  We justify some characterizations of the ground states of spin-1
Bose-Einstein condensates exhibited from numerical simulations. For
ferromagnetic systems, we show the validity of the single-mode approximation
(SMA). For an antiferromagnetic system with nonzero magnetization, we prove the
vanishing of the $m_F=0$ component. In the end of the paper some remaining
degenerate situations are also discussed. The proofs of the main results are
all based on a simple observation, that a redistribution of masses among
different components will reduce the kinetic energy.
