  We compare surface metrics for shape optimization problems with constraints,
consisting mainly of partial differential equations (PDE), from a computational
point of view. In particular, classical Laplace-Beltrami type based metrics are
compared with Steklov-Poincar\'e type metrics. The test problem is the
minimization of energy dissipation of a body in a Stokes flow. We therefore set
up a quasi-Newton method on appropriate shape manifolds together with an
augmented Lagrangian framework, in order to enable a straightforward
integration of geometric constraints for the shape. The comparison is focussed
towards convergence behavior as well as effects on the mesh quality during
shape optimization.
