The Classifications Of Gradient Ricci Almost Soliton And Some Related Problems On Riemannian Manifold

In this paper, we consider the following three problems:the classification of a gradient generalized quasi-Einstein manifolds; the classification of static space; Hamilton’s gradient estimates and Liouville theorems for porous medium equations.In Chapter one, we study a gradient generalized m-quasi-Einstein manifold. That is, there exist two smooth functions f,λ such that Rij+fij-m/1fifj=λ9ij, where m is a constant. We first obtain some rigidity results on compact gradient gener-alized m-quasi-Einstein manifolds. Then, we obtain some classifications for the special gradient generalized m-quasi-Einstein manifolds under the assumption that the Bach ten-sor is flat. In particular, we obtain some even stronger results in dimension three.In Chapter two, we treat both the static equation and the critical point equation in the same way. In both cases, on a given Riemannian manifold (Mn,g), there exists a smooth function f satisfying the equation fij= fRij+λ9ij, where Rij denotes the Ricci curvature and λ is a smooth function. We establish a new relationship to the above equation between the Cotton tensor, the 3-tensor D and the Weyl curvature tensor by f Cijk=-n-1/1Dijk+Wijklfl which is independent of the function A. By virtue of this relationship, we obtain some classifications under the assumption that the Bach tensor is flat, which generalize the results in [57]. In particular, for n=3, we prove that (Mn,g) is locally conformally flat if the Bach tensor is divergence-free.In Chapter three, we derive a local gradient estimate for positive solutions of the porous medium equation ut-△(up),1<p<1+(?) posed on (Mn,g) with the Ricci curvature bounded from below. Moreover, we also obtain a Liouville type theorem. In particular, the results obtained in this chapter generalize those in [66].