Local singularities of compact multiply warped Ricci flow solutions

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 ${\mathbb{R}}^k\times {\mathbb{S}}^{\mathrm{\ell }}$.