Canonical Metrics In K(?)hler Geometry And Sasakian Geometry

In this thesis,we study the canonical metrics in K(?)hler geometry and Sasakian geometry.In the first part,we study the existence and uniqueness of twisted K(?)hler-Ricci solitons on Fano manifolds.We prove that the existence of twisted K(?)hler-Ricci solitons is related to the properness of twisted modified Mabuchi K-energy.In the second part,we study the conical K(?)hler-Ricci solitons.Approximating by a sequence of twisted K(?)hler-Ricci solitons,we get that the properness of log Mabuchi K-energy implies the existence of conical K(?)hler-Ricci soliton.In the last part,we prove that the transverse Mabuchi K-energy is convex along the weak C2-geodesic.As a consequence,we get the uniqueness of compatible constant scalar curvature Sasakian metrics on Sasakian manifold.