Existence and uniqueness of $S^1$-invariant K\"ahler-Ricci solitons

Abstract

We use the momentum construction for $S^1$-invariant K\"ahler metrics as developed by Hwang-Singer to construct new examples of steady K\"ahler-Ricci solitons. We also prove that these solitons are unique in their K\"ahler class, provided the vector field and the asymptotic behaviour are fixed.