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

TL;DR

This paper classifies certain four-dimensional gradient Ricci solitons under specific curvature conditions, advancing understanding of their geometric structure.

Contribution

It provides new classification results for four-dimensional gradient steady and expanding Ricci solitons with particular curvature assumptions.

Findings

Classified gradient steady Ricci solitons with half-harmonic Weyl curvature under asymptotic conditions.

Classified gradient expanding Ricci solitons with half-harmonic Weyl curvature under asymptotic conditions.

Partially classified expanding Ricci solitons with half-nonnegative isotropic curvature.

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.