Pressure, double bubbles, and the isoperimetric problem
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.
Adjust the radii of two connected soap bubbles. The smaller bubble has higher internal pressure (ΔP = 4γ/R), so when connected, air flows from small to large. Click "Equalize pressure" to watch the small bubble deflate into the large one.
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.
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 = ∞). Drag the volume sliders to see how the interface curvature changes.
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.