Existence of polarized \(F\)-structures on collapsed manifolds with bounded curvature and diameter
The main result of this paper is the following. There exists a constant \(\delta(n,d)>0\) such that, if an \(n\)-dimensional compact Riemannian manifold \(M^n\) has bounded curvature, \(|K_{M^n}|\leq 1\), bounded diameter \(\text{diam}(M^n)\leq d\) and \(\text{Vol}(M^n)\leq\delta(n,d)\), then the associated sufficiently collapsible \(F\)-structure \(\mathcal F\) admits a polarized substructure. Let us recall that an \(F\)-structure \(\mathcal F\) is a generalization of torus action. By using this result, the authors prove the following weakened version of Gromov's conjecture: Theorem. There exists \(\delta(n,d)>0\) such that if \(M^n\) admits a metric with \[ |K_{M^n}|\leq 1,\;\text{diam}(M^n)\leq d,\;\text{Vol}(M^n)\leq\delta(n,d), \] then \(\text{Min Vol}(M^n)=0\), where \(\text{Min Vol}(M^n)=0\) is defined as the infimum of the volume of \(M^n\) taken over all metrics with bounded sectional curvature.
scientific article; zbMATH DE number 912458