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Four-dimensional complete gradient shrinking Ricci solitons

Huai-Dong Cao, Ernani Ribeiro, Detang Zhou

Abstract

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $\Bbb{R}^4,$ or $\Bbb{S}^{3}\times\Bbb{R}$, or $\Bbb{S}^{2}\times\Bbb{R}^{2}.$ In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

Four-dimensional complete gradient shrinking Ricci solitons

Abstract

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton or , or In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

Paper Structure

This paper contains 4 sections, 105 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Proof of Theorem \ref{['thmA']}
  4. Proof of Theorem \ref{['thmRIC']}

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • proof
  • proof
  • proof
  • proof
  • Remark 5