On the long-time behavior of immortal Ricci flows

Abstract

For an immortal Ricci flow on an $m$-dimensional $(m\ge 3)$ closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled by $t^{\frac{1}{2}}$, then any blowdown limit is an $m$-dimensional negative Einstein manifold, provided that Feldman-Ilmanen-Ni's ${\mathbf{\mu }}_{+}$-functional satisfies ${\lim }_{t\to \infty }t{\mathbf{\mu }}_{+}^{\prime }(t)=0$.

44 pages