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$4d$ steady gradient Ricci solitons with nonnegative curvature away from a compact set

Ziyi Zhao, Xiaohua Zhu

Abstract

In the paper, we analysis the asymptotic behavior of noncompact $κ$-noncollapsed steady gradient Ricci soliton $(M, g)$ with nonnegative curvature operator away from a compact set $K$ of $M$. In particular, we prove: any $4d$ noncompact $κ$-noncollapsed steady gradient Ricci soliton $(M^4, g)$ with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of $(M^4, g)$, which converges subsequently to a family of shrinking quotient cylinders.

$4d$ steady gradient Ricci solitons with nonnegative curvature away from a compact set

Abstract

In the paper, we analysis the asymptotic behavior of noncompact -noncollapsed steady gradient Ricci soliton with nonnegative curvature operator away from a compact set of . In particular, we prove: any noncompact -noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of , which converges subsequently to a family of shrinking quotient cylinders.
Paper Structure (7 sections, 18 theorems, 139 equations)

This paper contains 7 sections, 18 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. A splitting theorem
  3. Compact case of $(N, h(t))$
  4. Curvature decay estimate
  5. Estimate of level sets
  6. $4d$ steady Ricci solitons
  7. Proofs of the main results

Key Result

Theorem 2

Let $(M, g)$ be a $4d$ complete noncompact $\kappa$-noncollapsed steady gradient Ricci soliton with $\rm{Km}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$. Let $p_i\rightarrow \infty$ be any sequence in $M$ and let $\bar{g}(t)= h(t) +ds^2$ be the splitting limit flow of $(M,g_{p_i}(t)

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 2.1
  • proof
  • ...and 30 more