Dust patterns on vibrating plates and the equilibrium shapes of
phospholipid vesicles are both are governed by the bending energy W,
the integral of the squared mean curvature over an immersed surface in
3-space.  Though introduced 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.  Willmore
showed round spheres minimize among all closed surfaces, and
conjectured a particular torus is the W-minimizer among surfaces of
genus one, finally proven in 2012 by Fernando Coda Marques & Andre
Neves as a corollary to their work on minimal surfaces in the
3-sphere.  We discuss recent joint work with Peng Wang on the
W-minimizing and W-stability properties for higher genus surfaces that
project from minimal surfaces in the 3-sphere.  In particular, we
prove that all Lawson surfaces with area under 8π are W-stable.