Minimal hypersurfaces and bordism of positive scalar curvature metrics

摘要

Let (Y, g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume and are closed psc-manifolds equipped with stable minimal hypersurfaces and . Under natural topological conditions, we show that a psc-bordism gives rise to a psc-bordism between and equipped with the psc-metrics given by the Schoen-Yau construction.