  This paper investigates the numerical approximation of ground states of
rotating Bose-Einstein condensates. This problem requires the minimization of
the Gross-Pitaevskii energy $E$ on a Riemannian manifold $\mathbb{S}$. To find
a corresponding minimizer $u$, 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
$|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present
a precise characterization of the local convergence rates in a neighborhood of
each ground state $u$ and how these rates depend on the first spectral gap of
$E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$.
With this we establish the first convergence results for a Riemannian gradient
method to minimize the Gross-Pitaevskii energy functional in a rotating frame.
At the same, we refine previous results obtained in the case without rotation.
The major complication in our new analysis is the missing isolation of
minimizers, which are at most unique up to complex phase shifts. For that, we
introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u}
\mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic
convergence rates through a weighted eigenvalue problem. Afterwards, we link
the auxiliary iteration to the original Riemannian gradient method and bound
the spectrum of the weighted eigenvalue problem to obtain quantitative
convergence rates. Our findings are validated in numerical experiments.
