Abstract
<jats:p> We study 4d gradient steady Ricci solitons which are weak <jats:inline-formula> <jats:tex-math>\kappa</jats:tex-math> </jats:inline-formula> -solutions and exhibit <jats:inline-formula> <jats:tex-math>\mathrm{O}(3)</jats:tex-math> </jats:inline-formula> -symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact <jats:inline-formula> <jats:tex-math>\kappa</jats:tex-math> </jats:inline-formula> -solutions found in [Comm. Pure Appl. Math. 75, 1032–1073 (2022)]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows. </jats:p>