Studies On Vanishing Theorems Of Submanifolds In Riemannian Manifolds

In this paper,we mainly study the existence of non-trivial harmonic 1-forms in complete non-compact submanifolds in the Riemannian manifolds.Especially for a complete non-compact submanifold with constant mean curvature in hyperbolic space,if the length of tracelese second fundamental form is bounded,we get the vanishing theorems of nontrivial L2 harmonic 1-forms in the submanifolds.At the same time,under the condition of total curvature bounded,we obtained similar theorem in complete non-compact submanifolds in the Riemannian manifolds with lower bound sectional curvature.The paper consists of three sections:The first section is preliminary knowledge,it mainly introduces the related concepts and basic conclusions of the harmonic forms,as well as several types of inequalities involved in the proofs of the main theorems.In the second section,we mainly study the existence of L2 harmonic 1-forms in the complete noncompact submanifold with constant mean curvature in hyper-bolic space.Under the condition that the length of tracelese second fundamental form is bounded,we proves that there is no non-trivial L2 harmonic 1-form in the submanifold.As a result,such submanifold has only one end.In the third section,we investigate the existence problem of Lr harmonic 1-form in the complete non-compact submanifold of the Riemannian manifold with sectional curvature be bounded from lower.Under the condition that the total curvature of the submanifold is bounded,we prove the space of the L2 harmonic 1-forms has finite dimension.Furthermore,under the condition that the submanifold with bounded total curvature,then there exists no nontrivial L2 harmonic 1-form.