Hamilton's Type-III asymptotic soliton conjecture

A Ricci flow solution $g(t)$ on a manifold $M$ is Type-III if

$$g(t) \text{ exists } \forall t \geq 0 \qquad \text{and} \qquad \sup_{M \times [0,\infty)} t \|\operatorname{Rm}(t)\| < \infty.$$

Hamilton's Type-III asymptotic soliton conjecture. Type-III solutions asymptotically approach soliton metrics that are locally homogeneous.

The conjecture concerns the asymptotic behavior of Type-III Ricci flows and is motivated by Hamilton's work. The meaning of “asymptotically approach” is not precise here; evidence includes substantial progress by Lott in dimension three, while the higher-dimensional picture remains open.