# A Class of Weingarten Surfaces in Euclidean 3-Space

Authors: Yu Fu; Lan Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The class of biconservative surfaces in Euclidean 3-space𝔼3 are defined in (Caddeo et al., 2012)  by the equation A(gradH)=-HgradH for the mean curvature function H and the Weingarten operator A. In this paper, we consider the more general case that surfaces in 𝔼3 satisfying A(gradH)=kHgradH 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 𝔼3.


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## Body

## 1. Introduction

Letx:Mn→𝔼mbe an isometric immersion of a submanifoldMninto Euclidean (pseudo-Euclidean) space𝔼m. We denote byx,H→, andΔthe position vector, the mean curvature vector field, and the Laplace operator ofMn, respectively, with respect to the induced metric. The submanifoldMnin𝔼mis said to biharmonic if it satisfies the equationΔ2x=0. According to the well-known Betrami’s formulaΔx=-nH→, the biharmonic condition in Euclidean space𝔼mis also known as the equationΔH→=0.There is a well-known conjecture of  Chen [1].Chen’s Conjecture. The only biharmonic submanifolds of Euclidean spaces are the minimal ones.This conjecture has been proved by some geometers for some special cases. For example,  Chen proved that every biharmonic surface in the Euclidean 3-space𝔼3is minimal. Hasanis and Vlachos [2] proved that every biharmonic hypersurface in𝔼4is minimal, also  see [3]. For the general case, the conjecture is still open so far.The study of biharmonic submanifolds is nowadays a very active research subject. Many interesting results on biharmonic maps and submanifolds have been obtained in the last decade, see [1–13].Very recently, Caddeo et al.  introduced the notion of biconservative submanifolds in [14], which is a natural generalization of biharmonic submanifolds. It is interesting that biconservative submanifolds form a much bigger family of submanifolds including biharmonic submanifolds.Recall a well-known result; see, for instance, [3].Theorem A.
Letϕ:Mn→𝔼n+1be a hypersurface with mean curvature vectorH→=Hξ. Then,ϕis biharmonic if and only if the following equations hold:
(1)ΔH-H|A|2=0,2A(gradH)+nHgradH=0,
whereAis the shape operator of the hypersurface with respect to the unit normal vectorξ.Following the definition of Caddeo et. al. in [14], a hypersurfaceMnin an (n+1)-dimensional Euclidean space𝔼n+1is called biconservative if
(2)2A(gradH)+nHgradH=0.
In general, a submanifold is biconservative if the divergence of the stress bienergy tensor vanishes.In 1995, Hasanis and Vlachos, in [2], firstly studied biconservative hypersurfaces, which are also called H-hypersurfaces. The authors gave a classification of biconservative hypersurfaces in Euclidean 3-spaces and 4-spaces. Recently, Caddeo et al. [14] investigated biconservative surfaces in the three-dimensional Riemannian space forms. Moreover, 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 functionHof a non-CMC biconservative surface in a three-dimensional space formM3(c)satisfies the following relation:
(3)K=-3H2+c,
whereKandHdenote the Gaussian curvature and mean curvature of the surfaces, respectively. Clearly, the forementioned relation implies that all the biconservative surfaces in the Euclidean 3-space are linear Weingarten surfaces.A surface is called a Weingarten surface if there exists the Jacobi equationΨ(H,K)=0between the Gaussian curvatureKand the mean curvatureHon the surface. Weingarten surfaces were introduced by Weingarten in 1861 in the context of the problem of finding all surfaces isometric to a given surface of revolution. Along the history, they have been of interest for geometers. There is a great amount of literature on Weingarten surfaces, beginning with works of Chern, Hartman, Winter, and Hopf in the fifties of the last century. For a long time, many geometers tried to look for examples of linear Weingarten surfaces, for example, see [10].For surfaces in𝔼3, the biconservative condition is equivalent to the equation
(4)A(gradH)=-HgradH.
Observe from [14] that the forementioned equation corresponds to a class of linear Weingarten surfaces, which include CMC surfaces and a family of surfaces of revolution. From the view of equation, a natural idea is to extend this class of surfaces in order to search for more examples of Weingarten surfaces of revolution. Hence, from the view of geometry, we propose to study the surfaces in𝔼3satisfying a more general equation:
(5)A(gradH)=kHgradH,k∈ℝ.
We would like to call this new class of surfaces is generalized biconservative surfaces.In this note, we focus on the equation and study this class of surfaces in𝔼3. Precisely, we will prove that any generalized biconservative surface in Euclidean 3-space is a linear Weingarten surface satisfying a more general relationK=cH2for some constantc. A local classification of generalized biconservative surfaces in𝔼3is also obtained. Note that our method is slightly different from the method developed by  Caddeo et  al. in [14].


## 2. Preliminaries

Letx:M→𝔼3 be an isometric immersion of a surfaceMinto𝔼3. Denote the Levi-Civita connections ofM and𝔼3by∇and∇~, respectively. LetXandYdenote vector fields tangent toM, and letξbe a normal vector field. The Gauss and Weingarten formulas are given, respectively, by (cf. [8, 15])
(6)∇~XY=∇XY+h(X,Y),(7)∇~Xξ=-AξX,
whereh,Aare the second fundamental form and the shape operator. It is well known that the second fundamental formhand the shape operatorAare related by
(8)〈h(X,Y),ξ〉=〈AξX,Y〉.
The Gauss and Codazzi equations are given respectively by
(9)〈R(X,Y)Z,W〉=〈h(Y,Z),h(X,W)〉-〈h(X,Z),h(Y,W)〉,(∇XA)Y=(∇YA)X,
whereRis the curvature tensor of the Levi-Civita connection onM. The mean curvature vector fieldH→and the Gauss curvature ofMare given respectively, by
(10)H→=12traceh,K=detA.
As known from the Introduction, a surfaceMin𝔼3is biconservative if the mean curvature functionHsatisfies(11)A(gradH)+HgradH=0.
Motivated by the above equation for biconservative surfaces in𝔼3, we propose to the notion of generalized biconservative surfaces in𝔼3.Definition 1.
A surfaceM in Euclidean 3-space𝔼3is called generalized biconservative surface if the mean curvature functionHand the Weingarten operator satisfy a equation
(12)A(gradH)=kHgradH
for somek∈ℝ.Note that this class of surfaces include all the biconservative surfaces as a subclass whenk=-1.Clearly, all of the CMC surfaces in𝔼3are trivially generalized biconservative surfaces. This is also the case of biconservative surfaces. We are interested in the case of non-CMC surfaces in𝔼3.


## 3. The Characterizations of Generalized Biconservative Surfaces

In this section, let us focus on the situation of non-CMC generalized biconservative surfaces in𝔼3.Suppose thatgradH≠0on any pointp∈M. It follows from (12) thatgradHis a principal direction andkHis the corresponding principal curvature. We can choose a local orthonormal frame field{e1,e2}such thate1is parallel togradH. Therefore, we havee2(H)=0. Since (12) givesAe1=kHe1, it follows thatAe2=(2-k)He2. According to the Gauss equation, the Gaussian curvatureKis given by(13)K=k(2-k)H2,
which implies the following.Theorem 2.
The generalized biconservative surfaces in𝔼3are linear Weingarten surfaces.If we put∇Xe1=ω(X)e2, then∇Xe2=-ω(X)e1. Using the remark above, the Codazzi equation reduces to
(14)2(k-1)Hω(e1)=0,(2-k)e1(H)=2(k-1)Hω(e2).
SinceHis nonconstant, frome2(H)=0,one hase1(H)≠0. So, the second equation of (14) yieldsk≠1. Moreover, the first equation of (14) implies thatω(e1)=0. Without the loss of generality, one  assumes that H>0.According to the second equation of (14), one  divides it into the following two cases.Case A (k=2).
In this case, the surface is flat. Then, the second equation of (14) yieldsω(e2)=0as well. Choose the local coordinates onMas∂/∂s=e1and∂/∂t=e2. By applying the Gauss and Weingarten formulas (6) and (7) respectively, the immersion satisfies that
(15)xss=2Hξ,xst=0,xtt=0,(16)(ξ)s=-2H∂∂s,(ξ)t=0.By solving the second and third equations of (15), we obtain that
(17)x=c1t+α(s),
for a constant vectorc1and a curveα(s)in𝔼3. Substitute (17) into the first equation of (15) and the first equation of (16), respectively. Combining these equations, we obtain a three-order differential equation as follows:
(18)α′′′=H′Hα′′-4H2α′.
In order to solve the above equation, we introduce two functions,β(s)(vector-valued function) andu(s),by puttingβ=α′andu′=2H. Note thatu′is the nonzero principal curvature andu′is not constant.Denote by “·” the derivative with respect to the new variableu. With these symbols, (18) becomes
(19)β··+β=0,
whose solution is given by
(20)β(s)=c2cosu+c3sinu,
wherec2andc3are constant vectors in𝔼3. Consequently, by a suitable translation, the immersionxis given by
(21)x=c1t+c2∫scosuds+c3∫ssinuds.
Considering the metric of surfaces, we may choose that(22)c1=(1,0,0),c2=(0,1,0),c3=(0,0,1).
Hence, the surface can be expressed as
(23)x(s,t)=(t,∫scosu(s)ds,∫ssinu(s)ds).
Remark that the surface (23) is a cylinder, but not a circular cylinder, since the curvature of the curveαis not constant.Case B (k≠2,1).
Letu=H(2-k)/(2k-2). Sinceω(e1)=0, it follows from the second equation of (14) that[e1,ue2]=0. Therefore, there exist local coordinates(s,t)onMsuch that∂/∂s=e1and∂/∂t=ue2. Then, the metric tensor ofMis given by
(24)g=ds2+u2dt2.Sincee2(H)=0, we have thatu=H(2-k)/(2k-2)is a function depending only on the variables. Consequently, the Levi-Civita connection∇is given by the expressions
(25)∇∂/∂s∂∂s=0,∇∂/∂t∂∂s=u′u∂∂t,∇∂/∂t∂∂t=-uu′∂∂s,
and the second fundamental form is given by
(26)h(∂∂s,∂∂s)=ku(2k-2)/(2-k)ξ,h(∂∂s,∂∂t)=0,h(∂∂t,∂∂t)=(2-k)u2/(2-k)ξ.
Moreover, it follows from (25) and second fund form that the Gauss and Weingarten formulas (6) and (7) yield, respectively
(27)xss=ku(2k-2)/(2-k)ξ,xst=u′uxt,xtt=-uu′xs+(2-k)u2/(2-k)ξ,(28)(ξ)s=-ku(2k-2)/(2-k)∂∂s,(ξ)t=-(2-k)u(2k-2)/(2-k)∂∂t.
By (28), the compatibility condition of PDE system (27) is given by
(29)u′′=-k(2-k)u(3k-2)/(2-k).
Integrating on (29), we obtain
(30)u′2+(2-k)2u2k/(2-k)=C
for some integral constantC. Clearly, we haveC>0for nonconstant functionu. Solving the second equation of (27), the immersionxis given by
(31)x=u(s)a(t)+b(s)
for two vector-valued functionsa(t)andb(s)in𝔼3.Case B.1  (k = 0). In this case, (27) and (28) become
(32)xss=0,xst=u′uxt,xtt=-uu′xs+2uξ,(33)(ξ)s=0,(ξ)t=-2u-1∂∂t.
It follows from (30) that
(34)u=C-4s+C1
for some constantC1, andC>4. Substituting (31) into the first equation of (32), after a suitable translation, we obtain the immersion
(35)x(s,t)=u(s)a(t)+c1u(s)=u(s)a¯(t),
wherea¯(t)is another curve in𝔼3. Substituting (35) into the third equation of (32) and applying (33), we have the following three-order differential equation:
(36)a¯′′′(t)+(u′2+4)a¯′(t)=0.
By (30), the solution of (36) is given by
(37)a¯(t)=c2cos(Ct)+c3sin(Ct)+c4
for constant vectorci,i=2,3,4in𝔼3. Hence, the immersion becomes
(38)x(s,t)=u(s)a(t)+c1u(s)=u(s)(c2cos(Ct)+c3sin(Ct)+c4).
In view of the metric (24), one can obtain thatc2,c3,c4are mutual orthonormal and
(39)〈c2,c2〉=〈c3,c3〉=1C,〈c4,c4〉=4C(C-4).
After choosingc2,c3, andc4as
(40)c2=1C(1,0,0),c3=1C(0,1,0),c4=2C(C-4)(0,0,1),
the immersion can be expressed as
(41)x(u,t)=uC(cosCt,sinCt,2C-4).
Note that this surface is a cone.Case B.2
(k≠ 0,1,2). Substituting (31) into the first and third equations of (27), we have
(42)ua′′(t)=-uu′(u′a(t)+b′(s))+2-kku2(u′′a(t)+b′′(s)),
which is equivalent to
(43)a′′(t)=(-u′2+2-kkuu′′)a(t)-u′b′(s)+2-kkub′′(s).
Substituting (29) and (30) into the previous equation in succession, we have
(44)a′′(t)+Ca(t)=-u′b′(s)+2-kkub′′(s).
In view of (44), the two sides of the equation have different variables, respectively. Hence, we have
(45)a′′(t)+Ca(t)=c,(46)-u′b′(s)+2-kkub′′(s)=c
for some constant vectorcin𝔼3. Solving (45) gives
(47)a(t)=c1cos(Ct)+c2sin(Ct)+1Cc
for two constant vectorsc1andc2in𝔼3. Looking at (31), we may assume that c=0. In fact, the immersion can be rewritten as
(48)x=u(s)a¯(t)+b¯(s),
wherea¯(t)=c1cos(Ct)+c2sin(Ct)andb¯(s)=(1/C)u(s)c+b(s). Hence, the immersion becomes
(49)x=u(s)(c1cos(Ct)+c2sin(Ct))+b(s).
Solving (46) gives
(50)b′(s)=uk/(k-2)c3
for a constant vectorc3.One can compute from (49) that
(51)xs=u′(c1cos(Ct)+c2sin(Ct))+uk/(k-2)c3,xt=Cu(s)(-c1sin(Ct)+c2cos(Ct)).
It follows from the above expressions and the metric (24) that
(52)〈c1,c1〉=〈c2,c2〉=1C,〈c1,c2〉=〈c1,c3〉=〈c2,c3〉=0,1Cu′2+u2k/(k-2)〈c3,c3〉=1.
Combining the previous expression (52) with (30) gives
(53)〈c3,c3〉=(2-k)2C.
After a change of the variablet, we can assumeC=1. Hence, the three vectorsc1,c2,c3in𝔼3can be chosen as
(54)c1=(1,0,0),c2=(0,1,0),c3=(0,0,2-k).
Now, let us consider (50), which can be rewritten as
(55)b˙(u)u′(s)=uk/(k-2)c3,
whereb˙is the derivative ofbwith respect tou. By applying (30), (55) becomes
(56)b˙(u)=±1u2k/(k-2)-(2-k)2c3.
By solving (56)
(57)b(u)=±c3∫u1u2k/(k-2)-(2-k)2du+c4
for some constant vectorc4in𝔼3.Combining (49) with (54) and (57), and by a suitable translation, we obtain the immersion
(58)x(u,t)=(ucost,usint,f(u)),
where
(59)f(u)=∫u1(1/(2-k)2)u2k/(k-2)-1du.
In this case, the immersion is a surface of revolution with non-constant mean curvature.In summary, we have the following classification result.Theorem 3.
Letx:M→𝔼3be a nondegenerate generalized biconservative surface immersed in the 3-dimensional Euclidean space𝔼3. Then, the immersionx(M)is either a CMC surface or locally given by one of the following three surfaces: (1)
a cylinder given by(60)x(s,t)=(t,∫scosu(s)ds,∫ssinu(s)ds),where the functionu satisfies u′′(s)≠0;(2)
a cone given by(61)x(u,t)=uC(cosCt,sinCt,2C-4),whereC∈(4,+∞);(3)
a surface of revolution given by(62)x(u,t)=(ucost,usint,f(u)),wheref is defined as
(63)f(u)=∫u1(1/(2-k)2)u2k/(k-2)-1dufork∈(-∞,0)∪(0,1)∪(1,2)∪(2,+∞).


## 4. Some Examples of Generalized Biconservative Surfaces

In this section, we give some examples of generalized biconservative surfaces (3) in Theorem3, depending on different values fork.Example 1.
In the casek=-1, the functionf(u)can be integrated as
(64)f(u)=∫u1(1/9)u2/3-1du=92(u1/3u2/3-9+9ln(u1/3+u2/3-9))+C0
for some integral constantC0. Hence, by a suitable translation, the non-CMC biconservative surface in𝔼3(see also [14]) is given by
(65)x(u,t)=(ucost,usint,f(u)),
wherefis defined as
(66)f(u)=92(u1/3u2/3-9+9ln(u1/3+u2/3-9)).
See Figure 1.Figure 1
The non-CMC biconservative surfaces.Example 2.
Fork=-2, the functionf(u)can be integrated as
(67)f(u)=∫u1(1/16)u-1du=8u-16+C1
for some integral constantC1andu>16. We have a non-CMC generalized biconservative surface (after a suitable translation) in𝔼3, given by
(68)x(u,t)=(ucost,usint,8u-16).



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