Doughnut-shaped soap bubbles

Abstract: Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only responsible for conforming the soap bubble shape, the realized bubble surfaces are always minimal area ones. Here, we consider the problem of finding the axisymmetric minimal area surface enclosing a fixed volume $V$ and with a fixed equatorial perimeter $L$. It is well known that the sphere is the solution for $V=L^3/6\pi^2$, and this is indeed the case of a free soap bubble, for instance. Surprisingly, we show that for $V<\alpha L^3/6\pi^2$, with $\alpha\approx 0.21$, such a surface cannot be the usual lens-shaped surface formed by the juxtaposition of two spherical caps, but rather a toroidal surface. Practically, a doughnut-shaped bubble is known to be ultimately unstable and, hence, it will eventually lose its axisymmetry by breaking apart in smaller bubbles. Indisputably, however, the topological transition from spherical to toroidal surfaces is mandatory here for obtaining the global solution for this axisymmetric isoperimetric problem. Our result suggests that deformed bubbles with $V<\alpha L^3/6\pi^2$ cannot be stable and should not exist in foams, for instance.

• The paper shows that minimal area soap bubble surfaces constrained by fixed volume and equatorial perimeter can transition from spherical to toroidal (doughnut-shaped) forms when the enclosed volume is sufficiently small.
• Numerical analysis reveals a critical point where lens-shaped surfaces composed of spherical caps cease to be minimal, requiring a shift to toroidal minimal surfaces below a specific volume threshold.
• This research implies that unstable, deformed soap bubbles below this critical threshold should not be observed in stable foam structures and expands the understanding of how topological properties influence minimal surfaces under constraints.

Doughnut-shaped Soap Bubbles: A Variational Perspective

The paper by Deison Pr and Alberto Saa presents a compelling study of the geometric and topological aspects of soap bubbles constrained by specific boundary conditions. The focus lies in examining the axisymmetric surfaces that minimize area while enclosing a fixed volume with a fixed equatorial perimeter. This research explores the core of geometric optimization problems, exposing a nuanced interrelation between simple geometrical shapes and complex topological transitions.

Analysis and Results

The authors present a thorough exploration of the isoperimetric problem for soap bubbles, typically handled by spheres under free conditions. However, by introducing constraints on volume and perimeter, the study discovers a transition from spherical to toroidal, or doughnut-shaped surfaces. This transition becomes evident when the enclosed volume $V$ satisfies the inequality $V < \alpha L^3/6\pi^2$, with $\alpha \approx 0.21$. This finding contradicts common assumptions about soap bubbles, which usually conform to spheres due to surface tension acting predominantly.

The numerical results are pivotal in understanding this transition. The paper highlights a critical transition point wherein lens-shaped surfaces composed of spherical caps no longer represent minimal area solutions if $V$ decreases beyond the specified threshold. Instead, a mandatory shift to toroidal minimal surfaces takes precedence—a revelation that alters the conventional understanding of minimal surfaces under such constraints.

Implications

The implications of this research are twofold. Practically, the findings suggest that deformed soap bubbles, which cannot maintain stability below the critical volumetric threshold, should not be observed in stable foam structures. Theoretically, this paper expands the understanding of variational problems, providing insights into how topological properties can dictate minimal surfaces under constrained conditions. This bridges classical geometric problems with modern computational techniques, paving the way for future studies on more complex boundary conditions or multi-dimensional variational surfaces.

Future Directions

Potential future developments suggested by this research could include exploring higher-genus surfaces under varied constraints and further investigating the instability mechanisms leading to the breakdown of toroidal structures. This exploration may yield considerations relevant for practical applications in materials science, where controlling minimal surface formations could be critical.

By identifying the surface configuration that optimizes area under fixed volume and perimeter constraints, this paper enhances the comprehension of a subset of isoperimetric problems. It opens numerous opportunities to apply these principles in both theoretical contexts and real-world applications—ranging from understanding natural phenomena to designing materials with desired surface properties.