A Class of Weingarten Surfaces in Euclidean 3-Space

Yu Fu,Lan Li

doi:10.1155/2013/398158

Abstract

The class of biconservative surfaces in Euclidean 3-space𝔼3are defined in (Caddeo et al., 2012) by the equationA(grad H)=-H grad Hfor the mean curvature functionHand the Weingarten operatorA. In this paper, we consider the more general case that surfaces in𝔼3satisfyingA(grad H)=kH grad Hfor some constantkare 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 in𝔼3.

Highlights

Let x : Mn → Em be an isometric immersion of a submanifold Mn into Euclidean space Em
From the view of geometry, we propose to study the surfaces in E3 satisfying a more general equation: A = kH grad H, k ∈ R
We focus on the equation and study this class of surfaces in E3

Summary

Introduction

Let x : Mn → Em be an isometric immersion of a submanifold Mn into Euclidean (pseudo-Euclidean) space Em. The only biharmonic submanifolds of Euclidean spaces are the minimal ones This conjecture has been proved by some geometers for some special cases. Caddeo et al [14] investigated biconservative surfaces in the three-dimensional Riemannian space forms They proved that a biconservative surface in Euclidean 3-space is either a CMC (constant mean curvature) surface or a surface of revolution. This class of surfaces carry some interesting geometry. It was proved in [14] that the mean curvature function H of a non-CMC biconservative surface in a three-dimensional space form M3(c) satisfies the following relation:. Note that our method is slightly different from the method developed by Caddeo et al in [14]

Preliminaries

The Characterizations of Generalized Biconservative Surfaces

Some Examples of Generalized Biconservative Surfaces