Lp harmonic 1-forms on minimal hypersurfaces with finite index

Abstract Let N be a complete simply connected Riemannian manifold with sectional curvature K N satisfying − k 2 ≤ K N ≤ 0 for a nonzero constant k . In this paper we prove that if M is an n ( ≥ 3 ) -dimensional complete minimal hypersurface with finite index in N , then the space of L p harmonic 1-forms on M must be finite dimensional for certain p > 0 provided the bottom of the spectrum of the Laplace operator is sufficiently large. In particular, M has finitely many ends. These results can be regarded as an extension of Li–Wang (2002).