Free boundary hypersurfaces with nonpositive Yamabe invariant in mean convex manifolds

2014/06/17 by Barros, A., Cruz, C. Tiarlos · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1406.4221

Abstract

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface Σ 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 Σ is locally volume-minimizing in a manifold Mⁿ with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally M splits along Σ. In the case that the scalar curvature of M is at least -n(n-1) and Σ locally minimizes a certain functional inspired by [30], a neighborhood of Σ in M is isometric to ((-ε,ε)×Σ,dt²+e^2tg), where g is Ricci flat.

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