Ancient Ricci flows with nonnegative Ricci curvature

In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity – a noncollapsed $\mathbb{F}$-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed $\mathbb{F}$-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature ($\mathrm{Ric}\ge \epsilon Rg$), either it is compact, or every tangent flow at infinity is a Ricci flat cone.