On gradient Ricci solitons with symmetry

We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus, the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper “Rigidity of gradient Ricci solitons” to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.