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.
2010 Mathematics Subject Classification:
53A05, 53A07, 30F15Keywords
generalized Weingarten surfaces, holomorphic functions, Weierstrass-type representation, Appell surfaces
Author Details
Laredo Rennan Pereira Santos
Instituto Federal de Educação Ciência e Tecnologia de Goiás
Rua 64, Expansão Parque Lago, Formosa-Brasil
73813816, Goiás-GO, Brasil
e-mail: laredo.santos@ifg.edu.br
Armando Mauro Vasquez Corro
Instituto de Matemática e Estatística
Universidade Federal de Goiás
74001-970, Goiás-GO, Brasil
e-mail: corro@mat.ufg.br
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