Pluriclosed flow on generalized K\"ahler manifolds with split tangent bundle

We show that the pluriclosed flow preserves generalized K\"ahler structures with the extra condition $[J_+,J_-] = 0$, a condition referred to as "split tangent bundle." Moreover, we show that in this 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 $n=2$ 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 $n=2$, 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 K\"ahler geometry with split tangent bundle.Comment: to appear Crelle's Journa

Generalized Kahler Geometry and the Pluriclosed Flow

In prior work the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with B-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kahler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the B-field renormalization group flow also preserves generalized Kahler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.Comment: Final version, to appear in Nuc. Phys.

Pluriclosed flow on manifolds with globally generated bundles

We show global existence and convergence results for the pluriclosed flow on manifolds for which certain naturally associated tensor bundles are globally generated