Double bubble theorem

The double bubble theorem states that, in three-dimensional Euclidean space, the standard double bubble minimizes surface area among surfaces enclosing two regions of prescribed volumes separately.

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Standard double bubble: A configuration of three spherical or planar surfaces enclosing two adjacent regions and minimizing their total area. This is the configuration whose area the theorem proves minimal.
Hass–Hutchings–Schlafly double bubble proof: The 1995 proof that the standard double bubble minimizes area for two equal prescribed volumes in three-dimensional space. It established the conjecture in the equal-volume case before the full theorem.
Geometric measure theory: The study of geometric objects using measure-theoretic tools, including generalized surfaces and area. It supplies rigorous methods for defining and comparing candidate bubble boundaries.
Isoperimetric inequality: An inequality stating that a sphere encloses the greatest volume for a given surface area in Euclidean space. It treats one enclosed region, whereas the double bubble theorem treats two regions sharing a boundary.
Young–Laplace equation: An equation relating pressure differences across an interface to its curvature and surface tension. Constant pressure within each bubble requires each interface to have constant mean curvature.
Frank Morgan: An American mathematician known for work on geometric measure theory and minimal surfaces. His work helped develop the mathematical framework and proof strategy for the theorem.
Isoperimetric problem: The problem of finding shapes that enclose a prescribed volume with the least boundary area. The double bubble theorem is a two-region extension of this optimization problem.
Triple bubble conjecture: The conjecture that a particular standard cluster minimizes perimeter for three prescribed volumes in three-dimensional space. Adding a third region creates a harder cluster problem beyond the theorem’s two-region result.
Plateau's laws: Rules describing how soap films meet along curves and at junctions. The three interfaces meet along a curve at 120-degree angles, as soap-film junctions do.
Michael Hutchings: An American mathematician whose work includes symplectic geometry and geometric optimization. He co-authored the proof establishing the equal-volume double bubble result.