On gradient Ricci solitons with Symmetry

TL;DR

This paper investigates highly symmetric gradient Ricci solitons, proving the non-existence of non-trivial homogeneous and certain noncompact shrinking examples with nonnegative curvature, thereby advancing understanding of their geometric structure.

Contribution

It establishes the non-existence of non-trivial homogeneous and specific noncompact cohomogeneity one shrinking gradient Ricci solitons with nonnegative curvature.

Findings

No non-trivial homogeneous gradient Ricci solitons exist.

No noncompact cohomogeneity one shrinking gradient Ricci solitons with nonnegative curvature exist.

Main results extend rigidity understanding of symmetric Ricci solitons.

Abstract

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial 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.