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Local singularities of compact multiply warped Ricci flow solutions

James Isenberg, Dan Knopf, Zilu Ma, Natasa Sesum

Abstract

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$.

Local singularities of compact multiply warped Ricci flow solutions

Abstract

We demonstrate that any four-dimensional shrinking Ricci soliton , where is any two-dimensional complete noncompact surface and is a warped product metric over the base , has to be isometric to the generalized cylinder equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models .

Paper Structure

This paper contains 28 sections, 23 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Summary of our main results
  3. Outline of the paper
  4. The metrics we study
  5. Notational conventions
  6. Evolution by Ricci flow
  7. Evolution of the scalar curvature of the base
  8. Classifying warped product shrinking solitons on $\mathbb{R}^2\times\mathbb{S}^2$
  9. Uhlenbeck frame
  10. Standard formulas
  11. Warped product geometry
  12. Warped product shrinkers
  13. A new identity
  14. General elementary estimates
  15. $C^0$ estimates
  16. ...and 13 more sections

Key Result

Theorem 1

Let $(\mathcal{B}^2\times\mathbb{S}^2, g)$ be a noncompact and nonflat shrinking Ricci soliton, where $g = \check g + v^2 g_{\mathbb{S}^2}$ and $v : \mathcal{B}^2\to (0,\infty)$. Then $(\mathcal{B}^2\times\mathbb{S}^2, g)$ is isometric to the bubble sheet (generalized cylinder $\mathbb{R}^2\times\m

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm-classification']}
  • Lemma 7
  • ...and 33 more