A Bj\"orling Representation for Jacobi Fields on Minimal Surfaces and Soap Film Instabilities

Abstract

We develop a general framework for the description of instabilities on soap films using the Bj\"orling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localised if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schr\"odinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasise that the geometric nature of the Bj\"orling representation allows direct connection with instabilities observed in soap films.