K\"ahler-Ricci solitons with almost maximal symmetry

Abstract

This paper studies a non-trivial gradient K\"{a}hler-Ricci soliton, of complex dimension $n$, with an isometry group of dimension at least $n^2-1$. We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension $2^2$. In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily K\"{a}hler), the potential function is invariant by the action.