On a class of fully nonlinear flow in Kähler geometry

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In this paper, we study a class of fully nonlinear metric flow on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the corresponding Euler equation, which is fully nonlinear of Monge-Ampère type. As an application, we also discuss a complex Monge-Ampère type equation including terms of mixed degrees, which was first posed by Chen.