Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

TL;DR

The paper develops a symmetry-restricted theory for cohomogeneity-one gradient expanding Ricci solitons on $S^1\times\mathbb{R}^3$ and $S^2\times\mathbb{R}^2$, showing the existence of two-parameter soliton families with asymptotic cones over $S^2\times S^1$ and form $g=\mathrm{d}r^2+a(r)^2g_{S^1}+b(r)^2g_{S^2}$, governed by a trio of ODEs in $a,b,f$. By proving monotonicity, completeness, and conical asymptotics, the authors define a cohomogeneity-one expander degree $\deg^{\text{sym}}_{\text{exp}}$ via a map $F$ from initial data to asymptotic cone slopes $(a'_{\infty},b'_{\infty})$, and show $F$ is continuous and proper. They then compute the degree in the two topologies: $\deg^{\text{sym}}_{\text{exp}}(S^1\times\mathbb{D}^3)=1$ and $\deg^{\text{sym}}_{\text{exp}}(S^2\times\mathbb{D}^2)=0$, using homotopy arguments and obstructions from non-surjectivity, respectively. These results extend degree-theoretic approaches to expanding solitons in higher dimensions under symmetry, offering tools to construct solitons asymptotic to prescribed cones and informing Ricci-flow singularity analysis in dimension four and beyond.

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.

Theorems & Definitions (104)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1