Mean Curvature Flow

Surfaces evolving by mean curvature — soap films and singularity formation

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.

Interactive: Surface Evolution under MCF

Watch surfaces smooth out and shrink under mean curvature flow. 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.

Step: 0
Area: 0.000
Remaining: 100.0%
Low HH=0High H

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 = ΔF

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 = Hn reveals that MCF is simply the heat equation on the surface itself.

Gradient Flow of Area

dA/dt = -∫ H² dA ≤ 0
  • Area always decreases under MCF (unless H = 0 everywhere)
  • Rate of decrease equals the integral of H² over the surface
  • Fixed points (H = 0) are minimal surfaces — like soap films

Sphere Collapse

R(t) = √(R₀² - 2(n-1)t)
  • Spheres remain spheres but shrink to a point
  • Cylinders remain cylindrical but collapse
  • Convex surfaces become round before vanishing (Huisken 1984)

Interactive: The Dumbbell Neckpinch

The classic MCF singularity: a dumbbell-shaped surface develops a neckpinch. The thin neck has enormous mean curvature and shrinks far faster than the bulbous ends, pinching off to form a singularity before the rest of the surface can collapse.

Step: 0
Neck radius: 0.0000
Neck remaining: 100.0%
Low HHigh H (neck)

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.

Interactive: Area Functional Descent

MCF is the steepest descent for the area functional. For a sphere, area decreases linearly: A(t) = A(0) - 4πt, vanishing at the extinction time T = A(0)/(4π). Compare theoretical predictions with numerical evolution.

Time: 0.000
R=2: A = 50.3
R=1.5: A = 28.3
R=1: A = 12.6
Progress: 0%

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 surface points by ∂F/∂t = Hn, equivalent to the heat equation on the embedding
  • MCF is the gradient flow of surface area — area decreases monotonically at rate ∫H²dA
  • Convex surfaces become round and shrink to points (Huisken's theorem)
  • Non-convex surfaces can form neckpinch singularities — the dumbbell is the canonical example
  • Minimal surfaces (H = 0) are the fixed points — soap films are stationary under MCF

Next: Ricci Flow — extending geometric flows from surfaces to the intrinsic geometry of Riemannian manifolds, leading to the proof of the Poincaré conjecture.