The Other Pure Seminar (TOPS): An introduction to Constant Mean Curvature Surfaces (2 of 3)
Date: Tuesday 6,13,20 May, 10-11 am
Location: Lower Napier LG28
Speaker: John Rice
Title: An introduction to Constant Mean Curvature Surfaces
Abstract: Constant mean curvature (CMC) surfaces emerged at the beginning of the 1800’s as the key concept for explaining the effects of the (then) newly posited theory of surface tension. While spheres, as soap bubbles, are a common manifestation of CMC surfaces, we will see that there are other surfaces of revolution that are CMC. Called Delaunay surfaces after their discoverer, he showed that they are the only CMC surfaces of revolution. He gave an interesting geometric characterisation of them, whose explanation is quite unsatisfying.
The theory of CMC surfaces is remarkable for the breadth of mathematical concepts and techniques that it involves. We will touch on the use of complex function theory to show that any CMC surface that is a topological sphere must in fact be a round sphere. We will then introduce the use of Riemann surfaces and loop groups in generating large families, of immersed CMC surfaces. These are the technique of integrable systems. They have been successful in characterising immersed CMC surfaces that are topological tori.
How to deal with surfaces of higher genus is an open and challenging problem.