Geodesic
Generalized Weingarten hypersurfaces of radial support type
Serdica Mathematical Journal, Tome 48 (2022) no. 1–2, pp. 69-102.

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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.
Keywords: generalized Weingarten surfaces, holomorphic functions, Weierstrass-type representation, Appell surfaces, 53A05, 53A07, 30F15 
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     title = {Generalized {Weingarten} hypersurfaces of radial support type},
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Pereira Santos, Laredo Rennan; Corro, Armando Mauro Vasquez. Generalized Weingarten hypersurfaces of radial support type. Serdica Mathematical Journal, Tome 48 (2022) no. 1–2, pp. 69-102. http://geodesic.mathdoc.fr/item/SMJ2_2022_48_1_2_a1/