On the classification of gradient Ricci solitons

TL;DR

This paper classifies certain types of gradient Ricci solitons, showing that complete shrinking ones with vanishing Weyl tensor are quotients of standard models, and provides new proofs and classifications for expanding solitons.

Contribution

It offers a new proof of the classification of 3D shrinking gradient Ricci solitons and extends classification results to expanding solitons with specific curvature conditions.

Findings

Complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of standard models.

New proof of Hamilton-Ivey-Perel'man classification in 3D.

Classification of expanding gradient Ricci solitons with constant scalar curvature and decaying Weyl tensor.

Abstract

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor.