Eigenvalue estimates of minimal hypersurfaces with finite index in Riemannian manifolds

TL;DR

This paper investigates eigenvalue estimates for minimal hypersurfaces with finite index in Riemannian manifolds, extending previous theorems and generalizing Liouville type results for stable minimal hypersurfaces.

Contribution

It generalizes key theorems related to eigenvalues and stability of minimal hypersurfaces in Riemannian manifolds, building upon and extending prior results by Schoen, Yau, and Seo.

Findings

Extended eigenvalue estimates for minimal hypersurfaces with finite index.

Generalized Liouville type theorems for stable minimal hypersurfaces.

Connected previous results to broader classes of hypersurfaces.

Abstract

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})