Day 17 | LThMath Advent Calendar
Did you ever play with soap bubbles? Did you ever think about changing the circle with any other shape and see what was going to happen with the bubbles? Plateau studied not the actual soap bubbles, but the soap film.
A ring dipped into a basin of soap water comes out spanned by a shimmering, iridescent film - a thin flat disk. This is the minimal surface defined by a circle. Plateau observed that any loop of wire, no matter how bent, bounds at least one soap film. Replacing the wire by a curve, or contour, and the soap film by a surface turns a specific physical observation into a mathematical question: Does at least one minimal surface (the mathematical version of a soap film) span every conceivable closed curve in space?
The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory.