Studies On Vanishing Theorems For Harmonic Forms On Riemannian Manifolds

In this paper,we study the existence of nontrivial harmonic l-forms on Riemannian mani-folds,the paper is organized as follows.Firstly,we consider the vanishing theorems for L2 harmonic l-forms on submanifolds in hyperbolic space,and obtain that if the first eigenvalue satisfies certain conditions and some inequality holds on submanifolds then there is no nontrivial L2 harmonic l-forms,and we also obtain a Liouville theorem.Secondly,the vanishing theorems for p-harmonic l-forms on complete noncompact sub-manifold Mn in a sphere Sn+m with flat normal bundle are discussed.By applying Bochner-Weitzenbock formula,Kato inequality,Sobolev inequality and estimating on Ricci curvature,we prove that the dimension of the space of p-harmonic l-forms with finite LQ(Q>2)and total curvature is finite,which is some extension of the results of Han[16]and Zhu-Fang[41,42].In addition,if complete noncompact submanifolds have sufficiently small total curvature,we obtain that all p-harmonic l-forms with finite LQ(Q ? 2)must be trivial.Thirdly,we investigate p-harmonic l-forms on Riemannian manifolds with a weighted Poincare inequality,and get a vanishing type theorem,which shows that if we put some as-sumptions on the bound of the Weitzenbock curvature operator and the first eigenvalue of the Laplacian operator,then there is no nontrivial p-harmonic l-form(2 ? l ? n-2)with finite Lp norm on M,which generalizes previous results in Dung[9]and Vieira[36].We also deduce a vanishing type theorem for p-harmonic 1-forms on M.Finally,by applying Vieira's technique to smooth metric measure spaces,we prove some vanishing theorems under the assumptions of curvature operator with lower bound and the first eigenvalue of the f-Laplacian satisfies some conditions.Moreover,we obtain a Liouville theorem.