Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature (Q766127)

The authors give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface \(M\) in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, they prove that if a complete minimal hypersurface \(M\) has sufficiently small total scalar curvature then \(M\) has only one end. The authors also obtain a vanishing theorem for \(L^2\) harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, they provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.