Positive scalar curvature and minimal hypersurfaces

We show that the minimal hypersurface method of Schoen and Yau can be used for the ``quantitative'' study of positive scalar curvature. More precisely, we show that if a manifold admits a metric $g$ with $s_g \ge \vert T \vert$ or $s_g \ge \vert W \vert$, where $s_g$ is the scalar curvature of $g$, $T$ any 2-tensor on $M$ and $W$ the Weyl tensor of $g$, then any closed orientable stable minimal (totally geodesic in the second case) hypersurface also admits a metric with the corresponding positivity of scalar curvature. A corollary pertaining to the topology of such hypersurfaces is proved in a special situation.