A class of generalized special Weingarten surfaces

Abstract

In [Classes of generalized Weingarten surfaces in the Euclidean 3-space, Adv. Geom. 16(1) (2016) 45–55], the authors study a class of generalized special Weingarten surfaces, where coefficients are functions that depend on the support function and the distance function from a fixed point (in short EDSGW-surfaces), this class of surfaces has the geometric property that all the middle spheres pass through a fixed point. In this paper, we present a Weierstrass type representation for EDSGW-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we classify isothermic EDSGW-surfaces with respect to the third fundamental form parametrized by planar lines of curvature. Moreover, we give explicit examples of EDSGW-surfaces and isothermic EDSGW-surfaces.