Willmore flow

Named after the American differential geometer Tom Willmore, the Willmore flow corresponds to the $L^{2}$-gradient flow of the geometric energy

$e[{\mathcal {M}}]={\frac {1}{2}}\int _{\mathcal {M}}H^{2}\,\mathrm {d} A$

where $h$ stands for the mean curvature of the manifold ${\mathcal {M}}$. This flow leads to a evolution problem in differential geometry: the surface ${\mathcal {M}}$ is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion it is a fourth-order flow, since the variation of the energy contains fourth derivatives.