By Adam Murray, The University of Queensland
We can think of the Willmore energy as a measure of the roundness of a surface sitting in three-dimensional space. This is perhaps easy to imagine in the case of a sphere (which is of course, quite round by anyone’s reckoning), but how exactly can we measure the roundness of a donut or similarly ‘unspherical’ and perhaps hole-filled shapes? Returning to our sphere, we can actually see that this is, in some sense, the roundest shape possible, so it should serve as a model for the property of roundness itself. The Willmore energy can then be interpreted as the deviation from a sphere.
Why should we study this seemingly quite abstract property of roundness? How can it help us in the sciences? Of course, the study of pure mathematics is not always concerned with direct applications of its results, however understanding of the Willmore energy appears to be essential in understanding a number of aspects of the physical world. It naturally arises in many phenomena from physics to optics to biology. For example, Helfrich’s models for the red blood cell involve minimising the Willmore energy on the cell walls, subject to appropriate constraints. It also appears in physics in expressions of Hawking radiation – the black body radiation posited by Hawking in the 1970’s to be expelled by black holes, a topic which always excites the imagination.
One of the most famous problems involving the Willmore energy is the Willmore conjecture. We have long known that the sphere is the global minimiser, but what if we consider shapes that have one or more ‘holes’ in them, for example, a torus (read: donut)? This is exactly the contents of the Willmore conjecture. Willmore conjectured that the minimum possible energy for a torus was 2π2 , compared to the 4π for a sphere, and furthermore, that this number was obtained by a special torus known as the Clifford torus.
The conjecture remained unsolved for almost 50 years, with many distinguished mathematicians lending their expertise. Recently, André Neves and Fernando Codá Marques, using quite novel techniques, discovered a solution, namely a method known as Almgren-Pitts Min-max theory.
The resolution of a longstanding conjecture in mathematics is always an exciting event, as it usually signals the introduction of a number of innovative methods and tools that can not only be applied to other problems in similar areas, but to vastly different fields of mathematics.
It is with this view that I move into an honours year in geometry and geometric analysis, excited by the prospect of working with quite new developments in my chosen field.
Adam Murray was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.