What do the dust patterns on vibrating plates or the equilibrium shapes of phospholipid vesicles have in common? Both are governed by the bending energy W, the integral of the squared mean curvature over an immersed surface in 3-space. Although introduced over two centuries ago by Sophie Germain, W is now* named for Tom Willmore, who suggested the global problem of minimizing W for a fixed topological class of surfaces, and who proved round spheres minimize W among all closed surfaces. Willmore conjectured a particular torus is the W-minimizer among surfaces of genus one. This was proved in 2012 by Fernando Coda Marques and Andre Neves as a corollary to their work on minimal surfaces.
We will discuss recent joint with Peng Wang understanding the W-minimizing properties for higher genus surfaces that project from minimal surfaces in the 3-sphere, as well as even more recent joint work with Jonas Hirsch and Elena Mader-Baumdicker on W-unstable surfaces arising from complete minimal surfaces in 3-space, which have been used to illustrate topological problems like turning a sphere inside-out!
*For a pre-colloquium introduction to the Willmore bending energy and its twisted history:
704 Thackeray Hall