A weighted Reilly type integral formula for differential forms and its applications

TL;DR

This paper develops a weighted Reilly type integral formula for differential forms on smooth metric measure spaces and applies it to derive eigenvalue bounds, cohomology properties, and inequalities related to the weighted Hodge Laplacian and Steklov problems.

Contribution

It introduces a new weighted Reilly type integral formula for differential forms and applies it to establish eigenvalue bounds and cohomology properties in weighted geometric settings.

Findings

Lower bound for the spectrum of the weighted Hodge Laplacian on boundary

Lower bound for the first positive Steklov eigenvalue related to boundary curvature

Universal inequalities for eigenvalues of weighted Hodge Laplacian on submanifolds

Abstract

In this paper, we derive a weighted Reilly type integral formula for differential forms on a compact smooth metric measure space with boundary. As applications, a lower bound of the spectrum for the weighted Hodge Laplacian acting on differential forms on the boundary, and some special properties for p-th absolute cohomology space with respect to the lowest p-curvatures of the boundary have been obtained, respectively. Furthermore, we obtain a lower bound for the first positive eigenvalue of the Steklov eigenvalue problem on differential forms which is related to the lowest principal curvature of the boundary, and a comparison result between the eigenvalues of the Steklov eigenvalue problem and the Hodge Laplacian on the boundary. On the other hand, for closed submanifolds of weighted Euclidean space, we derive universal inequalities for the sum of eigenvalues with respect to the weighted Hodge Laplacian, which can be seen as a generalization of Levitin-Parnovski inequality.