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

Abstract

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.

Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold

“…It immediately follows from the above result that such M must have only one end. We remark that the upper bound of φ n is less than the upper bound in Corollary 1.1 of [2], which is nonetheless a generalization of [9] and [18]. Moreover, when the ambient space N has a pinched nonpositive sectional curvature, we immediately obtain the following.…”

“…(See [3,8,10,19,27] and references therein for more details.) It is well-known that the only complete orientable stable minimal surface in R 3 is a plane [4,7].…”