Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sⁿ⁺¹

Abstract

We present a complete description of all rotational linear Weingarten hypersurfaces in the Euclidean sphere S n+1 . These hypersurfaces are characterized by a linear relation aH1 +bH2 = c, where H1 and H2 stand for the first two symmetric functions of the principal curvature and a, b and c are real constants.

Linear Weingarten Hypersurfaces With Bounded Mean Curvature in the Hyperbolic Space

“…Thereafter, Shu [19] proved another rigidity theorems concerning linear Weingarten hypersurfaces with two distinct principal curvatures immersed in a space form ‫ޑ‬ n+1 c . We also point out that López [13], and Barros, Silva and Sousa [5,6] obtained descriptions related to rotational linear Weingarten surfaces in the Euclidean space and in the Euclidean sphere, respectively. Furthermore, the first and second authors [4] used the Hopf's strong maximum principle and an extension of a suitable maximum principle of Yau [21] due to Caminha [8] in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space ‫ވ‬ n+1 .…”