Higher order Willmore hypersurfaces in Euclidean space
For \(n>2\), \(x:M^n\to\mathbb{R}^{n+1}\) denotes an \(n\)-dimensional hypersurface; for \(0<i\leq n-1\), \(\sigma_i\) denotes the \(i\)-th mean curvature and \(Q_n=\sum_{i=0}^n(-1)^{i+1} {n \choose i}\sigma_1^{n-i}\sigma_i\). In [Differ. Geom. Appl. 25, No. 5, 543--551 (2007; Zbl 1138.53016)], the author showed that \(W_n(x)=\int_M Q_ndM\) is a conformal invariant, which he called the \textit{Willmore functional of \(x\)}, and he studied a variational problem related to that functional. The author calls an extremal hypersurface for the functional \(W_n\) an \textit{\(n\)-th order Willmore hypersurface}. In the case \(n=2\), \(W_2\) is the well-known Willmore functional \(\int_M(\sigma_1^2 -\sigma_2)^{n/2}\), studied by several authors. The purpose of the paper under review is to construct concrete examples of third-order Willmore hypersurfaces.
scientific article; zbMATH DE number 5629486
scientific article