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A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

Huai-Dong Cao, Junming Xie

Abstract

In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient steady (respectively, expanding) Ricci solitons with half-harmonic Weyl curvature. In addition, we obtain a partial classification of four-dimensional gradient expanding Ricci solitons with half-nonnegative isotropic curvature.

A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

Abstract

In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient steady (respectively, expanding) Ricci solitons with half-harmonic Weyl curvature. In addition, we obtain a partial classification of four-dimensional gradient expanding Ricci solitons with half-nonnegative isotropic curvature.

Paper Structure

This paper contains 3 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:cylin_half-harmonic']} and Theorem \ref{['thm:conical_half-harmonic']}
  3. Four-dimensional Ricci expanders with half-WPIC & half-PIC

Key Result

Theorem 1.1

Let $(M^4,g,f)$ be a four-dimensional complete, noncompact, asymptotically cylindrical gradient steady Ricci soliton with half-harmonic Weyl curvature. Then, $(M^4,g,f)$ is isometric to the Bryant soliton up to scalings.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Lemma 2.1: WWW:18
  • Lemma 2.2: WWW:18
  • proof : Proof of Theorem \ref{['thm:cylin_half-harmonic']}
  • proof : Proof of Theorem \ref{['thm:conical_half-harmonic']}
  • Remark 2.1
  • Definition 2.1
  • ...and 10 more