A Class of Weingarten Surfaces in Euclidean 3-Space

Abstract

The class of biconservative surfaces in Euclidean 3-space ${𝔼}^{\mathrm{3}}$ are defined in (Caddeo et al., 2012) by the equation $A(\text{grad\hspace{0.17em}}H)=-H\text{\hspace{0.17em}grad\hspace{0.17em}}H$ for the mean curvature function $H$ and the Weingarten operator $A$. In this paper, we consider the more general case that surfaces in ${𝔼}^{\mathrm{3}}$ satisfying $A(\text{grad\hspace{0.17em}}H)=kH\text{\hspace{0.17em}grad\hspace{0.17em}}H$ for some constant $k$ are called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in ${𝔼}^{\mathrm{3}}$.