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Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

Anusha M. Krishnan, Francesco Pediconi, Sammy Sbiti

Abstract

Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.

Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

Abstract

Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.
Paper Structure (20 sections, 29 theorems, 204 equations, 1 table)

This paper contains 20 sections, 29 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous ancient solutions to the Ricci flow
  4. Compact homogeneous spaces
  5. Collapsing tori of homogeneous ancient solutions
  6. Limit solitons for homogeneous ancient solutions
  7. Riemannian submersions on homogeneous torus bundles
  8. Ricci flow solutions on homogeneous torus bundles
  9. Existence of limit Einstein metrics
  10. NR-decompositions for the isotropy representation
  11. Normalizer-adapted decompositions
  12. Diagonalizing the Ricci tensor
  13. Existence of toral symmetries
  14. The submersion tensor and a formula for the Ricci eigenvalues
  15. Vanishing of the submersion tensor for ancient solutions
  16. ...and 5 more sections

Key Result

Theorem 1

Let $g(t)$ be a collapsed, ancient solution to the homogeneous Ricci flow on a compact ma-ni-fold and $\xi = \{t^{(n)}\}$ a sequence of times with $t^{(n)} \to -\infty$. Then, up to passing to a subsequence, the collapsing directions of the rescaled metrics $\frac{1}{|t^{(n)}|}g(t^{(n)})$ converge t

Theorems & Definitions (71)

  • Theorem 1: c.f. Ped19
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Remark 2.1
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['thm:main-coll.solitons']}
  • ...and 61 more