Geometry Of Ricci Solitons

Assume that M is a smooth Riemannian manifold and that g is the Riemannian metric of M.Let Rm,Ric,S be the Riemannian curvature tensor,Ricci curvature and scalar curvature respectively.We call that M is a Ricci soliton,if there exists a constant derivative.If the vector field V is the gradient of some smooth function f∈C^∞（M）,that is V=grad f,then Ric+?²f=ρg.Now we call that M is a gradient Ricci soliton,and the function f is called as the potential function.Ifρ=0,M is called a steady（gradient）Ricci soliton;Ifρ>0,M is called a shrinking（gradient）Ricci soliton;Ifρ<0,M is called a expanding（gradient）Ricci soliton.If V=0 or f=0,then M is an Einstein manifold.Then we see that Ricci soliton is a natural generalization of Einstein manifold.The concept of Ricci solitons was first introduced by Richard Hamilton in 1980s.It corresponds to the self-similar solutions of Ricci flow and plays an important role in the sigularities analysis of Ricci flow.We begin with the definition of Ricci solitons.Using the method of moving frame,some basic formulas of Ricci solitons are given.Then we get several fundamental results of Ricci solitons,including the positiveness of the scalar curvature,the relationship between scalar curvature and the relationship between distant function and scalar curvature.Based on these results,we obtain the estimate of the curvature of shrinking gradient Ricci solitons of dimension 4.For any shrinking gradient Ricci soliton of dimension 4 with bounded scalar curvature,there exists a constant c,such that|Rm|≤c S.And we have the lower estimate of the Riemannian curvature operator Rm≥-c（log（r+1））^-1/4,where r is the distance function to one fixed point.If the scalar curvature S converges to 0 at infinity,then we have S f≤c,which means the conial strucure of Ricci solitons.At last,a result about the estimate of the diameter of compact shrinking gradient Ricci solitons is given.