Vanishing theorems for p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincaré inequality

Vanishing theorem and finiteness theorem for $p$-harmonic $1$ form

“…In [8], Dung showed some vanishing type theorems for p-harmonic l-forms on such a manifoldwith a weighted Poincar inequality. Motivated by [8], in [7], Chao et al investigated p-harmonic l-forms on Riemannian manifolds with a weighted Poincar inequality, and we get a vanishing type theorem. In [1], Afuni studied the monotonicity of vector bundle valued harmonic form.…”

harmonic 1-forms on hypersurfaces with finite index

“…Recently, Sang and Thanh [25] proved that a complete noncompact stable minimal hypersurface with property (P ρ ) in Riemannian manifold N has no nontrivial L 2 harmonic 1-form if the sectional curvature of N satisfies K N (x) ≥ − (1−τ )ρ(x) (2n−1)(n−1) , 0 < τ ≤ 1 and ρ(x) 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.…”