## Abraham Frandsen and Dr. Michael Dorff, Department of Mathematics

The goal of this project was to explore a new problem in geometric optimization: isoperimetric surfaces with both boundary and volume constraints. The idea behind the problem is the following: what is the optimal way to enclose a given volume with a surface that must also span a given boundary? In this case, optimal means least possible surface area. If we were to consider only the volume constraint, we would have the classical isoperimetric problem—and by now it is well-known and mathematically proven that the best way to enclose a given volume is with a sphere. On the other hand, by considering only the boundary conditions, we find ourselves in the territory of minimal surface theory and the Steiner problem.

While isoperimetry and minimal surfaces are both well-studied areas in geometric optimization, little work had been done on my particular problem. Empirical solutions to the problem are not difficult to obtain—one needs only to construct the desired boundary, dip into a soapy solution to produce a soap film spanning the boundary, and then use a straw to blow a bubble in the midst of the film. The resulting soap surface is nearly minimal with respect to surface area, and allows one to make mathematical conjectures based on its visual appearance. The first mathematical results, however, appeared in the paper “Isoperimetric Surfaces with Boundary,”^{1} in which the authors employed calibration techniques to construct their minimal surfaces. As a subsequent paper showed^{2}, the results obtained in this first paper were limited to surfaces generated by uniform polytopes meeting certain conditions. My work sought to improve on these results by considering a broader class of surfaces.

Happily, in collaboration with Neil Steinburg and Donald Sampson, I was able to successfully carry out this objective. We now have a complete description of all piecewise-spherical equitent surfaces in R2 and R3. As an illustration, there are precisely six structurally distinct equitant surfaces with spherical faces in R3 that can be realized as soap films, and each of these corresponds in an important way with a regular deltahedron. Our results also give a complete description of piecewise-spherical equitant surfaces in all dimensions, with the added caveat that the surfaces have simplificial vertex figures. Our results have been published in the Pacific Journal of Mathematics under the title “Isoperimetric Surfaces with Boundary, II.”^{3}

As is always the case with research in mathematics, one has very little control from the outset over how the research progresses. In my case, I achieved my stated goals for the project more quickly than anticipated. This allowed me extra time to work on other problems in isoperimetry and equitent surfaces. These additional efforts yielded no results of significance, but laid a groundwork for future research into equitent surfaces with negatively-curved faces, as well as a metacalibration proof of the isoperimetric inequality in spaces of constant Gaussian curvature.

I found several helpful tools and activities throughout the course of my research. Most invaluable was the time spent with my collaborators, where we would discuss ideas, explore new avenues, and polish proofs. Computer software tools such as Mathematica (and its online counterpart, www.wolframalpha.com) and Surface Evolver were important for facilitating computations and understanding equitent surfaces at a more qualitative level. Access to mathematical journals and texts also proved vital for gaining background knowledge as well as specialized knowledge pertinent to the research. Overall, I found that the tools and resources at my disposal provided by BYU were very satisfactory.

In addition to the published paper referenced above, I also prepared and delivered a 12-minute presentation on the project for the BYU Spring Research Conference. The interested reader is encouraged to read our published article, keeping in mind, of course, that any polished mathematical text conceals hours and hours of work and research.

### References

- Rebecca Dorff and Donald Sampson et al. Isoperimetric surfaces with boundary. Proc. Amer. Math. Soc., 139:4467-4473, 2011.
- Jacob Ross, Donald Sampson, and Neil Steinburg. Soap film realization of isoperimetric surfaces with boundary. Involve, In Press.
- Abe Frandsen, Donald Sampson, and Neil Steinburg. Isoperimetric surfaces with boundary, II. Pacific Journal of Mathematics, 259,2:307-313,2012.