Collaborative Research: FRG: Minimal Surfaces, Moduli Spaces and Computation

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The global theory of minimal surfaces in space is in a phase ofexplosive growth. Many new methods of constructing completeembedded minimal surfaces have recently been found; in place of adearth of examples just a few years ago, we now have a quitevaried collection of surfaces, including infinite families. Abasic problem is to classify these examples, i.e. collect theminto families with common properties and understoodlimits. Fruitful approaches have recently been developed thatcombine numerical simulation with methods from the theory ofgeometric structures on surfaces and classical complex analysis,notably Teichmuller theory. Some of the problems the team willattack are: Are there embedded minimal surfaces with oneheliciodal end and arbitrary genus? Is the classical Scherksurface the unique desingularization of a pair of planes? Ofwhat families is the Scherk surface the limit point? At the sametime, the group hopes to make progress on simulation of minimalsurfaces. For example, we hope to set up a library of Weierstrassrepresentations of minimal surfaces which is reproducible, fullydocumented, and useful as a research tool.A guiding philosophy in many areas of science, from physics tobiochemistry to ecology, is that nature is maximally efficient;indeed, many explanations of natural phenomena have at theirfoundation the assumption that the phenomenon has optimized someor several of its features in the expression we witness. At itsbase, this philosophical principle is mathematical in nature: wesearch for principles in science that can be formulated asextremal problems. In mathematics, we can make this assumption ofoptimality very rigorous by expressing it as an equation. Thisleaves us with the problem of understanding all of the solutionsof that equation. In this project, we aim to study one very richtype of optimization problem, the minimal surface problem, whichis already known to have a number of quite subtlecharacteristics. (A minimal surface is one for which each smallpiece has less area than any other surface with the sameboundary.) The study of these surfaces has its origins inphysical problems studied first by Euler; then, a century later,the problem also arose in the studies of the behavior of rotatingdroplets and soap films by F. Plateau. Today the applicationsrange from cosmology to the understanding of the structure ofstable periodic structures in compound copolymers. As in manyother optimization problems, for the minimal surface problem, wedo not have much general information about solutions to theequation expressing extremality. At present though, we do have awide variety of examples which help to guide our intuition, andwhich we are beginning to organize. It is thus a good modelproblem, enriching our understanding of all optimizationproblems.