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

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

The global existence and convergence of the Calabi flow on Cn/Zn+iZn\mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n

In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on $X$ when the dimension is 2, partially confirming Chen's conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat K\"ahler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture