Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the rigidity of Ricci shrinkers

Yu Li, Wenjia Zhang

TL;DR

This work addresses the rigidity of Ricci shrinkers in the pointed-Gromov-Hausdorff topology, focusing on generalized cylinders $\bar{M}=N^n\times\mathbb{R}^{m-n}$ where $N$ is $H$-stable with an obstruction of order $3$. The authors establish a rigidity inequality of mixed orders for these non-compact cylinders and develop a contraction-extension framework to promote local rigidity to global rigidity, including quotients. As applications, they prove rigidity of the generalized cylinder (and quotients) and deduce the uniqueness of tangent flows for compact Ricci flows when one tangent flow is such a cylinder. The results advance the understanding of singularity models in Ricci flow, providing a local-to-global mechanism via elliptic estimates on non-compact shrinkers and $\mathbb{F}$-convergence. Key tools include an elliptic-Sobolev theory for non-compact shrinkers, a detailed study of infinitesimal solitonic deformations, and a precise decomposition of rigidity spaces along with the contraction-extension strategy.

Abstract

In this paper, we establish the rigidity of the generalized cylinder $N^n \times \mathbb R^{m-n}$, or a quotient thereof, in the space of Ricci shrinkers equipped with the pointed-Gromov-Hausdorff topology. Here, $N$ is a stable Einstein manifold that has an obstruction of order $3$. The proof is based on a quantitative characterization of the rigidity of compact Ricci shrinkers, a rigidity inequality of mixed orders on generalized cylinders, and the method of contraction and extension. As an application, we prove the uniqueness of the tangent flow for general compact Ricci flows under the assumption that one tangent flow is a generalized cylinder.

On the rigidity of Ricci shrinkers

TL;DR

This work addresses the rigidity of Ricci shrinkers in the pointed-Gromov-Hausdorff topology, focusing on generalized cylinders where is -stable with an obstruction of order . The authors establish a rigidity inequality of mixed orders for these non-compact cylinders and develop a contraction-extension framework to promote local rigidity to global rigidity, including quotients. As applications, they prove rigidity of the generalized cylinder (and quotients) and deduce the uniqueness of tangent flows for compact Ricci flows when one tangent flow is such a cylinder. The results advance the understanding of singularity models in Ricci flow, providing a local-to-global mechanism via elliptic estimates on non-compact shrinkers and -convergence. Key tools include an elliptic-Sobolev theory for non-compact shrinkers, a detailed study of infinitesimal solitonic deformations, and a precise decomposition of rigidity spaces along with the contraction-extension strategy.

Abstract

In this paper, we establish the rigidity of the generalized cylinder , or a quotient thereof, in the space of Ricci shrinkers equipped with the pointed-Gromov-Hausdorff topology. Here, is a stable Einstein manifold that has an obstruction of order . The proof is based on a quantitative characterization of the rigidity of compact Ricci shrinkers, a rigidity inequality of mixed orders on generalized cylinders, and the method of contraction and extension. As an application, we prove the uniqueness of the tangent flow for general compact Ricci flows under the assumption that one tangent flow is a generalized cylinder.
Paper Structure (7 sections, 59 theorems, 431 equations, 2 figures)

This paper contains 7 sections, 59 theorems, 431 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rigidity of the compact Ricci shrinkers
  3. Elliptic estimates on non-compact Ricci shrinkers
  4. Rigidity inequality for generalized cylinders
  5. Proof of the main theorem
  6. Uniqueness of the tangent flow
  7. Growth of eigentensors on perturbed Ricci shrinkers

Key Result

Theorem 1.1

A compact Ricci shrinker $(N^n,\bar{g},\bar{f})$ is rigid if and only if it satisfies the rigidity inequality of some order $k$, i.e., there exist positive constants $\epsilon$ and $C$ such that the following property holds: For any $g \in C^{2,\alpha}(S^2(N))$ with $\|g- \bar{g} \|_{C^{2,\alpha}} <

Figures (2)

  • Figure 1: Three equivalent definitions
  • Figure 2: Characterization of the rigidity

Theorems & Definitions (123)

  • Theorem 1.1
  • Theorem 1.2: Main Theorem
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5: Uniqueness of the tangent flow
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 113 more