functional until it either reaches a minimum (H = 0) or collapses entirely.

Famous minimal surfaces — the catenoid, helicoid, Costa surface — are all stationary solutions of MCF. If you perturb them slightly, MCF will push them back toward equilibrium (if they are stable) or away from it (if they are unstable, like the catenoid).

Key Takeaways

  • Mean curvature flow moves each surface point at a speed equal to its mean curvature.
  • It lowers surface area as fast as possible; minimal surfaces (H = 0) stay put.
  • Convex surfaces become round and shrink to points (Huisken).
  • Non-convex surfaces can pinch off at a thin neck, as in the dumbbell.

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