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Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

Ziyi Zhao, Xiaohua Zhu

Abstract

Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $κ$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact split limit Ricci flow of type I arising from the blow-down of $(M, g)$, if there is an $(n-1)$-dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that $(M^n,g)$ with $\rm{Rm}\geq 0$ must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows $(M,g_{p_i}(t); p_i)$ of $(M,g)$ converges subsequently to a family of shrinking quotient cylinders.

Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

Abstract

Let be a complete noncompact -noncollapsed steady Ricci soliton with and away from a compact set of . We prove that there is no any -dimensional compact split limit Ricci flow of type I arising from the blow-down of , if there is an -dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that with must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows of converges subsequently to a family of shrinking quotient cylinders.
Paper Structure (7 sections, 9 theorems, 45 equations, 1 figure)

This paper contains 7 sections, 9 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A decay estimate of curvature
  4. A classification of split compact ancient solutions
  5. Proofs of main results
  6. Proof of Theorem \ref{['noncompact-ancient-solution']}
  7. Proofs of Corollary \ref{['notype1-2']} and Theorem \ref{['Zhao-Z-high-dim']}

Key Result

Theorem 1.1

Let $(M^n, g)$$(n\ge 4)$ be a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative curvature operator. Suppose that there exists a sequence of $p_i\in M$$( \to \infty)$ such that the rescaled Ricci flows $(M,g_{p_i}(t); p_i)$ of $(M,g)$ converge subsequently to a family of

Figures (1)

  • Figure 1:

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 3.1
  • ...and 2 more