Research in Classical Minimal Surface Theory

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DMS-0405836Title: Research in classical minimal surface theoryPI: William H. Meeks, University of Massachusetts (Amherst)ABSTRACTProposed Research Project AbstractIn this proposal the researcher will study the geometry, asymptoticbehavior, conformal structure and topology of properly embeddedminimal surfaces in three-dimensional Euclidean space. One of the maingoals of the proposal is to classify all of the properly embedded minimal surfaces which can be parametrized by domains in the Euclidean plane andto describe the asymptotic geometry of all finite genus examples.Related theoretical techniques concerning compactness, regularity andconvergence of minimal surfaces of locally bounded genus will beinvestigated as well. One hoped for application of this research isto classify all smooth finite group actions on the three-dimensionalsphere. As an outgrowth of his recent joint manuscript with CharlesFrohman on the topological classification for minimal surfaces, theresearcher proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted.Classical minimal surface theory has its roots in 18-th and 19-thcentury mathematics. Minimal surfaces are the firstimportant examples of what is called the calculus of variations, firstdescribed by Euler around 1735. Physically minimal surfaces can bemodeled locally as soap films on wires or by surfaces of least-area relative to their local boundaries. Minimal surfaces play an importantrole as a tool in the study of three-dimensional topology andRiemannian geometry. The subject of minimal surfaces has a broad impact in mathematics andphysical sciences. Minimal surfaces are stationary fluid interfaces,sotheir shapes arise in many physical problems. The work in thisproposal would classify the possible physical shapes which could occuras infinite interfaces. Many of the known examples of minimal surfacesare observed physically, so it is of interest to have a rigoroustheorem which predicts the shapes which can occur.The research proposed here strongly impacts the area of classicaldifferential geometry of surfaces in three-dimensional Euclideanspace. As is well known to geometers, minimal surface theory has beenand continues to be one of the principal tools for proving theorems ingeneral relativity and three-dimensional topology. One well-knownsuch application is Schoen and Yau's proof of the Positive MassConjecture. Recent work of Gabai on the Generalized Smale Conjectureshows the continued importance of minimal surfaces inthree-dimensional topology. Most of the research proposed here is related to andmotivated by the hope that it will lead to a positive solution of thePitts-Rubenstein Conjecture and to the classification ofthree-manifolds with finite fundamental group. This hoped fortopological application to one of the outstanding classificationproblems in mathematics has its roots in previous joint research bythe researcher, Peter Scott, Charles Frohman and S. T. Yau. In partbecause of the important connections with other areas of mathematicsand the ease in which it is possible to make computer graphicspictures of beautiful classical examples, minimal surfaces continue tobe one of the principal topics for popular science articles and publicscience exhibits. Thus, indirectly, the exciting research problemsoutlined in this proposal help bring many young scientists andmathematicians to the frontiers of research.