Abstract

For each $n\ge 3$, we construct a “pancake-like” ($O(2)\times O(n-1)$-invariant with positive curvature operator and bounded “girth”) ancient Ricci flow, which generalises the two-dimensional “ancient sausage” solution. We also establish explicit asymptotics backwards in time. This solution is new even in dimension three and opens new avenues into the analysis of ancient solutions. Indeed, the symmetry ansatz does not reduce the problem to the analysis of a system of ODEs; instead we obtain a coupled system of parabolic PDEs with one space variable, which we use to establish the invariance under Ricci flow of certain inequalities on the curvature and its spatial derivatives (which does not follow from Hamilton’s tensor maximum principle) and exploit the evolution of volume, Myers’ theorem and the differential Harnack inequality.

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