FINAL REPORT: DOE-FG03-95ER25250

Abstract

The research conducted in this project concerns the geometry of extremal surfaces, embedded minimal surfaces in particular. The methods include geometric analysis, computational simulation, mathematical visualization and software development. Minimal surface research stands at the intersection of partial differential equations, calculus of variations, complex function theory and topology. Advances in this area are often---as is the case with our research---tied to the development and implementation of computational methods and tools of mathematical visualization. Understanding the structure of the space of minimal surfaces has been important in applications from cosmology to structural engineering, as well as other applied areas including polymer physics. The subject has benefited from the discovery of new examples by the use of computation, examples far beyond the range current theoretical construction techniques. Not only are these surfaces important for the understanding of equilibrium morphology via inter-material dividing surfaces, they arise in the study of grain boundaries and dislocations. These same examples are in turn signposts for the further theoretical development in mathematics. This research project has made fundamental advances in the study of equilibrium interfaces. Carrying on the parent project that was based at the University of Massachusetts, we have: Proved the existence of large families of periodic minimal surfaces that serve as models for compound polymers. Developed software to simulate the transmission electron microscopy of the nanostructure of block copolymers, and in the understanding of materials whose structure was previously not known. Pioneered the use of numerical approximation and image simulation for minimal and CMC surfaces in the theoretical investigation of these variationally define equilibrium interfaces. Developed and maintained an archival site and model libraries This website was one of the first such sites and has served as a model for others. We have proved the existence of an embedded minimal surface of genus one with one helicoidal end. This is the first properly embedded minimal surface with infinite total curvature and finite topology to be found since the helicoid was shown to minimal by Muesnier in the 1770's. During the period that this project was carried out at MSRI, we provided support and consultation for mathematicians who had interests in using our graphical software and expertise in their research. In addition, Hoffman organized, initiated or acted as MSRI coordinator for a variety of scientific meetings and workshops on topics ranging from visualizat