The Vanishing Theorems And Finiteness Theorems Of L~p Harmonic Forms

This paper mainly study the dimension of the space of Lp(p>1)harmonic forms on complete noncompact Riemannian manifold Mm.Imposing restrictions on the curvature of Mm for the technique of Bochner formula,we get the integral inequality of the norm of harmonic forms by using the cutoff function,divergence theorem,Sobolev inequality and other basic inequalities.Furthermore,we obtain the vanishing theorems and the finiteness theorems of Lp harmonic forms by the condition of LP.The main contents is as follows:1.Studying the Lp p-harmonic r-forms on complete non-compact Riemannian manifold Mm with weighted Poincare inequality.Applying Bochner formula by assuming that the Weitzenbock curvature of Mm has an appropriate lower bound.By using the technique of cutoff function,the divergence theorem,weighted Poincare inequality and the assumption of the lower bound of the first eigenvalue of the Laplacian of Mm.We get the integral inequality of the norm |w| of p-harmonic r-form(1≤r≤m)and then obtain |w|=0 by properties of Lp,that is,the vanishing theorem of Lp p-harmonic r-forms on Mm.2.Considering the Lp harmonic 1-forms on complete non-compact minimal hypersurfaces Mm in spheres.Assume that Mm has finite index,we get x0∈Mm,r0∈ R such that the inequality of the norm of Lp harmonic 1-forms holds on Bx0(r0+1)by the technique of Sobolev inequality,index iteration,cutoff function.Furthermore,we obtain the finiteness theorems of Lp harmonic 1-forms on Mm by the inequality of the dimensional estimation of the Lp harmonic forms.3.Studying the Lp harmonic 1-forms on complete non-compact submanifolds Mm in spheres.Assuming that the total curvature(Lm-norm of the traceless sec-ond fundamental form)or the maximal norm of traceless second fundamental form of Mm,respectively,bounded from above by certain constants.Applying Bochn-er formula for harmonic 1-forms and the estimation of the Ricci curvature of the submanifolds in spheres,combing the technique of cutoff function,the divergence theorem and the Sobolev inequality,we prove the vanishing theorem of Lp harmonic 1-forms on Mm.