Generalized Weingarten hypersurfaces of radial support type

Abstract

In this paper we introduce a class of Weingarten hypersurfaces called generalized Weingarten hypersurfaces of radial support type (in short, RSGW hypersurfaces). These hypersurfaces are parameterized using a new technique consisting in obtain locally any hypersurface in Euclidean space as an envelope of a sphere congruence wherein the other envelope is contained in a sphere. We extend the definition of the classical Appell surfaces to hypersurfaces and we characterize both Appell and RSGW hypersurfaces in terms of a same harmonic function in the sphere. For two-dimensional case, we provide a Weierstrass-type representation for RSGW surfaces and from this we get a Weierstrass-type representation for the classical Appell surfaces. These representations depend on two holomorphic functions in such a way that a same pair of functions provides examples in each class. Using it, we give a classification for the rotational cases for both classes. Furthermore is provided a necessary and sufficient condition on the holomorphic data of these classes so that they are parameterized by lines of curvature. We also prove that a compact, complete RSGW surface is a sphere.