EXISTENCE OF POLARIZED F-STRUCTURES ON COLLAPSED MANIFOLDS WITH BOUNDED CURVATURE AND DIAMETER

We study the class of collapsed Riemannian n-manifolds with bounded se ctional curvature and diameter. Our main result asserts that there is a constant; delta(n, d) > 0, such that if a compact n-manifold has bou nded curvature, K-Mn less than or equal to 1, bounded diameter, diam(M (n)) less than or equal to d and sufficiently small volume, Vol(M(n)) less than or equal to delta(n, d), then it admits a mixed polarized F- structure. As a consequence, inf(g) Vol(M(n), g) = 0: where the infimu m is taken over all metrics with \K-(Mn,K-g)\ less than or equal to 1. This assertion carl be viewed as a weakened version of Gromov's ''cri tical volume'' conjecture.