Cohomogeneity One Expanding Ricci Solitons and the Expander Degree — UC Berkeley

We consider the space of smooth gradient expanding Ricci soliton structures on $$S^1 \times \mathbb {R}^3$$ S 1 × R 3 and $$S^2 \times \mathbb {R}^2$$ S 2 × R 2 which are invariant under the action of $$\text {SO}(3) \times \text {SO}(2)$$ SO ( 3 ) × SO ( 2 ) . In the case of each topology, there exists a 2-parameter family of cohomogeneity one solitons asymptotic to cones over the link $$S^2 \times S^1$$ S 2 × S 1 , as constructed in [1, 2], and [3]. By analyzing the resultant soliton ODEs, we reconstruct the 2-parameter families in each case and provide an alternate proof of conicality. Analogous to [4], 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.