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 L 2β 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 L 2 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 K N satisfying that −k 2 ≤ K N ≤ 0 for some constant k.

Vanishing theorems for $$L^{2}$$ L 2 harmonic forms on complete Riemannian manifolds

“…After the paper was submitted the author became aware that this result was proved for hypersurfaces in Riemannian manifolds with non-negative sectional curvature by Kim and Yun [12] and Dung and Seo [9]. Our technique to find the lower bound of the Ricci curvature of the hypersurface is more elementary.…”

$L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality

“…Dung and Seo in [6] proved a similar vanishing theorem for L 2 harmonic 1-forms on complete noncompact submanifolds under the same assumption as in [3] except for the condition that the lower bound of λ 1 (M ) depends on Φ 2 L n . In the second part of this paper, motivated by the above results, we prove the following nonexistence result of L p harmonic 1-forms on a complete noncompact submanifold with property (P ̺ ), assuming that the total curvature of the submanifold is sufficiently small instead of the assumption of δ-stability.…”

“…Tam and Zhou in [23] showed that a complete (n − 2)/n-stable minimal hypersurface in the Euclidean space is either a hyperplane or a catenoid if its second fundamental form satisfies some decay conditions. Dung and Seo in [6] proved the following vanishing theorem.…”

“…Besides, Dung and Seo in [6] considered the problem on a complete δ-stability hypersurface in a Riemannian manifold with nonnegative sectional curvature. First, we recall the definition of δ-stability which is a generalization of the usual stability.…”

$L^2$-harmonic $p$-forms on submanifolds with finite total curvature

“…Lemma 2.2. ( [13,22,23]) Let M n be a complete submanifold with flat normal bundle in S n+l , ω be a…”