$L^p$ harmonic $1$-forms and first eigenvalue of a stable minimal hypersurface

Keomkyo Seo

Vol. 268 (2014), No. 1, 205–229

DOI: 10.2140/pjm.2014.268.205

Abstract

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.

Keywords

minimal hypersurface, stability, first eigenvalue, $L^p$ harmonic $1$-form, Liouville type theorem

Mathematical Subject Classification 2010

Primary: 53C42

Secondary: 58C40

Milestones

Received: 9 October 2012

Accepted: 29 April 2013

Published: 21 May 2014