Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

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 Weitzenböck 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 Weitzenböck 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.

21 pages, no figure