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...

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

In this article, we study the geometry of compact quasi-Einstein manifolds with boundary. We establish sharp boundary estimates for that improve some previous results. Moreover, obtain a characterization theorem such in terms surface gravity components, which leads to new geometric inequality. addition, prove estimate (possibly disconnected) Brown–York mass.

We show that Einstein equations are compatible with the presence of massive point particle idealization and find the corresponding two parameter family of solutions. They are complete defined by the bare mechanical mass M > 0 and the Keplerian mass m > 0 (m < M) of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable ra...