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

Abstract

In this paper, we obtain some vanishing and finiteness theorems for L p p-harmonic 1-forms on a locally conformally flat Riemmannian manifolds which satisfies an integral pinching condition on the traceless Ricci tensor, and for which the scalar curvature satisfies pinching curvature conditions or the first eigenvalue of the Laplace-Beltrami operator of M is bounded by a suitable constant. Recently, H. Z. Lin [11], investigated the L 2 harmonic 1-form on locally con-518 2010 Mathematics Subject Classification. 53C21; 53C25.

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

“…Considering the integral on M is identical to that of M, we prefer to integrate function about s on M in the subsequent computations. According to Bochner formula for p harmopnic 1 form [18]:…”

“…Han [18] studied obtain some vanishing and finiteness theorems for L p p-harmonic 1forms on a locally conformally flat Riemmannian manifolds . In [15], Han studied Liouville theorem for p harmonic 1 form on submaniold in sphere.…”

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

“…However, not all higher-dimensional manifolds have locally conformally flat structure, and giving classification of locally conformally flat manifolds is important as well as difficult. However, under various geometric conditions, there are substantial research results on the classification of conformally flat Riemannian manifolds (see [2,5,6,14,18,22,29] for details).…”

“…The results of L 2 harmonic forms make L 2 theory on manifolds clearer and easier to understand as compared to general L p theory (see [26]). 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.…”