Riemannian geometry provides a vast toolbox for the numerical treatment of nonlinear optimization problems. Alas, some infinite dimensional problems, e.g., those related to curvature energies of immersed surfaces, can hardly be put into a Riemannian context. Still, many techniques from Riemannian optimization perform very well even in a suitable Banach manifold setting–at least experimentally.

In this talk, we demonstrate the efficiency of the method of gradient descent for the minimization of the Willmore energy. Instead of L²–bilinear forms (which lead to the parabolic Willmore flow), we utilize a H²–bilinear form to define gradients. Although the Willmore energy is not defined for general surfaces of class H², we show that in the space of immersions of class W^2,p with p > 2, the gradient vector field is well-defined and induces an ordinary differential equation. In case of equality constraints, we also provide conditions for the existence of the projected gradient flow.

Host: Keenan Crane