The existence of polarized F-structures on volume collapsed 4-manifolds

By the work of [CG2] and [CG3], the Riemannian manifolds whose injectivity radii are small everywhere relative to the bound on sectional curvature admit positive rank F-structures. We will prove, in dimension four, that if the volumes of the manifolds are also smaller than a positive constant, then the manifolds actually admit polarized F-structures. In particular, this result implies an affirmative answer to the Gromov's Gap conjecture on vanishing minimal volume in dimension four.