Geometry & Topology

In this talk, we will discuss uniqueness and symmetry results for steady gradient Ricci solitons that are asymptotically quotient-cylindrical. Under a rigidity assumption, we show that the steady solitons of Bryant and Appleton are unique among solitons with the same asymptotics. In dimension 4, we show that rigidity implies strong symmetry and obtain a partial classification of asymptotically quotient-cylindrical steady solitons. Our proofs rely on a symmetry principle which also generalizes to expanding solitons and Ricci-flat ALE spaces.