On the Geometry of Minimal Surfaces

Sigmundur Gudmundsson

(Lund University)

A lecture given at the Lund Mathematical Society on 20. 2. 1996.

Fix a closed curve in 3-space find a surface of minimal area having the given curve as boundary. If the curve is planar the solution to this problem is rather simple i.e. the plane containing the curve. But as soon as we force the curve from being planar this becomes an interesting mathematical problem.

This was first studied by Lagrange in 1760. He proved that if a graph, of a function of two variables, is minimal then the function satisfies a heavily non-linear partial differential equation. But his solutions only gave planes.

The first non-trivial examples of minimal surfaces, the catenoid and the helicoid, were found by the French Jean Baptiste Marie Meusnier in 1776. It then took nearly 60 years until in 1835 the German Heinrich Ferdinand Scherk discovered some more. They gave him the highly respected Prize of the Paris Academy of Science.

In the middle of the 19th century the Belgian physicist Joseph Antoine Ferdinand Platau made several interesting experiments with soap films. They led him to the conjecture that to each simple closed curve there exists a corresponding minimal surface. A mathematically rigorous solution to this so called "Plateau-Problem" was first found by the American Jesse Douglas in 1931. Five years later he was awarded the Fields medal for his achievement.

In 1861 Weierstrass found an important link between the theory of minimal surfaces and that of holomorphic functions. He found a representation formula which can be used to construct any such surface in terms of two holomorphic functions. Despite this fact relatively few minimal surfaces with nice geometrical and topological properties were known as late as the early 1980's.

Recent development in computer graphics has changed this completely, revitalised the area and turned it into a thriving industry. Using geometrical arguments it is now possible to get hands on the holomorphic data needed in the Weierstrass formula and construct many beautiful examples which solve the original problem of Lagrange.

In this talk we revise the fundamentals of surface theory, describe the Weierstrass Representation Formula and use it for constructing some of the fascinating minimal surfaces. This will be a combination of Complex Analysis and introductory Differential Geometry.