On $H^2$-gradient Flows for the Willmore Energy

We show that the concept of $H^2$-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class $W^{2,p}$ from a compact, $n$-dimensional manifold into Euclidean space, provided that $p \geq 2$ and $p>n$. We also discuss why this is not true for Sobolev class $H^2=W^{2,2}$. In the case of equality constraints, we provide sufficient conditions for the existence of the projected $H^2$-gradient flow and demonstrate its usability for optimization with several numerical examples.