Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

Abstract

Abstract We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition [ J + , J - ] = 0 [J_{+},J_{-}]=0 , a condition referred to as “split tangent bundle.” Moreover, we show that in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a number of a priori estimates for this equation, including a general estimate in dimension n = 2 n=2 of Evans–Krylov type requiring a new argument due to the nonconvexity of the equation. The main result is a long-time existence theorem for the flow in dimension n = 2 n=2 , covering most cases. We also show that the pluriclosed flow represents the parabolic analogue to an elliptic problem which is a very natural generalization of the Calabi conjecture to the setting of generalized Kähler geometry with split tangent bundle.