On stable complete hypersurfaces with vanishing \(r\)-mean curvature
The \(r\)-mean curvature of a hypersurface \(x: M^n\to\mathbb{R}^{n+1}\) is \[ H_r= {n\choose r}^{-1} \sum_{i_1<\cdots< i_r} k_{i_1}\cdots k_{i_r}, \] where \(k_{i_1}\cdots k_{i_r}\) are the principal curvatures of \(x\). It is proven that there exists no complete orientable, proper, stable hypersurface \(x: M^n\to \mathbb{R}^{n+1}\), of finite total curvature \(n\geq 3\), with \(H_{n-1}= 0\) and \(H_n\neq 0\) everywhere.
scientific article; zbMATH DE number 2114240