By Shuhui He
Suppose you are holding a wire frame in the shape of a helix with a straw in the middle connecting the ends of the helix and you dip it into a soap solution. Can you imagine how the soap film will spread across the wire frame when you pull it out? The resulting surface will be a helicoid and this is because the soap film tends to span in such a way that it minimises its surface area. Many other surfaces such as Enneper’s surface, the catenoid and Scherk’s doubly-periodic surface can be created by dipping wire frames of various shapes into a soap solution. These surfaces are called minimal surfaces.
Minimal surfaces are beautiful geometric objects with interesting properties. They can be precisely formulated in a differential geometry context. They are surfaces that have vanishing mean curvature, suggesting that every point on the surface is locally similar to a saddle. Their minimising propertymakes them relevant in a variety of contexts, including the soap film examplesabove as well as black hole horizons in general relativity. However, finding examples of minimal surfaces and classifying them was not achieved until the canonical Weierstrass representation was developed.
In my project, I first investigated examples of minimal surfaces and developed an understanding of elementary minimal surface theory. These groundings were then used to achieve the main goal of this project, which was to establish the Weierstrass representation for minimal surfaces. This representation formulamiraculously produces every possible minimal surface and provides an effective way to classify them.Inputting two analytic functions (technically one holomorphic function and one meromorphic function satisfying a certain property at its poles) to the formula will result in one minimal surface. This neat formulation connects two apparently distinct fields of mathematics, that is, differential geometry of submanifolds in three-dimensional Euclidean space and complex analysis. Inspired by this connection, our future study involves investigating connections between minimal surfaces and biharmonic surfaces via a Weierstrass-like representation.
Shuhui He was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.