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A soap film has zero mean curvature at any point, and the area of any soap film bordered by wire is the surface of least area that spans the wire. What is the maximum total surface area of soap film that can be bordered by a connected frame formed by a unit length of wire?

For example, with a pair of tangent equal-sized circles, three surfaces are formed (below) and when the angle between the circles approaches $180^\circ$ the total surface area, including both sides of each soap film, approaches ($\pi$ lovers may replace with circle constant of choice) $\frac{1}{2\tau}+\frac{4}{\tau^2}$, which is an improvement on $\frac{1}{\tau}$ for a circle of radius $\frac{1}{\tau}$. Is it possible to do any better than that?

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Angela Pretorius
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  • I am not sure I understand your usage of "spans the wire". Normally I would take it to mean a surface $S$ such that $\partial S$ is the wire $W$. But your example is a closed (non-smooth) surface. Removing both disks would decrease its area and make $\partial S=W$ true. – ˈjuː.zɚ79365 Jun 10 '13 at 14:24
  • The zero mean curvature condition means that there is no pressure difference, and the soap film is not a boundaryless bubble. – Angela Pretorius Jun 11 '13 at 05:45

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