Moduli and Limits of Minimal Surfaces

Indiana University, Bloomington IN

Investigators

Abstract

Abstract Award: DMS-0505557 Principal Investigator: Matthias Weber This research project aims to combine two powerful new methods to investigate moduli spaces of complete, properly embedded minimal surfaces in euclidean space and their limits: Flat cone metrics are a geometric way to represent Riemann surfaces together with a (possibly multivalued) meromorphic 1-form, giving immediate information about the periods of the form. In combination with Teichmuller theory, cone metrics have been applied to existence and classification problems of mimimal surfaces. Noded surfaces are natural limits of Riemann surfaces under conformal pinching of curves. They have been used to construct families of minimal surfaces that degenerate to a suitable noded limit, using the implicit function theorem. We aim for a description of the noded limits of minimal surfaces as geometric limits of flat cone metrics. This adds a third type of limit to the formerly considered geometric and conformal limits. The conformal limits ignore the minimal surface information and consider only the Riemann surface limit, while the geometric limit retain the minimal surface nature but loose conformal and topological information by rescaling the surfaces in space. The new cone metric limit will incorporate both types of information. We hope that this description will lead to new examples and classification results. This research will be backed by numerical and graphical experiments based on a minimal surface library currently under development. Minimal surfaces are mathematical abstractions of 2-dimensional shapes that arise at different scales in nature: We are all familiar with soap film experiments, but such surfaces also have been observed at the nano scale as interfaces between block copolymers. Their mathematical properties are important for understanding the physical nature of new fabrics. The physical goal to minimize surface tension translates into a mathematical equation which has been of interest for over 250 years: The minimal surface equation is just at the border between what we understand by general theory and what we only can analyze numerically. Any advance at this point will most likely have its effects on other equations from mathematical physics and engineering. The methods which are being used to investigate minimal surfaces range from geometric analysis to numerical mathematics. Recent theoretical advances from partial differential equations and Teichmuller theory allow us to study 'extreme' minimal surfaces which comparable to soap films that nearly break under deformations. Understanding these extreme surfaces not only helps us to analyze the examples we have by breaking them apart into simpler pieces but also allows the construction of exciting new surfaces by putting suitable pieces together. The computer experiments we conduct require elaborate symbolic manipulations of the formulas involved, high-precision numerical computations to get accurate 3-dimensional surface data, and high performance computer graphics to visualize the actual surfaces.