On the long-time behavior of immortal Ricci flows

TL;DR

This paper investigates the long-term behavior of immortal Ricci flows on closed manifolds, establishing convergence to orbifolds or Einstein manifolds under specific curvature, diameter, and functional conditions.

Contribution

It provides new convergence results for immortal Ricci flows, linking bounded curvature and diameter to orbifold limits, and characterizing blowdown limits as Einstein manifolds under certain conditions.

Findings

Bounded curvature and diameter imply convergence to orbifolds.

Type-III flows with controlled diameter growth have Einstein blowdown limits.

The $oldsymbol{oldsymbol{ u}_+}$-functional condition influences the limit behavior.

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 $\boldsymbol{\mu}_+$-functional satisfies $\lim_{t\to \infty} t\boldsymbol{\mu}_+'(t)=0$.