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

Abstract

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harmonic 1-forms on hypersurfaces with finite index

“…If we assume further that index(M) = 0 (i.e. M is stable) in Theorem 1.4, then H 1 (L 2p (M)) is trivial[6].Proof of Theorem 1.6. Let K N ≥ −kρ, where k < 4p(n−1)−2(n−2)−(n−1) √ n−1p 2 p 2 (n−1)(2n−2+n √ n−1)…”

“…Moreover, Dung and Seo [9] studied the same topic on a complete -stability hypersurface in a Riemannian manifold with nonnegative sectional curvature. The first author and Lv [6] also investigated the nonexistence of nontrivial harmonic 1-form of a complete -stable hypersurface with weighted Poincaré inequality in a Riemannian manifold with sectional curvature bounded below by a nonpositive function. Most recently, without the stability assumption, Choi and Seo [7] proved the following finiteness theorem.…”

“…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.…”

$L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

“…For L p harmonic 1-forms, Han et al [18] obtained some vanishing and finiteness theorems for L p p-harmonic 1-forms on a locally conformally flat Riemannian manifold with some assumptions. Analogously, there is substantial research indicating that the topologies of the submanifolds is closely associated with L p harmonic 1-forms; see [4,7,8,15,17,19,22] and the references therein.…”