The complete hyper-surfaces with zero scalar curvature in mathbb{R }^{n+1}

Let $M^n$ be a complete and noncompact hyper-surface immersed in $R^{n+1}$ . We should show that if $M$ is of finite total curvature and Ricci flat, then $M$ turns out to be a hyperplane. Meanwhile, the hyper-surfaces with the vanishing scalar curvature is also considered in this paper. It can be shown that if the total curvature is sufficiently small, then by refined Kato’s inequality, conformal flatness and flatness are equivalent in some sense. And those results should be compared with Hartman and Nirenberg’s similar results with flat curvature assumption.