Complete Noncompact Submanifolds Of Manifolds With Negative Curvature

In this paper,given an n-dimensional(n≥6)complete simply connected Rieman-nian manifold N with negative sectional curvature,we investigated the m-dimensional(m≥5)complete noncompact submanifold M immersed in N.Based on the technique of Bochner,imposing restrictions on the squared norm of the second fundamental form of M,the norm of its weighted mean curvature vector |H_f| and the weighted real-valued function f,we get the nonexistence of L~pharmonic 1-form by using Young’s inequality,Sobolev inequality,divergence theorem and other methods.Besides,the conclusion that M has only one end was successfully obtained.Furthermore,we ob-tained two Liouville type theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.This article consists of three parts.The first part introduces the background and significance of this study,and gives the main conclusions of this article.The second part shows the basic needed,notations and relevant conclusions.In the third part,based on the idea of Wang-Xia in[42],using methods such as Young’s inequality,Sobolev inequality and Bochner technique mentioned in the above paragraph,the proofs of main conclusions in this article will be shown.