The Vanishing Theorem And The Dimension Finite Theorem Of Harmonic Form In Riemannian Manifold

In this parper,we mainly study the existence of nontrivial p-harmonic l-forms and the space of the p-harmonic l-forms has finite dimensions on Riemannian manifolds.The paper is organized as follows.In part one,we consider the vanishing theorems for p-harmonic 1-forms and finite theorems for the space of p-harmonic 1-forms on locally conformal flat manifolds Nn.We prove that the space of the L2β p-harmonic 1-forms on Nn has finite dimension,which is an extension of the results of Lin[26].Inspired by the works of Li,Feng,Zhao[21,22]and Han[13],we focus on the vanishing theorem or finite theorem on locally conformally flat Riemannian manifolds which have zero index or finite index of some Schrodinger operators.These results generalise that of Han[13]and Li,Feng,Zhao[21,22].In part two,we investigate p-harmonic l-forms on Riemannian manifolds Vn with positively Yamabe invariant,and get vanishing type theorems.Firstly,under the assumption that various Ln/2-integral condition on the curvature,we have that all L2β p-harmonic l-forms must be trivial,which extends the results of Lin[29].In addition,we put some assumption on the zero index of Schrodinger operators and finite Ln/2-integral or non-negative condition on Scalar curvature,then there is no nontrivial L2β p-harmonic l-forms.In part three,The L2β p-harmonic l-forms on non-compact complete submanifold Mn with flat normal bundle immersed in a Hadamard manifold is discussed.We obtain a vanishing type theorem,which shows that if submanifold Mn have sufficiently small total curvature and(Pρ)property,then there is no nontrivial L2β p-harmonic l-form in submanifolds Mn,which generalizes result in Wang-Chao-Wu-Lv[42].The finite theorems of the space of L2β p-harmonic l-forms in submanifolds Mn with finite index of Schrodinger operators is also discussed,which is extensions of the results of Lin[27].