Boundary Singularities Produced by the Motion of Soap Films

TL;DR

This paper investigates how the topology of soap films influences the location and nature of boundary singularities during instability, combining experimental, computational, and theoretical approaches to understand the role of initial geodesics and deformation paths.

Contribution

It provides new insights into the dependence of singularity type on surface topology and initial conditions, supported by experiments and numerical simulations.

Findings

Singularity location depends on deformation path and initial geodesics.

Narrowest necks with shortest geodesics are fastest-moving during instability.

Conjecture: linked initial geodesics lead to boundary singularities, unlinked lead to bulk singularities.

Abstract

Recent work has shown that a M\"obius strip soap film rendered unstable by deforming its frame changes topology to that of a disk through a 'neck-pinching' boundary singularity. This behavior is unlike that of the catenoid, which transitions to two disks through a bulk singularity. It is not yet understood whether the type of singularity is generally a consequence of the surface topology, nor how this dependence could arise from an equation of motion for the surface. To address these questions we investigate experimentally, computationally, and theoretically the route to singularities of soap films with different topologies, including a family of punctured Klein bottles. We show that the location of singularities (bulk or boundary) may depend on the path of the boundary deformation. In the unstable regime the driving force for soap-film motion is the mean curvature. Thus, the narrowest part of the neck, associated with the shortest nontrivial closed geodesic of the surface, has the highest curvature and is the fastest-moving. Just before onset of the instability there exists on the stable surface a shortest closed geodesic, which is the initial condition for evolution of the neck's geodesics, all of which have the same topological relationship to the frame. We make the plausible conjectures that if the initial geodesic is linked to the boundary then the singularity will occur at the boundary, whereas if the two are unlinked initially then the singularity will occur in the bulk. Numerical study of mean curvature flows and experiments support these conjectures.