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No compact split limit Ricci flow of type II from the blow-down

Ziyi Zhao, Xiaohua Zhu

TL;DR

This work analyzes blow-down limits of noncompact, $bc$-noncollapsed steady gradient Ricci solitons with nonnegative curvature away from a compact set, using PerelmanbL's $$-geodesic framework. It shows that any compact split ancient $$-solution of codimension one arising from such a blow-down must be of type I, ruling out type II compact split limits in all dimensions $n  4$. The authors develop an $$-center localization, derive curvature decay estimates for $$-centers, and construct shrinking Ricci soliton limits via normally rescaled flows, leveraging a curvature-control program adapted to the steady-soliton setting. The main result yields a clean classification: from blow-downs, either all codimension-one split limits are compact ancient $$-solutions of type I or they are noncompact, with a corollary validating a conjecture in ZZ-4d and extending it to higher dimensions. The paper also proves a linear decay rate for the scalar curvature in this setting, connecting blow-down analysis with sharp curvature decay and soliton limit behavior.

Abstract

By Perelman's $\mathcal L$-geodesic theory, we study the blow-down solutions on a noncompact $κ$-noncollapsed steady gradient Ricci soliton $(M^n, g)$ $(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature away from a compact set of $M$. We prove that any compact split ancient solution of codimension one from the blow-down of $(M, g)$ is of type I. The result is a generalization of our previous work from $n=4$ to any dimension.

No compact split limit Ricci flow of type II from the blow-down

TL;DR

This work analyzes blow-down limits of noncompact, -noncollapsed steady gradient Ricci solitons with nonnegative curvature away from a compact set, using PerelmanbL's -geodesic framework. It shows that any compact split ancient -solution of codimension one arising from such a blow-down must be of type I, ruling out type II compact split limits in all dimensions . The authors develop an -center localization, derive curvature decay estimates for -centers, and construct shrinking Ricci soliton limits via normally rescaled flows, leveraging a curvature-control program adapted to the steady-soliton setting. The main result yields a clean classification: from blow-downs, either all codimension-one split limits are compact ancient -solutions of type I or they are noncompact, with a corollary validating a conjecture in ZZ-4d and extending it to higher dimensions. The paper also proves a linear decay rate for the scalar curvature in this setting, connecting blow-down analysis with sharp curvature decay and soliton limit behavior.

Abstract

By Perelman's -geodesic theory, we study the blow-down solutions on a noncompact -noncollapsed steady gradient Ricci soliton with nonnegative curvature operator and positive Ricci curvature away from a compact set of . We prove that any compact split ancient solution of codimension one from the blow-down of is of type I. The result is a generalization of our previous work from to any dimension.
Paper Structure (10 sections, 18 theorems, 155 equations)

This paper contains 10 sections, 18 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{L}$-length and $\mathcal{L}$-geodesic
  4. $\mathcal{L}$-length associated to a steady Ricci soliton
  5. Location of $\ell$-center
  6. Curvature estimates for $\ell$-center
  7. Construction of shrinking Ricci solitons
  8. Proofs of main theorems
  9. Proofs of Theorem \ref{['main-notype2']} and Corollary \ref{['alter-coro']}
  10. Proof of Theorem \ref{['compact-linear-decay']}

Key Result

Theorem 1

Let $(M^n, g)$$(n\ge 4)$ be a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton which satisfies where ${\rm Rm}$ and ${\rm Ric}$ denote the curvature operator and Ricci curvature of $(M, g)$, respectively. Then any compact split ancient solution $(N, h(t))$ in (splitt-solution) from the blow-down of $(M, g)$ is of type I. In the other words, there is no $(n-1)$-dimensional compact sp

Theorems & Definitions (33)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Proposition 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 2.1
  • proof
  • ...and 23 more