Classification of singularities in the complete conformally flat Yamabe flow

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\mathrm{\partial }}{\mathrm{\partial }t}g=-Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t=0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T<\mathrm{\infty }$. We assume that $(M,g(\cdot ,t))$ satisfies (i) the injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time $t=0$ and the scalar curvature bounded at each time-slice. We show that the singularity the flow develops at time $T$ is always of type I.