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 sg ≥ |T | or sg ≥ |W |, where sg is the scalar curvature of 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) hyp...

Let (M,g) be a compact oriented 4-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, with its standard Fubini-Study metric.