Eigenvalue estimates on weighted manifolds

TL;DR

This paper develops new eigenvalue estimates for the Hodge Laplacian on weighted manifolds, unifying previous results and applying them to geometric problems like minimal hypersurfaces.

Contribution

It introduces generalized eigenvalue bounds for the Hodge Laplacian on weighted manifolds, extending existing literature and connecting to geometric applications.

Findings

Unified eigenvalue estimates for weighted manifolds

Relation between Jacobi operator eigenvalues and Hodge Laplacian spectrum

Applications to (f)-minimal hypersurfaces

Abstract

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for (f)-minimal hypersurfaces and the spectrum of the Hodge Laplacian.