Positive scalar curvature and minimal hypersurfaces

Abstract

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 g with s g ≥ | T | s_g \ge \vert T \vert or s g ≥ | W | s_g \ge \vert W \vert , where s g s_g is the scalar curvature of g g , T T any 2-tensor on M M and W W the Weyl tensor of g 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.