  We study the convergences of three projected Sobolev gradient flows to the
ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed
as the gradient flows of the Gross-Pitaevskii energy functional with respect to
the $H^1_0$-metric and two other equivalent metrics on $H_0^1$, including the
iterate-independent $a_0$-metric and the iterate-dependent $a_u$-metric. We
first prove the energy dissipation property and the global convergence to a
critical point of the Gross-Pitaevskii energy for the discrete-time $H^1$ and
$a_0$-gradient flow. We also prove local exponential convergence of all three
schemes to the ground state.
