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 W2,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.