On Ricci solitons of cohomogeneity one

Recent Progress on Ricci Solitons

In recent years, there has seen much interest and increased research activities in Ricci solitons. Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton’s Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons. The concept of Ricci solitons was in...

On the Completeness of Gradient Ricci Solitons

A gradient Ricci soliton is a triple (M, g, f) satisfying Rij +∇i∇jf = λgij for some real number λ. In this paper, we will show that the completeness of the metric g implies that of the vector field ∇f .

On the Fundamental Groups of Ricci Solitons

On the classification of gradient Ricci solitons

We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones Sn, S × R, and Rn. This gives a new proof of the Hamilton-Ivey-Perel’man classification of 3dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of Hn, H × R, Rn, S ×...

Cohomogeneity One Actions on Noncompact Symmetric Spaces of Rank One

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CHn, n ≥ 3. For the quaternionic hyperbolic spaces HHn, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificat...