Abstract:We generalize a parametrization obtained by A.\,V.\ Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a~class of oriented hypersurfaces $M^n$, $n\geq 2$, in Euclidean space satisfying a~relation $\sum_{r=1}^{n} (-1)^{r+1}rf^{r-1} { n \choose r}H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty}(M^n;\mathbb{R})$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerr\'e minimal surfaces). We characterize these hypersurfaces in terms of harmonic applications. Also, we classify the Weingarten hypersurfaces of the spherical type of rotation and we give explicit examples.
Keywords: Weingarten hypersurface; Laguerre minimal surface; $r$th mean curvature; Laplace--Beltrami operator
DOI: DOI 10.14712/1213-7243.2020.024
AMS Subject Classification: 53C42 53A35