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 introduced by Hamilton [64] in mid 80’s. They are natural generalizations of Einstein metrics. Ricci solitons also correspond to selfsimilar solutions of Hamilton’s Ricci flow [62] and often arise as limits of dilations of singularities in the Ricci flow [66, 10, 25, 91]. They can be viewed as fixed points of the Ricci flow, as a dynamical system, on the space of Riemannian metrics modulo diffeomorphisms and scalings. Ricci solitons are of interests to physicists as well and are called quasi-Einstein metrics in physics literature (see, e.g., [50]). In this paper, we survey some of the recent progress on Ricci solitons as well as the role they play in the singularity study of the Ricci flow. This paper can be regarded as an update of the article [13] written by the author a few years ago.