Local singularities of compact multiply warped Ricci flow solutions

TL;DR

This paper classifies four-dimensional shrinking Ricci solitons with warped product structures, showing they are isometric to a standard cylinder, and constructs examples of Ricci flow solutions forming generalized cylinder singularities.

Contribution

It proves a classification result for certain warped Ricci solitons and constructs examples of Ricci flow solutions with generalized cylinder singularities.

Findings

Any four-dimensional shrinking Ricci soliton with warped product structure is isometric to a standard cylinder.

Constructs explicit Ricci flow solutions forming generalized cylinder singularities.

Provides rigorous examples of singularity formation in Ricci flow.

Abstract

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$.