Complete λ-hypersurfaces with constant norm of the second fundamental form

Abstract

In this paper, we introduce a new divergence theorem on a complete proper λ-hypersurface. By the new divergence theorem we classify λ-hypersurfaces under the conditions that the squared norm of the second fundamental form S and the 3-order mean curvature $f_3$ are constant. In particular, when $\lambda =0$, the self-shrinker (i.e., 0-hypersurface) is either a hyperplane $R^n$ passing through the origin, a cylinder $S^k\left(\sqrt{k}\right)\times R^{n-k},1\le k\le n-1$, or a round sphere $S^n\left(\sqrt{n}\right)$ with center at the origin.