| The investigation of complete minimal immersed hypersurfaces in Riemannianmanifold has flourished in the last century and a much better understanding of theirglobal geometric and topological structures has been obtained. In this paper, wemainly study the end and L2-harmonic one forms on a submanifold in a Riemannianmanifold. The paper is divided into five parts.In the first chapter, we introduce some related definitions and results for laterparts.In the second chapter, we study the structure of a complete noncompactnon-totally geodesic minimal hypersurface. Under some condition, we obtain thehypersurface has only one end and there exists no nontrivial L2-harmonic one formson the hypersurfaces.In the third chapter, according to the concepts of stable, δ-stable and harmonicstable, we come up with a new definition-δ-harmonic stable. In this chapter, wemainly study L2-harmonic one forms on the complete minimal hypersurfaces in Rn+1,under the assumption of-harmonic stablity, we obtain that there exists no nontrivialL2-harmonic one forms on the hypersurfaces.In the fourth chapter, we mainly study the L2-harmonic one forms of completehypersurfaces in a Riemannian manifold of nonnegative curvature, under somecondition, we obtain that there exists no nontrivial L2-harmonic one forms on thehypersurfaces.In the fifth chapter, according to the definition of super-stable, we generalize anew definition-super harmonic stable. We obtain that there exists no nontrivialL2-harmonic one forms on the complete Riemannian manifold Nn+p withnon-negative sectional curvature. |
