Compact hypersurfaces with constant higher order mean curvatures
Let \(k_ i\), \(i=1,...,n\) be the principal curvatures of an orientable hypersurface \(M^ n\) of the Euclidean space \(R^{n+1}\), with some fixed orientation. The mean curvature of order r, \(H_ r\) is defined by the identity \[ (1+tk_ 1)...(1+tk_ n)=1+\left( \begin{matrix} n\\ 1\end{matrix} \right)H_ 1T+\left( \begin{matrix} n\\ 2\end{matrix} \right)H_ 2t^ 2+...\quad +\left( \begin{matrix} n\\ n\end{matrix} \right)H_ nt^ n \] for any real number t. In this paper it is proved that the sphere is the only embedded compact hypersurface in the Euclidean space with \(H_ r\) constant for some \(r=1,...,n\).
scientific article; zbMATH DE number 4102001