Free boundary hypersurfaces with nonpositive Yamabe invariant in mean convex manifolds

TL;DR

This paper investigates free boundary hypersurfaces with nonpositive Yamabe invariant in mean convex manifolds, providing area and volume estimates, and demonstrating splitting results under certain curvature and minimality conditions.

Contribution

It establishes new estimates and splitting theorems for free boundary hypersurfaces with nonpositive Yamabe invariant in manifolds with scalar curvature bounds.

Findings

Derived area and volume bounds for hypersurfaces

Proved splitting results under minimality and curvature conditions

Identified local geometric structures near hypersurfaces

Abstract

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $\Sigma$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $\Sigma$ is locally volume-minimizing in a manifold $M^n$ with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally $M$ splits along $\Sigma$. In the case that the scalar curvature of $M$ is at least $-n(n-1)$ and $\Sigma$ locally minimizes a certain functional inspired by [30], a neighborhood of $\Sigma$ in $M$ is isometric to $((-\varepsilon,\varepsilon)\times\Sigma,dt^2+e^{2t}g)$, where $g$ is Ricci flat.