Minimal Surface with a Cavity of Given Perimeter

Abstract

We pose a variant of the classical minimum surface problem inspired by a simple experiment with soap films: to find the surface of least area containing a cavity of given perimeter. We show that the equilibrium surface is governed by a system of two equations one of which is the zero mean curvature condition. The other equation states that the curvature of the cavity's contour is constant and that its principal normal lies in the plane tangential to the surface. A gradient descent simulation confirms the analytical equilibrium conditions and yields configurations qualitatively consistent with experiment.