Abstract: In this talk we investigate the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy functional on a Hilbert manifold. To find a In corresponding minimizer, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density of a ground or excited state of the system. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state and how these rates depend on spectral gaps. Our findings are validated in numerical experiments.