Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

TL;DR

This survey explores the geometric properties, asymptotic behaviors, and recent classification results of four-dimensional gradient Ricci solitons, which are important in understanding Ricci flow singularities.

Contribution

It provides a comprehensive overview of examples, geometric aspects, and recent classification and rigidity results specific to four-dimensional gradient Ricci solitons.

Findings

Examples of four-dimensional gradient Ricci solitons

Asymptotic behaviors of these solitons

Recent classification and rigidity results

Abstract

[Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.