On the locally conformally flat hyper-surfaces with nonnegative scalar curvature in R^5

Abstract

This thesis studies the geometry and topology of manifolds from an extrinsic point of view. Suppose M^4 is a complete non-compact locally conformally flat hyper-surface with non-negative scalar curvature immersed in R^5. Given some conditions on the second fundamental form and the mean curvature, we should show that if the L^4 norm of the mean curvature of M is bounded by some constant which does not depend on the manifold M, then M is embedded in R^5. This result should be a generalization of S. M uller and V. Sver ak's result on two dimensional manifolds which immersed in R^n.