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Some rigidity results on shrinking gradient Ricci soliton

Jianyu Ou, Yuanyuan Qu, Guoqiang Wu

Abstract

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$, giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.

Some rigidity results on shrinking gradient Ricci soliton

Abstract

Suppose is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature is isometric to a finite quotient of , giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.

Paper Structure

This paper contains 5 sections, 16 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Structure of shrinking gradient Ricci soliton satisfying condition A
  4. Rigidity via nonnegative sectional curvature
  5. constant scalar curvature

Key Result

Theorem 1.1

Let $(M^n, g, f)$ be a shrinking gradient Ricci soliton satisfying condition A, then the universal cover of $M$ is isometric to $\mathbb{R}^k\times \mathbb{N}^{n-k}$, where $\mathbb{N}$ is an $n-1$ dimensional shrinking gradient Ricci soliton with positive Ricci curvature.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Petersen-Wylie2
  • Theorem 1.5: FR16
  • Theorem 1.6: Cheng-Zhou
  • Theorem 2.1: Cao-Zhou
  • Theorem 2.2: Naber, Naber
  • Definition 3.1
  • Proposition 3.2
  • ...and 7 more