Inequalities for eigenvalues of the weighted Hodge Laplacian

TL;DR

This paper derives universal inequalities for the eigenvalues of the weighted Hodge Laplacian on compact self-shrinkers, extending classical inequalities and establishing their optimality in eigenvalue order.

Contribution

It generalizes Yang-type and Levitin-Parnovski inequalities to the weighted Hodge Laplacian, providing optimal bounds based on Cheng and Yang's recursion formula.

Findings

Established universal eigenvalue inequalities for the weighted Hodge Laplacian.

Extended classical eigenvalue inequalities to a weighted geometric setting.

Proved the optimality of these inequalities in eigenvalue order.

Abstract

In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.