Schoen’s scalar-curvature volume-comparison conjecture

Prove that every closed manifold admitting a hyperbolic metric g_0 satisfies Vol(M,g)\geq Vol(M,g_0) for every Riemannian metric g with scalar curvature Sc_g\geq -n(n-1), in all dimensions.

Background

The paper considers whether a lower bound on scalar curvature analogous to the Ricci-curvature entropy comparison can force the hyperbolic metric to minimize volume. Specifically, for a closed hyperbolic n-manifold (M,g_0), the desired comparison is Sc_g\geq -n(n-1) \Longrightarrow Vol(M,g)\geq Vol(M,g_0).

The authors explain that the volume comparison is related to an entropy bound and state that Schoen proposed it in every dimension. The paper proves a local entropy result in dimension three and constructs higher-dimensional local entropy saddles, but it does not establish the global volume comparison conjecture in all dimensions.