On the long time behavior of ancient homogeneous Ricci flows

TL;DR

This paper establishes a precompactness result for invariant metrics on compact homogeneous spaces and shows that ancient homogeneous Ricci flows converge to gradient shrinking Ricci solitons through blow-down sequences.

Contribution

It introduces a precompactness theorem for invariant metrics without injectivity radius bounds and demonstrates convergence of ancient homogeneous Ricci flows to solitons.

Findings

Precompactness theorem for invariant metrics

Convergence of ancient Ricci flows to gradient shrinking solitons

No injectivity radius bounds required

Abstract

We prove a precompactness theorem for invariant metrics on compact homogeneous spaces without injectivity radius bounds, assuming uniform bounds on the diameter and on all derivatives of the curvature tensor. As a consequence, we prove that every ancient homogeneous Ricci flow on a compact manifold admits a blow-down sequence that converges to a gradient shrinking Ricci soliton.