Classes of Weingarten Surfaces in S^2xR

TL;DR

This paper characterizes special classes of Weingarten surfaces in a conformally flat space modeled on S^2×R, linking them to minimal and EDSGHW surfaces in Euclidean space, with explicit representations.

Contribution

It introduces two classes of Weingarten surfaces in the Radial Model of S^2×R, connecting them to known minimal and EDSGHW surfaces in Euclidean space, and provides their Weierstrass representations.

Findings

First class surfaces correspond to minimal surfaces in R^3.

Second class surfaces correspond to EDSGHW surfaces in R^3.

Both classes admit Weierstrass-type representations.

Abstract

In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature $K=1$ if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\mathbb{S}}^2\times {\mathbb{R}}$, so we consider it as the {\em Radial Model} of ${\mathbb{S}}^2\times {\mathbb{R}}$. We obtain two classes of Weingarten surfaces in the {\em Radial Model}, which satisfy $\tilde{K}_E+\tilde{H}^2-\tilde{K}=0 $ and $2\tilde{K}_E-\tilde{K}=0 $, where $\tilde{K}$ is the Gaussian curvature, $\tilde{H}$ is the mean curvature and $\tilde{K}_E$ is the extrinsic curvature. Moreover, by using the relations between the curvatures of the {\em Radial Model} and the curvatures with respect to the euclidean metric ([CPS]), we prove that first class the Weingarten surfaces in {\em Radial Model} corresponds, up to isometries, to the minimal surfaces in $\mathbb{R}^3$, and second class corresponds to EDSGHW - surfaces in Euclidean space $ \mathbb{R} ^ 3$(\cite{DC}). Consequently these two classes of surfaces have a Weierstrass type representation depending on two holomorphic functions.