| Let (M,ω0) be a compact Kahler manifold without boundary,and c1(M) be its first Chern class.The Calabi conjecture proved by Yau[15] is one of the most important work in Kahler geometry,that is to say:for every θ∈2πc1(M),there must be a unique Kahler metric ω>∈[ω0],which satisfies Ric(ω)=θ,where Ric(ω) is the Ricci form of the metric ω. The existence of Kahler-Einstein metric is also one of the most important researches in Kahler geometry,that is,when2πc1(M)=k[ω0],whether there will be a unique Kahler metric ω∈[ω0],such that Ric(ω)=kω,where k is a constant.The case of k≤0is worked out by Aubin and Yau.In case of k>0,there exists some obstacles,S.T.Yau,G.tian,W.Y.Ding, Futaki,Bando and Mabuchui have do a lot of work,in particular,the existence of Kahler-Einstein metric can be equivalent to some enery function which proved by G.Tian [12] is very important.A nature question,when2πc1(M)-k[ω0]=[α]≠0,for every θ∈[α],if there exists an unique Kahler metric ω∈[ω0],such that Ric(ω)=kω+θ.We call this question be the existence of the generalized Kahler-Einstein metric.In case of k≤0,we can do it by Yau’s results of complex Monge-mpere equation.When k>0,there also exists obstacles,X.Zhang and X.W.Zhang have considered this case in [17].Through researching the complex Monge-Ampere equation,the necessary and sufficient condition of the existence of Kahler Einstein metric can be obtained under some conditions.The Kahler Ricci flow is first researched by H.D.Cao.He proved the Calabi-Yau theorem again by the parabolic method.In recent years,the convergence of Kahler Ricci flow must be the main object of geometry analysis.Perelman,G.Tian, X.X.Chen,X.P.Zhu,X.H.Zhu and Phong have do a lot of work on this side.In this paper,we research the existence of Kahler-Einstein metric by the parabolic method.In the case of k>0(we consider k=1)and the closed real (1,1)form θ≥0,we research the generalized Kahler Ricci flow (?)gij/(?)t=—Rij+kgij+θijIn the second chapter of the article,we generalize the Perelman W function [6] of Kahler Ricci flow,and give a lower bound of the generalized W function under the given function space using the Logarithmic Sobolev inequality in paper [161,and then,we define the μ function.By the semi-positivity of θ,we proof that the generalized W function increases along the evolution equations and the μ function increases with the use of theory on P.D.E.solutions.In the third chapter of this paper,by method in [9],we first consider the evolution equation of the generalized Ricci potential where a(t)=1/V∫M UC-udV.We also prove that the function a(t)is uniformly bounded and give the uniformly lower bound of R一trgθ and u.Then we prove that the upper bound of R—trgθ and|(?)u|2can be controlled by the generalized Ricci potential u(t)through the semi-positivity of θ,and the upper bound of u can be bounded by diam(M,g(t)).At last,we generalize the Perelman noncollapsing theorem and bound the diam(M,g(t)) uniformly.In the last chapter,we first proof that‖(?)u‖c0(t+2)and‖R—n—trgθ‖c0(t+2) can be controlled by‖u‖c0(t)when‖u‖c0(t)is sufficient small.Then we generalize the Poincare inequality,and using it,‖(?)u‖2·‖(?)u‖c0n can control‖u‖c0n+1.Combined with‖(?)u‖2→0as t→+∞,we get the convergence of‖R—n—trgθ‖c0and‖u‖c1.The estimates and convergence obtained above will play an important role in the research of generalized Kahler Ricci flow,... |
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