Minimal hypersurfaces and bordism of positive scalar curvature metrics

TL;DR

This paper extends Schoen-Yau's results on positive scalar curvature metrics from closed minimal hypersurfaces to those with free boundary and explores psc-bordism relations between such hypersurfaces within psc-manifolds.

Contribution

It establishes an analogous positive scalar curvature result for free boundary minimal hypersurfaces and links psc-bordisms of manifolds to those of their minimal hypersurfaces.

Findings

Stable minimal hypersurfaces with free boundary admit psc-metrics.

psc-bordisms induce psc-bordisms between minimal hypersurfaces.

The results connect geometric measure theory, conformal geometry, and scalar curvature topology.

Abstract

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 $(Y_0,g_0)$ and $(Y_1,g_1)$ are closed psc-manifolds equipped with stable minimal hypersurfaces $X_0 \subset Y_0$ and $X_1\subset Y_1$. Under natural topological conditions, we show that a psc-bordism $(Z,\bar g) : (Y_0,g_0)\rightsquigarrow (Y_1,g_1)$ gives rise to a psc-bordism between $X_0$ and $X_1$ equipped with the psc-metrics given by the Schoen-Yau construction.