As picture below , I am fuzzy with the red line. As definition 1.5.1, assuming $g_{ij}(t)$ is expanding breather,then for some $t_1<t_2 \text{ and }\alpha>0$,$\alpha g_{ij}(t)$ and $g_{ij}(t)$ are diffeomorphic. Then $(M,g_{t_1})$ and $(M,g_{t_2})$ are diffeomorphic. Then $\lambda(g_{ij}(t_1))=\lambda(g_{ij}(t_2))$.Then,expanding breather is a steady gradient soliton. And likely, shrink breather is steady gradient soliton.
The picture below is from 203rd page of this paper.
