Soap Bubbles & Constant Mean Curvature

Pressure, double bubbles, and the isoperimetric problem

Soap Bubbles & Constant Mean Curvature

A bubble traps air at higher pressure than the room outside. That pressure bends the film into a surface whose curvature is the same everywhere — and it makes small bubbles swallow themselves.

Bubble Pressure & Young–Laplace

Set the two radii, then click "Equalize pressure". The smaller bubble has the higher pressure, so it deflates into the larger one.

Bubble 1
R =1.000 m
ΔP = 4γ/R =0.1000 Pa
Bubble 2
R =1.800 m
ΔP = 4γ/R =0.0556 Pa
Go deeper:The Young–Laplace Equation

While soap films have zero mean curvature, soap bubbles enclose a volume of air at higher pressure. The pressure difference across the film is balanced by surface tension, producing surfaces of constant mean curvature (CMC) — not zero, but uniform.

The Young–Laplace equation ΔP = 2γH relates pressure difference to mean curvature. A sphere is the simplest CMC surface, but there are many others — and the double bubble conjecture (proved in 2002) tells us exactly what shape minimizes area for two given volumes.

For a soap bubble of radius R, the pressure difference between inside and outside is ΔP = 4γ/R (the factor of 4 accounts for the two surfaces of the film). Smaller bubbles have higher pressure — which is why a small bubble connected to a large one will deflate into it.

In general, the Young–Laplace equation relates the pressure jump across any interface to its mean curvature: ΔP = 2γH. For a sphere with H = 1/R, this gives ΔP = 2γ/R per surface.

Double Bubble

Drag the volume sliders and watch the shared wall bend toward the larger bubble. Make the volumes equal and the wall goes flat.

Bubble 1
R₁ = 0.620
Bubble 2
R₂ = 0.492
Interface
R₃ = 2.387
Bubble 1
Bubble 2
Shared wall (1/R₃ = 1/R₁ − 1/R₂)
Go deeper:Double Bubble

Two bubbles sharing a wall. The shared interface has curvature satisfying 1/R₃ = 1/R₁ − 1/R₂. When the volumes are equal, the wall is flat (R₃ = ∞).

CMC Surface Gallery

Step through the sphere, unduloid, nodoid, and Wente torus. Each one has the same mean curvature at every point.

H = 1/R everywhere — the most symmetric CMC surface
Go deeper:CMC Surface Gallery

Beyond spheres, there is a rich family of constant mean curvature surfaces. The Delaunay surfaces (unduloids, nodoids) are CMC surfaces of revolution, and the Wente torus (1986) proved that CMC tori exist — a surprise to the mathematical community.

Delaunay surfaces: Complete classification of CMC surfaces of revolution.

Key Takeaways

  • CMC surfaces: Constant mean curvature, not zero — balanced by pressure.
  • Young–Laplace: ΔP = 2γH relates pressure to curvature.
  • Small bubbles lose: Higher internal pressure drives air into larger bubbles.
  • Double bubble conjecture: The standard double bubble minimizes area for two volumes (proved 2002).

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