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On the long time behavior of ancient homogeneous Ricci flows

Anusha M. Krishnan, Francesco Pediconi

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.

On the long time behavior of ancient homogeneous Ricci flows

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.
Paper Structure (10 sections, 20 theorems, 67 equations)

This paper contains 10 sections, 20 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ancient homogeneous Ricci flows
  4. Convergence in the absence of injectivity radius estimates
  5. Preparatory definitions for convergence
  6. Equivariant Gromov--Hausdorff convergence of compact homogeneous spaces
  7. Geometry of collapsed compact homogeneous spaces
  8. Convergence to a shrinking Ricci soliton
  9. Locally homogeneous gradient Ricci solitons
  10. Symmetrization along the collapsed directions

Key Result

Theorem 1

Every ancient homogeneous Ricci flow on a compact manifold admits a blow-down sequence that converges locally in the $\mathcal{C}^{\infty}$-topology to a gradient shrinking Ricci soliton.

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: c.f. Gli03, Theorem 3; Lott07, Theorem 1.4; BLS19, Corollary 2
  • Definition 2.7
  • ...and 41 more