K\"ahler-Ricci solitons with almost maximal symmetry

Gradient Kähler-Ricci solitons are special metrics on complex manifolds satisfying an equation that combines the Ricci curvature with a potential function, serving as self-similar solutions under the Ricci flow. The paper examines cases where these solitons possess a very large symmetry group, specifically when the dimension of the isometry group is at least n squared minus one for complex dimension n. Under this condition, the authors show that the symmetry group must act with orbits of codimension one, a setup called cohomogeneity one. This structure permits reduction to a simpler model using Sasakian geometry, which connects the Kähler structure to an associated odd-dimensional contact geometry. In the special case of complex dimension two, the symmetry is forced to be maximal, reaching exactly dimension four. Separately, the paper establishes that for any non-trivial gradient Ricci soliton (Kähler or not) whose isometry group acts by cohomogeneity one, the potential function remains constant on the orbits and is thus invariant under the group action. These findings constrain the possible shapes of highly symmetric solitons and provide an ansatz for further study.