Donald Sampson and Dr. Gary Lawlor, Mathematics Education

Goal/Purpose

My project was to further develop the technique of metacalibration, a new method of minimization proof, in order to prove the triple bubble conjecture: that the standard triple bubble is the least surface area way to separately enclose three given volumes.

Importance of Project

Over the past few decades, mathematicians have become increasingly interested in “multiple bubble problems.” These problems ask which figure among all those that separately contain a given number of volumes has the least surface area. It is conjectured that the minimizer is the standard shape that soap bubbles form when clumped together. Unfortunately, this has been proven only in some few cases. The traditional approach has been to use calculus of variations to isolate properties of the area-minimizing figure and compare all possible figures of this type. Some advancements in planar multiple bubble problems were made by Frank Morgan, who showed that perimeter-minimizing figures consist of circular arcs meeting at vertices of degree three, forming 120° angles [3]. This reduced the argument to listing all combinatorial types meeting these requirements. Using this result, students of the 1990 SMALL group under Frank Morgan proved that the so-called “standard double bubble” was perimeter-minimizing among all figures separately enclosing two fixed areas [1]. This method was also employed by Wacharin Wichiramala, whose doctoral dissertation proves the corresponding result for three separated areas [5]. Unfortunately, this approach is marred by an ever-increasing combinatorial complexity. For example, Wichiramala’s dissertation had to consider fifty-four possible configurations in order to prove minimization of the standard triple bubble. This complexity proves to be a significant barrier to further results.

The three-dimensional analogs of these multiple bubble problems have proven even more difficult to solve. Schwarz first proved in 1884 that the sphere was surface area minimizing for figures that enclose a fixed volume [4], but was not until 2002 that Frank Morgan and others proved the double bubble conjecture in R3 [2]. The final proof relied largely on the fact that the area-minimizing figure must be a surface of revolution. The triple bubble conjecture in space remains unproven. With no special symmetry, the triple bubble conjecture appears to be exceedingly difficult, if not impossible, to prove using this method.

My Project

In my research I further developed a new method of proof called metacalibration in order to tackle the as-yet-unproven triple bubble conjecture. Metacalibration is a generalization of previously used calibration techniques and was developed at BYU by my adviser, Dr. Gary Lawlor. While calibration techniques have proven very successful in fixed boundary problems, their applicability to fixed volume constraints has previously been limited. Metacalibration is a new approach to calibration proof that overcomes these limitations. This shift in application allows metacalibration to handle a wider range of problems, including the fixed area or volume conditions of multiple bubble problems. Metacalibrations were first used to give a simple new proof of the planar isoperimetric inequality. These results have been subsequently generalized to show boundary minimization of the n-sphere and to prove the planar double bubble conjecture.

Progress

My project set out to tackle the triple bubble conjecture as a series of sub-problems. So far, I have tackled and solved the first two of these problems. The first was developing a new way to overcome the problems of developing metacalibrations for problems in three or more dimensions. This was done using specialized differential forms to generalize our previous two-dimensional results into arbitrary dimension. These differential forms (a generalization of vector field integration) are defined in such a way that allows us to compare competitor figures directly by using Stoke’s theorem to prove the necessary inequalities.

Using this approach, I also completed the next step: generalizing our proof of the double bubble conjecture to higher dimensions. This proof combined the principles found in my previous proof of the double bubble conjecture in the plane and the specialized differential form approach to metacalibrations developed in the first sub-problem. Considering that the original proof of the double bubble conjecture in R3 took some twelve years to develop, producing a new and simpler proof of this result was a sizable achievement which we plan to soon submit for publication in a peer-reviewed journal. My fellow researchers and I have also been invited to give a presentation on these results at the Joint Meetings of the MAA and AMS in January.

In my future research activities, I plan to tackle the last obstacles to proving the triple bubble conjecture in space. While the triple bubble conjecture in the plane was initially proven in 2002, the proof given by Wichiramala does not extend into higher dimensions. A metacalibration approach, however, is more likely to generalize. Using previously noted relationships between the minimizing figure and particular differential forms, I intend to prove the planar triple bubble conjecture in a metacalibration framework.

Proving the triple bubble conjecture in the plane using metacalibration techniques will overcome a significant hurdle to proving the triple bubble conjecture in space. Expanding this result to higher dimensions by producing the appropriate specialized map will constitute the final sub-problem. Combining this specialized map with the differential form approach will put metacalibrations in a position to prove the triple bubble conjecture. While the ultimate goal of this project is to prove the triple bubble conjecture, it is hoped that the advancements made in metacalibrations theory will provide a base from which to tackle other unsolved problems in mathematics, and to simplify the proofs of those that have already been solved.

Citations

1. Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in R2 uniquely minimizes perimeter, Pacific J. Math. 159 (1993) 47-59.
2. Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Annals of Math. 155 (2002) 459-489.
3. Frank Morgan, Soap bubbles in R2 and in surfaces, Pacific J. Math. 165 (1994) 347-361.
4. Alan Siegel, A Historical Review of the Isoperimetric Theorem in 2-D, and its place in Elementary Plane Geometry. <http://www.cs.nyu.edu/faculty/siegel/SCIAM.pdf> (20 Oct 2008).
5. Wacharin Wichiramala, Proof of the planar triple bubble conjecture, J. Reine Angew. Math. 567 (2004) 1-49.