Estimation of almost Ricci-Yamabe solitons on static spacetimes

Ricci Fall-off in Static, Globally Hyperbolic, Non-singular Spacetimes

What restrictions are there on a spacetime for which the Ricci curvature is such as to produce convergence of geodesics (such as the preconditions for the Singularity Theorems) but for which there are no singularities? We answer this question for a restricted class of spacetimes: static, geodesically complete, and globally hyperbolic. The answer is that, in at least one spacelike direction, the...

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...

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption we derive a differential equality of Yamabe constant Y (g (t)) where g (t) is a solution of the Ricci flow on a closed n-manifold. As an application we show that when g (0) is a Yamabe metric at time t = 0 and Rgα n−1 is not a positive eigenvalue of the Laplacian ∆gα for any Yamabe metric gα in the conformal class [g0], then d dt ∣∣ t=0 Y (g (t)...

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