Ancient Ricci flows of bounded girth

TL;DR

This paper constructs new ancient Ricci flows with positive curvature and bounded girth, analyzing their asymptotic behavior, and introduces novel invariance conditions not derived from Hamilton's maximum principle.

Contribution

It provides the first known example of such Ricci flows in dimension three and develops new invariance conditions for curvature under symmetry.

Findings

Constructed a pancake-like ancient Ricci flow with positive curvature

Determined the asymptotic limits of the flow backwards in time

Introduced invariance conditions on curvature not based on Hamilton's maximum principle

Abstract

For each $n\ge 3$, we construct a 'pancake-like', $O(2)\times O(n-1)$-invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in dimension three. The construction hinges on the Ricci flow invariance of certain conditions on the curvature and its spatial derivatives under this symmetry regime, whose proof does not follow from Hamilton's tensor maximum principle.