Eigenvalue estimates of minimal hypersurfaces with finite index in Riemannian manifolds

The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (cite{SchoenYau1976}). Recently, Seo (cite{Seo14}) generalized Theorem 1.7 (cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (cite{SchoenYau1976}) and 1.8 (cite{Seo14})