The Vanishing Theorems And Finiteness Theorems Of P-harmonic Forms On Complete Riemannian Manifolds

In this thesis,we mainly study the dimension of space of p-harmonic forms on complete,noncompact,simply connected locally conformal flat Riemannian manifolds.Mainly using Bochner technique,the cutoff function method and some inequalities,we obtain some vanishing theorems and finiteness theorem of p-harmonic forms.The main results of this thesis are as follows:In the first part,we prove some vanishing theorems of p-harmonic forms on a complete,noncompact,simply connected locally conformally flat Riemannian manifold.Firstly,we assume that the scalar curvature of the manifold Mⁿ is nonnegative,and the L^n/2 norm of the traceless Ricci tensor is less than some positive constant.By using Kato inequality,Duzaar Fuchs cutoff function,we obtain that the norm of all L^p p-harmonic t-forms on the manifold are 0,that is,there are no nontrivial L^p p-harmonic forms on the manifold.Secondly,we assume that the dimension of the manifold is even and the scalar curvature of the manifold is nonnegative,then there are no nontrivial L^q p-harmonic m-forms on M^2m.Finally,we assume that the scalar curvature of the manifold Mⁿ is not positive and the L^n/2 norm of the Ricci curvature tensor is less than some positive constant,then there are no nontrivial L^β p-harmonic forms on the manifold.In the second part,we study the relationship between the index of the Schrodinger operator L=Δ+n²/2|E| and the L^q p-harmonic forms on a locally conformal flat Riemannian manifold.We get some vanishing theorems and finiteness theorems by assuming some conditions on the index of the Schrodinger operators L on complete manifolds.Firstly,we assume that the index of the Schrodinger operator L is 0,the scalar curvature of the manifold is not positive,and the L^n/2 norm of the scalar curvature is less than some positive constant.By using the Cauchy-Schwarz inequality,integration by parts,H（?）lder inequality and Sobolev inequality,we obtain that there are no nontrivial L^β p-harmonic t-forms on the manifold.Secondly,we assume that the scalar curvature of the manifold is non-positive,the index of the Schrodinger operator L is finite and the L^n/2 norm of the scalar curvature is finite.By using Moser iteration,we obtain that the dimension of the space of L^β p-harmonic t-forms on the manifold is finite.