Bounded cohomology and the Cheeger isoperimetric constant

We study equivalent conditions for the Cheeger isoperimetric constant of Riemannian manifolds to be positive. We first give a proof of Gromov’s assertion for locally symmetric spaces with infinite volume, which states that the existence of a bounded primitive of the Riemannian volume form is equivalent to the positivity of the Cheeger isoperimetric constant. Furthermore, under the assumption of pinched negative sectional curvature, we obtain another equivalent condition in terms of bounded cohomology classes. This generalizes Soma’s result (Duke Math J 88(2):357–370, 1997 ) for hyperbolic 3-manifolds to R -rank one locally symmetric spaces.