We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in . We study the relationship of the existence of the flow and associated static metrics to topological information on the underlying complex manifold. Solutions to the static equation are automatically Hermitian-symplectic, a condition we define herein. These static metrics are classified on K3 surfaces, complex toroidal surfaces, nonminimal Hopf surfaces, surfaces of general type, and class VII+ surfaces. To finish, we discuss how the flow may potentially be used to study the topology of class VII+ surfaces.
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