Existence of polarized F-structures on collapsed manifolds with bounded curvature and diameter

We study the class of collapsed Riemannian n-manifolds with bounded sectional curvature and diameter. Our main result asserts that there is a constant. δ (n, d) > 0, such that if a compact n-manifold has bounded curvature, |K _Mn| ≤ 1, bounded diameter, diam(Mⁿ) ≤ d and sufficiently small volume, Vol(Mⁿ) ≤ δ (n, d), then it admits a mixed polarized F-structure. As a consequence, inf_g Vol(Mⁿ . g) = 0, where the infimum is taken over all metrics with |K _{(Mn, g)} | ≤ 1. This assertion can be viewed as a weakened version of Gromov's "critical volume" conjecture.