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Bach-flat gradient steady Ricci solitons

Huai-Dong Cao, Giovanni Catino, Qiang Chen, Carlo Mantegazza, Lorenzo Mazzieri

Abstract

In this paper we prove that any $n$-dimensional ($n\ge 4$) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6] and [9].

Bach-flat gradient steady Ricci solitons

Abstract

In this paper we prove that any -dimensional () complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6] and [9].

Paper Structure

This paper contains 5 sections, 19 theorems, 67 equations.

Table of Contents

  1. The results
  2. Background material
  3. Proof of Theorem \ref{['teo1']}
  4. Proof of Theorem \ref{['teo3D']}
  5. Further remarks

Key Result

Theorem 1.1

For $n\geq 4$, let $(M^{n},g_{ij}, f)$ be a complete gradient steady Ricci soliton with positive Ricci curvature such that the scalar curvature $R$ attains its maximum at some interior point. If in addition $(M^{n},g_{ij}, f)$ is Bach-flat, then it is isometric to the Bryant soliton up to a scaling

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • ...and 16 more