Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

TL;DR

This paper establishes new lower bounds for the first eigenvalue of the Hodge-Laplacian on immersed Riemannian manifolds, leading to rigidity and homology sphere results, with optimal estimates in constant curvature cases.

Contribution

It provides novel extrinsic eigenvalue bounds for the Hodge-Laplacian considering the pull back Weitzenb"{o}ck operator, extending previous results and including optimal estimates.

Findings

Derived extrinsic lower bounds for the first eigenvalue.

Proved rigidity results and a homology sphere theorem.

Showed optimality of estimates in constant curvature scenarios.

Abstract

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenb\"{o}ck operator (defined in Section 2) of $\bar M$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results and a homology sphere theorem. Second, when the pull back Weitzenb\"{o}ck operator of $\bar M$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of $M$ and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then $M$ is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates and homology sphere theorems are optimal when $\bar M$ has constant curvature.