K\"ahler-Ricci solitons with almost maximal symmetry

TL;DR

This paper classifies certain K"ahler-Ricci solitons with high symmetry, showing they have cohomogeneity one actions and maximal symmetry in complex dimension two.

Contribution

It demonstrates that high-symmetry gradient K"ahler-Ricci solitons have cohomogeneity one actions and maximal symmetry in complex dimension two.

Findings

Isometry group acts by cohomogeneity one.

In dimension two, solitons have maximal symmetry with isometry group of dimension 4.

Potential function invariance under cohomogeneity one action in Ricci solitons.

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.