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

TL;DR

This paper investigates properties of stable minimal hypersurfaces in negatively curved Riemannian manifolds, providing eigenvalue estimates, topological implications, and conditions for stability.

Contribution

It offers new eigenvalue bounds, topological results based on scalar curvature, and stability criteria for minimal hypersurfaces in negatively curved spaces.

Findings

First eigenvalue estimate for Laplace operator on stable minimal hypersurfaces.

Proves that small total scalar curvature implies a single end.

Vanishing of $L^2$ harmonic 1-forms under certain curvature conditions.

Abstract

We give an estimate of the first eigenvalue 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, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We 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 we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.