$SU(2)$-invariant steady gradient Ricci solitons on four-manifolds

TL;DR

This paper constructs new and simpler examples of steady gradient Ricci solitons on four-manifolds with specific symmetries, expanding the known family of such geometric structures using advanced mathematical techniques.

Contribution

It introduces a new family of $SU(2)$-invariant steady gradient Ricci solitons on four-manifolds and simplifies existing constructions for $U(2)$-invariant solitons on similar manifolds.

Findings

Constructed a new family of $SU(2)$-invariant steady gradient Ricci solitons.

Provided simpler constructions of existing $U(2)$-invariant solitons.

Extended the class of known complete steady gradient Ricci solitons on four-manifolds.

Abstract

Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram $\mathbb{Z}_4\subset U(1)\subset SU(2)$. We also provide simpler constructions of the existing $U(2)$-invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams $\mathbb{Z}_n\subset U(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton's non-collapsed solitons for $n\ge 3$.