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On Bach-flat gradient shrinking Ricci solitons

Huai-Dong Cao, Qiang Chen

Abstract

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cylinder $S^3\times R$. More generally, for n>4, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $R^n$ or the product $N^{n-1}\times R$, where $N^{n-1}$ is Einstein.

On Bach-flat gradient shrinking Ricci solitons

Abstract

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton or the round cylinder . More generally, for n>4, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton or the product , where is Einstein.

Paper Structure

This paper contains 5 sections, 19 theorems, 82 equations.

Table of Contents

  1. The results
  2. Preliminaries
  3. The covariant 3-tensor $D_{ijk}$
  4. The proof of Theorem 1.1 and Theorem 1.2
  5. Gradient Ricci solitons with vanishing $D_{ijk}$

Key Result

Theorem 1.1

Let $(M^4, g_{ij}, f)$ be a complete Bach-flat gradient shrinking Ricci soliton. Then, $(M^4, g_{ij}, f)$ is either (i) Einstein, or (ii) locally conformally flat, hence a finite quotient of either the Gaussian shrinking soliton $\mathbb R^{4}$ or the round cylinder $\mathbb S^3\times \mathbb R$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Corollary 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 19 more