L²HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

Abstract

Abstract. In the present note, we deal with L 2 harmonic 1-forms on complete submanifolds with weighted Poincaré inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for L 2 harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and Vitório.

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

“…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. More precisely, We have the following vanishing theorem which is an extension of Theorem 1.2 in [3] and Theorem 1.5 in [4]. Theorem 1.7.…”

harmonic 1-forms on hypersurfaces with finite index

“…Recently, Sang and Thanh [25] proved that a complete noncompact stable minimal hypersurface with property ( ) in Riemannian manifold N has no nontrivial harmonic 1-form if the sectional curvature of N satisfies and satisfies certain growth condition. Motivated by [4, 5, 6, 9, 25], we can obtain an another improvement of Theorem 1.1. More precisely, we have the following theorem.…”