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

TL;DR

This paper studies the pluriclosed flow on generalized K"ahler manifolds with split tangent bundle, proving long-term existence in dimension two and relating it to a generalized Calabi conjecture.

Contribution

It demonstrates that the pluriclosed flow preserves split tangent bundle structures and reduces to a scalar potential flow, providing new a priori estimates and long-term existence results.

Findings

Flow preserves split tangent bundle structures.

Established long-time existence in dimension two.

Connected flow to a generalized Calabi conjecture.

Abstract

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.