Seminars & Colloquia Calendar

A gradient Ricci soliton is a complete Riemannian manifold (M, g), together with a smooth potential function f,  such that its Ricci tensor satisfies the equation  \(Ric + Hess f = lambda g\), for some constant \(lambda\).  Ricci solitons are important geometric objects because they are natural extensions of Einstein metrics and they model singularity formations in the Ricci flow.  

The classification of locally conformally flat (LCF) gradient shrinking Ricci solitons was completed around 2009, and the rigidity of shrinking solitons with harmonic Weyl curvature was proved during 2011-2013 by the combined works of Fernandez-Lopez & Garcia-Rio [2011] and Munteanu & Sesum [2013]. However, for steady solitons, while the LCF ones were classified in 2012, the rigidity problem for steady solitons has been open until very recently. In this talk, I shall discuss the rigidity of gradient steady Ricci solitons and report on my joint work with Jiangtao Yu, a Ph.D student, and also the work of another Ph.D student, Fengjiang Li.