Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

Abstract

We consider the space of smooth gradient expanding Ricci soliton structures on $S^1\times {\mathbb{R}}^3$ and $S^2\times {\mathbb{R}}^2$ which are invariant under the action of $\text{SO}(3)\times \text{SO}(2)$. In the case of each topology, there exists a $2$-parameter family of cohomogeneity one solitons asymptotic to cones over the link $S^2\times S^1$, as constructed by Nienhaus-Wink and Buzano-Dancer-Gallaugher-Wang. By analyzing the resultant soliton ODEs, we reconstruct the $2$-parameter families in each case and provide an alternate proof of conicality. Analogous to work of Bamler and Chen, we define a notion of expander degree for these cohomogeneity one solitons through a properness result. We then proceed to calculate this cohomogeneity one expander degree in the cases of the specific topologies.

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