Boundary singularities produced by the motion of soap films

Abstract

Recent work has shown that a Möbius 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 the 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.minimal surfaces | topological transitions | systoles T he study of singularities, be it their dynamic evolution (1-3) or their location and structure in a system at equilibrium (4, 5), has a long history. The main focus of research has been either on systems without boundaries, where both the static and dynamic cases are fairly well understood, or on finite systems with prescribed boundaries displaying static singularities, as for example defects in liquid crystals (6) and superfluids (7), Langmuir monolayers (8), and quantum field theories (9). Work on dynamic aspects of these bounded systems is limited, particularly in the case of singularity formation accompanying a topological change (10). For instance, to our knowledge, there is no mathematical tool available to predict from initial conditions whether a singularity in a bounded system will occur in the bulk or at the boundary.In a previous paper (11) we found that a soap film in the shape of a Möbius band spanning a slowly deforming wire would change its topology through a singularity that occurs at the boundary. This simple example provided a first model to investigate the dynamical aspects of the formation of a boundary singularity from the moment the system becomes unstable, and until a new stable configuration is reached. While progress was made in the particular case of the Möbius band, left unanswered were questions of greater generality: (i) Are there any other configurations of films spanning...

Instability of a Möbius Strip Minimal Surface and a Link with Systolic Geometry

“…2c represent the vectorial displacement field ψ cn along the systolic curve (discussed below). The pattern of displacements shown is qualitatively identical to that observed in experiment [16,18], where the interface collapses towards the frame.…”

“…3a, superimposed on the surface. As in other examples studied to date [18], much of this systolic curve lies on the catenoidlike neck of the surface. Figure 3b shows, on a contour plot of the unstable eigenfunction ψ c , that C M passes very near the maximum of the unstable mode.…”

“…For example, the catenoid instability leads to a singularity in the bulk of the film, whereas a Möbius strip minimal surface transforms to a disc through a boundary singularity [16][17][18]. Moreover, at least in these two cases, we observed that the location of the singularity (bulk or boundary) can be determined by studying the topology of the systolic curve on the surface [1,19].…”

“…In the absence of analytical results on the shape of such a geodesic, we employ a numerical scheme introduced recently [18]. This method is based on the use of the curve length L as a Lyapunov function in a gradient flow relaxational scheme.…”

“…This approximation shares with the full numerical result the feature of tracking the ridge in the Gaussian curvature and passes near the maximum of the marginal eigenfunction. Previously [18], we conjectured that if the local systolic curve C is linked to the boundary ∂S then the singularity that occurs when the minimal surface becomes unstable will be on ∂S, otherwise it will be in the bulk. The present results lead to the additional conjecture that when a neck on a minimal surface S becomes unstable the local systolic curve C passes through the neighborhood of the maximum of the unstable mode.…”