Steady conformally Kähler Ricci solitons with isoparametric potential

Abstract

We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are $U(2)$-invariant nuts on $\mathbb R^4$, including the Bryant and Cao solitons, and bolts on $\mathbb{S}^2 \times \mathbb R^2$ and on $\mathcal O(-m)\to\mathbb{CP}^1$. For each $m\ge3$ there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.