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

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 $$\mathbb{Z}$_n\subset U(1)\subset SU(2)$forany$n\in $\mathbb{N}$$, including Appleton’s non-collapsed solitons for $n\ge 3$.