The existence of polarized \(F\)-structures on volume collapsed 4- manifolds (Q1308873)

An \(F\)-structure on a manifold \(M\) is a family of local torus actions on \(M\) satisfying certain consistency conditions on overlaps. An \(F\)- structure is said to be polarized if all isotropy groups of the local torus actions are of finite order. The minimal volume of \(M\) is the infimum of volumes of metrics whose sectional curvature satisfies \(| K|\leq 1\). \textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 23, 309-346 (1986; Zbl 0606.53028)] and \textit{K. Fukaya} [ibid. 28, No. 1, 1- 21 (1988; Zbl 0652.53031)] show that if \(M\) admits a polarized \(F\)- structure then its minimal volume is zero. The author proves that the minimal volume of a 4-manifold is either 0 or uniformly bounded away from 0, by finding a polarized \(F\)-structure whenever the minimal volume is sufficiently small.