Lp harmonic 1-forms and first eigenvalue of a stable minimal hypersurface

We estimate the bottom of the spectrum of the Laplace operator on a stable minimal hypersurface in a negatively curved manifold. We also derive various vanishing theorems for L p harmonic 1-forms on minimal hypersurfaces in terms of the bottom of the spectrum of the Laplace operator. As consequences, the corresponding Liouville type theorems for harmonic functions with finite L p energy on minimal hypersurfaces in a Riemannian manifold are obtained.