Mean Curvature: How Surfaces Bend, Explained Simply

Mean curvature is the average of the two principal curvatures: H = (kappa_1 + kappa_2)/2. Where Gaussian curvature multiplies them, mean curvature adds them, measuring the surface's overall tendency to bulge toward or away from its normal, like the average curving of a soap bubble.

Nature minimises it: soap films on wire frames settle into zero-mean-curvature shapes called minimal surfaces, since any nonzero average bending could relax to reduce area. Mean curvature also drives curvature flow, used to smooth noisy shapes in imaging and materials science.

Key Concepts

• Mean curvature H: average of principal curvatures, units 1/length; sign depends on the normal's direction.
• Minimal surface: H = 0 everywhere, e.g. a soap film; area-stationary, not necessarily area-minimising.
• CMC surfaces: constant nonzero H; soap bubbles are CMC, with pressure jump proportional to H (Young-Laplace law).
• Umbilic point: kappa_1 = kappa_2, equal bending every way; every point of a sphere or plane.
• Relation: H averages, K = kappa_1 kappa_2 multiplies; together they recover both principal curvatures.

Formulas

• From the forms: H = (eG - 2fF + gE)/(2(EG - F^2)).
• Sphere radius R: |H| = 1/R.
• Cylinder radius R: |H| = 1/(2R), from principal curvatures 1/R and 0.
• Young-Laplace: pressure jump Delta p = 2 gamma H across a film (gamma = surface tension).

Worked Example

Mean curvature of z = x^2 + y^2 at (1, 0, 1); reuse E = 5, F = 0, G = 1, e = g = 2/sqrt(5), f = 0.

1. Numerator: eG + gE = (2/sqrt(5))(1) + (2/sqrt(5))(5) = 12/sqrt(5).
2. Denominator: 2(EG - F^2) = 10.
3. H = (12/sqrt(5))/10 = 6 sqrt(5)/25 ≈ 0.537.
4. Check: K = (eg - f^2)/(EG - F^2) = (4/5)/5 = 0.16; principal curvatures ≈ 0.895 and 0.179 sum to 2H ≈ 1.074 and multiply to 0.16. Consistent.

Common Misconceptions

• H = 0 is not flat. A soap film on a twisted frame has H = 0 yet curves visibly; it is area-stationary, not planar.
• Zero average allows large opposing bends, e.g. +2 and -2 on a saddle-shaped minimal surface.
• The sign is conventional: reversing the normal flips H; only magnitude, or sign relative to a stated normal, is meaningful.

Real-World Applications

• Soap films and bubbles: films are minimal (H = 0), bubbles are CMC, shaped by the Young-Laplace law.
• Biology: cell membrane shapes follow mean-curvature energy, explaining red blood cell forms.
• Architecture: Frei Otto's tensile roofs use minimal-surface geometry for large spans with little material.

Mean curvature H = (kappa_1 + kappa_2)/2 averages a surface's principal bendings; on z = x^2 + y^2 at (1,0,1) it is 6 sqrt(5)/25 ≈ 0.537. Vanishing H gives the minimal surfaces of soap films, constant H the bubbles. It is the extrinsic counterpart to Gaussian curvature's intrinsic product.