Optimal curvature estimates for homogeneous Ricci flows

TL;DR

This paper establishes uniform curvature bounds for homogeneous Ricci flows, classifies solution types based on extinction and longevity, and proves a gap theorem for Ricci-flatness on homogeneous spaces.

Contribution

It introduces new uniform curvature estimates for homogeneous Ricci flows and a gap theorem for Ricci-flatness, advancing understanding of solution behaviors and geometric structure.

Findings

Solutions with finite extinction time are Type I.

Immortal solutions are Type III.

Ancient solutions are Type I.

Abstract

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local $W^{2,p}$ convergence result, which holds without symmetry assumptions.