  In this paper, we first show that a complete shrinking breather with Ricci
curvature bounded from below must be a shrinking gradient Ricci soliton. This
result has several applications. First, we can classify all complete
$3$-dimensional shrinking breathers. Second, we can show that every complete
shrinking Ricci soliton with Ricci curvature bounded from below must be
gradient -- a generalization of Naber's result. Furthermore, we develop a
general condition for the existence of the asymptotic shrinking gradient Ricci
soliton, which hopefully will contribute to the study of ancient solutions.
