Computer Graphics in Minimal Surface Theory

Abstract

We give a summary of the computer-aided discoveries in minimal surface theory. In the later half of the 20th century, the global properties of complete minimal surfaces of finite total curvature were investigated. Proper embeddedness of a surface is one of the most important properties amongst the global properties. However, before the early 1980s, only the plane and catenoid were known to be properly embedded minimal surfaces of finite total curvature. In 1982, a new example of a complete minimal surface of finite total curvature was found by C.J. Costa. He did not prove its embeddedness, but it was seen to satisfy all known necessary conditions for the surface to be embedded, and D. Hoffman and W. Meeks III later proved that the surface is in fact embedded. Computer graphics was a very useful aid for proving this. In this paper we introduce this interesting story.