# Willmore Day at Durham

Willmore surface sculpture

Last week’s Yorkshire Durham Geometry Day, held at Durham, celebrated the work and life of Tom Willmore. Willmore was a differential geometer who spent most of his academic career at the University of Durham. As I was giving a talk in a school I missed the first two talks of the day which was a shame as the first explained a recent proof of the long-standing Willmore Conjecture.

For a surface we can define the notion of curvature in a number of ways. For Willmore surfaces we use mean curvature so let’s define that. At a point $p$ of a surface $S$ take a normal vector. Any non-zero vector in the line perpendicular to the tangent plane to the surface will do. Next take any plane that contains this normal. The intersection of this plane with the surface is a curve. Take the curvature of this curve at the point $p$, call this curvature $c_1$. (The curvature of a curve at a point is $1/r$ where $r$ is the radius of the circle that best approximates the curve at that point.) Next take the plane containing the normal that is at right angles to the first one. We get another curve and hence another curvature, call it $c_2$. Take the mean of these two curvatures, $\frac{1}{2} (c_1+c_2)$. This is the mean curvature at $p$; we will denote it by $H(p)$. We can do this at every point and so we get a function $H:S\to R$ given by $H(p)$. An alternative viewpoint is given on the Wikipedia page.

Now we integrate the square of this function over the whole of the surface. The Willmore energy of a surface $S$, denoted $W(S)$, is the number $W(S)=\int _S H^2$. That is we integrate over the whole surface the square of the mean curvature. So for every surface we have a number.

If I deform a surface slightly, then the Willmore energy is likely to change, maybe up, maybe down. A surface is a Willmore surface if any small deformation of the surface cause the Willmore energy to increase. I.e., out of all nearby surfaces this surface has minimal Willmore energy. If we take surfaces with the topology of a sphere, for example a perfectly round sphere, an ellipsoid or an egg, then $W(S)\geq 4\pi$. The minimum occurs for the perfectly round sphere.

What if we move from spheres to tori? The Willmore Conjecture is that for surfaces with the topology of the torus we have $W(S)\geq 2\pi ^2$. This conjecture has resisted proof for many years and the recently announced proof looks good to the experts (sometimes people make dubious claims for proofs of famous problems, see NP versus P and the Riemman Hypothesis for examples of this). The proof is rather technical and can be found on the arXiv: Willmore Conjecture paper

For pure mathematicians the main interest of Willmore surfaces is that you get rather good looking surfaces. However, Willmore energy is of use in applied mathematics as it can be used to describe bending energy, for example in biological problems on the elasticity of membranes.

The sculpture is inspired by a four-lobed Willmore torus and as such is not a Willmore surface, one can see this clearly due to the flat looking parts of the sculpture. A picture of a four-lobed torus can be found at GeometrieWerkstatt. The plaque near the sculpture makes it clear that the shape is only inspired by a Willmore surface as can bee seen below.

Plaque

A video was made of the public lecture by Franz Pedit. I’ll link to that when it appears on the web.