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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 dimensionn  =  2n=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 dimensionn  =  2n=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.

Pluriclosed flow on generalized Kähler manifolds with split tangent bundle