Degeneration of shrinking Ricci solitons

TL;DR

This paper investigates the geometric limits of shrinking Ricci solitons, showing that their Gromov-Hausdorff limits are smooth manifolds outside a small singular set, thus advancing understanding of their degeneration behavior.

Contribution

It proves that Gromov-Hausdorff limits of shrinking Ricci solitons are smooth manifolds outside a codimension at least 2 singular set, clarifying their degeneration structure.

Findings

Limits are smooth manifolds outside a codimension at least 2 set

The limit spaces satisfy a shrinking Ricci soliton equation

Provides a detailed description of degeneration of shrinking Ricci solitons

Abstract

Let $(Y,d)$ be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, $Y$ is a smooth manifold satisfying a shrinking Ricci soliton equation.