Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold

Abstract

Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2≤n≤6, we prove that if M satisfies the δ-stability inequality (0<δ≤1), then there is no nontrivial L2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that -k2≤KN≤0 for some constant k. © 2014 Elsevier Inc.