Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces
BBOUR’S THEOREM AND HELICOIDAL SURFACES WITH CONSTANTMEAN CURVATURE IN THE BIANCHI-CARTAN-VRANCEANU SPACES
RENZO CADDEO, IRENE I. ONNIS, AND PAOLA PIU
Abstract.
In this paper we generalize a classical result of Bour concerning helicoidal surfaces inthe three-dimensional Euclidean space R to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian -manifolds whose metrics have groups of isome-tries of dimension or , except the hyperbolic one. In particular, we prove that in a BCV-spacethere exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface;then, by making use of this two-parameter representation, we characterize helicoidal surfaces whichhave constant mean curvature, including the minimal ones. Introduction and preliminaries
Helicoidal surfaces in the Euclidean three dimensional space R are invariant under the action ofthe -parameter group of helicoidal motions and are a generalization of rotation surfaces. Sincethe beginning of differential geometry of surfaces much attention has been given to the surfaces ofrevolution with constant Gauss curvature or constant mean curvature (CMC-surfaces). The sur-faces of revolution with constant Gauss curvature seem to have been known to Minding (1839, [16]),while those with constant mean curvature have been classified by Delaunay (1841, [10]). Helicoidalminimal surfaces were studied by Scherk in 1835 (see [26] and, also, [29]), but it is rather recent theclassification of the helicoidal surfaces in R with nonzero constant mean curvature, given by DoCarmo and Dajczer in [12].The starting point of their work in [12] is a result of Bour about helicoidal surfaces in R (see [5],p. 82, Theorem II), for which he received the mathematics prize awarded by the Académie desSciences de Paris in 1861 . Bour proved that there exists a -parameter family of helicoidal surfacesisometric to a given helicoidal surface in R . For this, firstly he obtained orthogonal parameters ( u, t ) on a helicoidal surface M for which the families of u -coordinate curves are geodesics on M parametrized by arc length, and the t -coordinate curves are the trajectories of the helicoidal motion.Such parameters are called natural parameters and the first fundamental form with respect to themtakes the form ds = du + U ( u ) dt . Reciprocally, given the natural parameters ( u, t ) on M anda function U ( u ) , Bour determined a -parameter family of isometric helicoidal surfaces that haveinduced metric given by ds = du + U ( u ) dt , that includes rotation surfaces. An exposition ofBour’s results about the theory of deformation of surfaces can be found in the Chapter IX of [9]. Mathematics Subject Classification.
Key words and phrases.
Helicoidal surfaces, constant mean curvature surfaces, BCV-spaces, Bour’s theorem.Work supported by Fondazione di Sardegna (Project STAGE) and Regione Autonoma della Sardegna (ProjectKASBA). The problem that sometimes bears the name of Bour was proposed in 1861 by the Académie des Sciences andconsists in determining all the surfaces that are isometric to a given surface ( M, ds ) . E. Bour demonstrated thateach helicoidal surface is applicable to a surface of revolution, and that the helices on the first surface correspondto the parallels on the second. Bour’s work [5] contains several theorems on ruled and minimal surfaces; but in itsprinted version this work does not include the complete integration of the problem’s equations in the case of surfacesof revolution; in fact, it is this result that enabled Bour to win the Academy’s grand prix. a r X i v : . [ m a t h . DG ] F e b y using the result of Bour, in [12] Do Carmo and Dajczer established a condition for a surface ofthe Bour’s family to have constant mean curvature. Also they obtained an integral representation(depending on three parameters) of helicoidal surfaces with nonzero constant mean curvature, thatis a natural generalization of the representation for Delaunay surfaces, i.e. CMC rotation surfaces,given by Kenmotsu (see [15]).In [25] the authors obtain a generalized Bour’s theorem for helicoidal surfaces in the products H × R and S × R , and use it to determine all isometric immersions in these spaces that give the surfaceswhich are helicoidal and have the same constant mean curvature.In regard to the study of CMC helicoidal surfaces in BCV-spaces, in [13] and in [17, 20] the authorsuse the equivariant geometry to classify the profile curves of these surfaces in the Heisenberg group H and in H × R , respectively. The case of rotational minimal and constant mean curvature surfacesin the Heisenberg group is treated in [6]. J. Ripoll in [23, 24] classified the CMC invariant surfacesin the -dimensional sphere S and also in the hyperbolic -space H .The aim of this paper is to generalize the results obtained in [12] and [25]. The paper is organizedas follows. Section 2 is devoted to give a short description of the Bianchi-Cartan-Vranceanu spacesand the helicoidal surfaces in these spaces. In Section 3 we establish a Bour’s type theorem forhelicoidal surfaces in the BCV-spaces (see Theorem 3.1) and, as an immediate consequence of thisresult, we have that every helicoidal surface in a BCV-space can be isometrically deformed into arotation surface through helicoidal surfaces. Moreover, Corollary 3.4 refers to the particular case ofisometric rotation surfaces.In Section 4 we use standard techniques of equivariant geometry, in particular the Reduction The-orem of Back, do Carmo and Hsiang (see [2]), to deduce a differential equation that the function U ( u ) must satisfy in order that a helicoidal surface of the Bour’s family determined by U ( u ) hasconstant mean curvature. We solve this equation by making a transformation of coordinates, treat-ing separately the case of the space forms R and S from the other BCV-spaces. In this way, weobtain Theorem 4.3 that provides a description, in terms of natural parameters, of all helicoidalsurfaces of constant mean curvature in a BCV-space, including the minimal ones. We conclude byshowing that in R these results give a natural parametrization of all the helicoidal minimal surfacesobtained by Scherk in [26].2. Helicoidal surfaces in Bianchi-Cartan-Vranceanu spaces
A Riemannian manifold ( M , g ) is said to be homogeneous if for every two points p and q in M , thereexists an isometry of M , mapping p into q . The classification of -dimensional simply connectedhomogeneous spaces is well-known and can be summarized as follows. First of all, the dimensionof the isometry group must be equal to , or (see [3] or [14]). Then, if the isometry group isof dimension , M is a complete real space form, i.e. the Euclidean space E , a sphere S ( k ) , or ahyperbolic space H ( k ) . If the dimension of the isometry group is , M is isometric to SU(2) , thespecial unitary group, to (cid:94)
SL(2 , R ) , the universal covering of the real special linear group, to Nil ,the Heisenberg group, all with a certain left-invariant metric, or to a Riemannian product S ( k ) × R or H ( k ) × R . Finally, if the dimension of the isometry group is , M is also isometric to a simplyconnected Lie group with a left-invariant metric, for example that called SOL , one of the Thurston’seight models of geometry [27]. n explicit classification of -dimensional homogeneous Riemannian metrics based on the dimensionof their isometry group was first given by Luigi Bianchi in 1897 (see [3] or [4]). Later Élie Cartanin [7] and Gheorghe Vranceanu in [28] proved that all the metrics whose group of isometries hasdimension or , except the hyperbolic one, can be represented in a concise form by the followingtwo-parameter family of metrics(2.1) g κ,τ = dx + dy B + (cid:18) dz + τ ydx − xdyB (cid:19) , for κ, τ ∈ R , and B = 1 + κ x + y ) , ( x, y, z ) ∈ R , positive. Thus, the family of metrics g κ,τ , thatcan rightfully be named the Bianchi-Cartan-Vranceanu metrics (BCV metrics) consists of allthree-dimensional homogeneous metrics whose group of isometries has dimension or , except forthose of constant negative sectional curvature . In the following we shall denote by N κ,τ the opensubset of R where the metrics g κ,τ are defined.With respect to (2.1) we have the following globally defined orthonormal frame(2.2) E = B ∂∂x − τ y ∂∂z , E = B ∂∂y + τ x ∂∂z , E = ∂∂z and, also, Proposition 2.1 ([21, 22]) . The isometry group of g κ,τ admits the basis of Killing vector fields (2.3) X = (cid:16) − κ y B (cid:17) E + κ xy B E + 2 τ yB E ,X = κ xy B E + (cid:16) − κ x B (cid:17) E − τ xB E ,X = − yB E + xB E − τ ( x + y ) B E ,X = E . Therefore, the group of isometries of the BCV-spaces contains the helicoidal subgroup, whose infin-itesimal generator is the Killing vector field given by X = − y ∂∂x + x ∂∂y + a ∂∂z , a ∈ R . We consider the surfaces in N κ,τ which are invariant under the action of the one-parameter group ofisometries G X of g κ,τ generated by X . For convenience, we shall introduce cylindrical coordinates x = r cos θ,y = r sin θ,z = z, with r ≥ and θ ∈ (0 , π ) . In these coordinates the metric (2.1) becomes g κ,τ = dr B + r (cid:16) τ r B (cid:17) dθ + dz − τ r B dθdz, where B = 1 + κ r . Moreover, the Killing vector field X takes the form X = ∂∂θ + a ∂∂z and a set of two invariant functions is ξ = r, ξ = z − a θ. hus, the orbit space of the action of G X can be identified with(2.4) B := N κ,τ /G X = { ( ξ , ξ ) ∈ R : ξ ≥ } and the orbital distance metric of B is given by(2.5) (cid:101) g = dξ B + ξ dξ ξ + ( a B − τ ξ ) , where B = 1 + κ ξ .Now, consider a helicoidal surface M (with pitch a ) that, locally, with respect to the cylindricalcoordinates, can be parametrized by(2.6) ψ ( u, θ ) = ( ξ ( u ) , θ, ξ ( u ) + a θ ) , and suppose that the profile curve ˜ γ ( u ) = ( ξ ( u ) , ξ ( u )) is parametrized by arc-length in ( B , (cid:101) g ) , sothat(2.7) ξ (cid:48) B + ξ ξ (cid:48) ξ + ( a B − τ ξ ) = 1 . Therefore from ψ u = ξ (cid:48) (cid:16) cos θB E + sin θB E (cid:17) + ξ (cid:48) E ,ψ θ = ξ B (cos θ E − sin θ E ) + (cid:18) a − τ ξ B (cid:19) E = X it follows that the coefficients of the induced metric of the helicoidal surface are given by(2.8) E ( u ) = 1 + ξ (cid:48) ( u ) (cid:18) a B ( u ) − τ ξ ( u ) B ( u ) ω ( u ) (cid:19) , F ( u ) = ξ (cid:48) ( u ) (cid:18) a B ( u ) − τ ξ ( u ) B ( u ) (cid:19) , and G = ξ ( u ) B ( u ) + (cid:18) a − τ ξ ( u ) B ( u ) (cid:19) = ω ( u ) , where ω ( u ) is the volume function of the principal orbit.3. A Bour’s type theorem
In this section, we show that every helicoidal surface in a BCV-space admits a reparametrization bynatural parameters and, conversely, given a positive function U , it is possible to find a -parameterfamily of isometric helicoidal surfaces associate with it that are parameterized by natural parameters. Theorem 3.1.
In the BCV-space N κ,τ there exists a two parameter family of helicoidal surfacesthat are isometric to a given helicoidal surface of the form (2.6) and that includes a rotation surface.More precisely, for a given positive function U ( u ) and arbitrary constants m (cid:54) = 0 and a , the helicoidalsurfaces (2.6) whose profile curve ˜ γ ( u ) = ( ξ ( u ) , ξ ( u )) is given by (3.1) ξ ( u ) = 2 (cid:115) m U − a (1 + √ ∆) − τ m U ,ξ ( u ) = (cid:90) m U (4 + κ ξ )4 ξ (cid:114) ξ − m U U (cid:48) (4 + κ ξ )
16 ∆ du, ith (3.2) θ ( u, t ) = tm + (cid:90) (4 τ − a κ ) ξ − a m U ξ (cid:114) ξ − m U U (cid:48) (4 + κ ξ )
16 ∆ du, where ∆( u ) = (1 − a τ ) + ( m U ( u ) − a )(4 τ − κ ) , are all to each other isometric and have first fundamental form given by du + U ( u ) dt .Proof. From (2.8) we have that the induced metric of a helicoidal surface (2.6), with pitch a , isgiven by(3.3) g ψ = E ( u ) du + 2 F ( u ) du dθ + ω ( u ) dθ = du + ω ( u ) (cid:18) dθ + ξ (cid:48) ( u ) a B ( u ) − τ ξ ( u ) B ( u ) ω ( u ) du (cid:19) , where ω ( u ) = ξ ( u ) B ( u ) + (cid:18) a B ( u ) − τ ξ ( u ) B ( u ) (cid:19) . Now we introduce a new parameter t = t ( u, θ ) that satisfies:(3.4) dt = dθ + ξ (cid:48) ( u ) a B ( u ) − τ ξ ( u ) B ( u ) ω ( u ) du. As the Jacobian | ∂ ( u, t ) /∂ ( u, θ ) | is equal to , it follows that ( u, t ) are local coordinates on ahelicoidal surface M and, also, that we can write (3.3) as(3.5) g ψ = du + ω ( u ) dt . We now observe that the u -coordinate curves are parametrized by arc length and also that aregeodesics of M (see [19]) which are orthogonal to the t -coordinate curves, i.e. the helices. Conse-quently, the local parametrization ψ ( u, θ ( u, t )) is a natural parametrization of the helicoidal surface M .Conversely, given a function U ( u ) > , we want to determine functions θ, ξ , ξ of ( u, t ) such that(3.6) du = dξ B + ξ dξ ξ + ( a B − τ ξ ) , ± U ( u ) dt = (cid:112) ξ + ( a B − τ ξ ) B (cid:20) dθ + B ( a B − τ ξ ) ξ + ( a B − τ ξ ) dξ (cid:21) , where B = 1 + κ ξ .From the first equation of (3.6) we have that ξ i = ξ i ( u ) , i = 1 , . Then, from the second, we obtain(3.7) ∂θ∂u = − B ( u ) [ a B ( u ) − τ ξ ( u )] ξ ( u ) + ( a B ( u ) − τ ξ ( u )) ξ (cid:48) ( u ) ,∂θ∂t = ± B ( u ) U ( u ) (cid:112) ξ ( u ) + ( a B ( u ) − τ ξ ( u )) , where B ( u ) = 1 + κ ξ ( u ) .Therefore, ∂ θ∂t∂u = 0 nd hence there exists a constant m (cid:54) = 0 such that(3.8) ± B ( u ) U ( u ) (cid:112) ξ ( u ) + ( a B ( u ) − τ ξ ( u )) = 1 m . Thus the second equation of system (3.6) becomes(3.9) dθ = dtm − B ( u ) ( a B ( u ) − τ ξ ( u )) ξ ( u ) + ( a B ( u ) − τ ξ ( u )) dξ . If we consider the function f := 1 /ξ , equation (3.8) can be written as ( a − m U ) (cid:107)∇ f (cid:107) κ,τ + (1 − a τ ) (cid:107)∇ f (cid:107) κ,τ + τ − κ/ and, therefore, (cid:107)∇ f (cid:107) κ,τ = 1 − a τ + √ ∆2 ( m U − a ) , with ∆ = (1 − a τ ) + ( m U − a )(4 τ − κ ) . As (cid:107)∇ f (cid:107) κ,τ = f + κ/ , we conclude that(3.10) ξ = 4( m U − a )(1 + √ ∆) − τ m U . Then, differentiating (3.8) and using (3.10), we get(3.11) m B U U (cid:48) = √ ∆ ξ ξ (cid:48) and hence(3.12) ( ξ (cid:48) ) B = m B U U (cid:48) ξ ∆ . Therefore, taking into account the first equation of system (3.6), we obtain dξ = m B U ξ (cid:18) − ξ (cid:48) B (cid:19) du = m B U ξ (cid:18) ξ − m B U U (cid:48) ∆ (cid:19) du . Thus, as B = 1 + κ ξ , it turns out that(3.13) ξ ( u ) = (cid:90) m U (4 + κ ξ )4 ξ (cid:114) ξ − m U U (cid:48) (4 + κ ξ )
16 ∆ du.
Also, from (3.9) we have(3.14) θ ( u, t ) = tm + (cid:90) (4 τ − a κ ) ξ − a m U ξ (cid:114) ξ − m U U (cid:48) (4 + κ ξ )
16 ∆ du.
Consequently, the natural parametrization of the helicoidal surface (2.6) with given first fundamentalform g ψ = du + U ( u ) dt can be calculated by means of equations (3.10), (3.13) and (3.14). (cid:3) emark . If κ = 0 = τ , the BCV-space is the Euclidean space R and Theorem 3.1 becomes theclassical one ([5], p. 82, Theorem II) due to Bour. In fact, in this case the functions B and ∆ areconstant and equal to ; thus we obtain(3.15) ξ ( u ) = (cid:112) m U − a ,ξ ( u ) = (cid:90) m Um U − a (cid:112) m U − a − m U U (cid:48) du,θ ( u, t ) = tm − am (cid:90) √ m U − a − m U U (cid:48) U ( m U − a ) du. Remark . The family of helicoidal surfaces Ψ( u, t ) := ψ [ U,m,a ] ( u, t ) in the BCV-space N κ,τ ob-tained in the Theorem 3.1 depends on two parameters m (cid:54) = 0 and a , and for m = 1 and a = a itcontains the original helicoidal surface. Also, when m = 1 and a = 0 , we obtain a rotational surfaceisometric to the given helicoidal surface. Therefore, by varying the constant a from a = 0 to a = a ,we get an isometric deformation from a rotational surface to a given helicoidal surface. Example . In R , we consider the function U ( u ) = √ u + d , d ∈ R , d ≥ a . If we suppose m = 1 ,from the formulas (3.15) we obtain that ξ ( u ) = (cid:112) u + d − a ,ξ ( u ) = (cid:112) d − a (cid:90) √ u + d u + d − a du,θ ( u, t ) = t − a (cid:112) d − a (cid:90) du √ u + d [ u + d − a ] . By varying a from a = 0 to a = d we have the classical isometric deformation of the catenoid ψ [ U, , ( u, t ) = (cid:18)(cid:112) u + d cos t, (cid:112) u + d sin t, d cosh − (cid:16)(cid:114) u d + 1 (cid:17)(cid:19) into the helicoid ψ [ U, ,d ] ( u, t ) = ( u cos t, u sin t, t + b ) , b ∈ R , that are minimal surfaces. Also, theintermediate helicoidal surfaces are all minimal and their natural parametrization is given by ξ ( u ) = (cid:112) u + d − a ,ξ ( u ) = (cid:112) d − a cosh − (cid:16)(cid:114) u d + 1 (cid:17) + a arctan (cid:16) a u √ d − a √ u + d (cid:17) ,θ ( u, t ) = t − arctan (cid:16) a u √ d − a √ u + d (cid:17) , < a < d. Such surfaces are also called second Scherk’s surfaces (see [11] and, also, the Example 3).
Example . In the Heisenberg space H equipped with the metric g κ,τ with κ = 0 and τ = 1 / , weconsider the function U ( u ) = ( u + 2) / . If we suppose that m = 1 , from formulas (3.1) we get(3.16) ξ ( u ) = (cid:113)(cid:112) u + 4 u + 8 (1 − a ) + 2 ( a − ,ξ ( u ) = (cid:90) (2 + u )2 ξ ( u ) (cid:115) ξ ( u ) − u ( u + 2) u + 4 u + 8 (1 − a ) du,θ ( u, t ) = t + (cid:90) ξ ( u ) − a ( u + 2) ξ ( u ) (cid:115) ξ ( u ) − u ( u + 2) u + 4 u + 8 (1 − a ) du. n particular, for a = 1 / we obtain the curve ˜ γ ( u ) = ( (cid:112) u + 1 , ( u + arctan u ) / , the profile curve of the helicoidal catenoid , that is a helicoidal minimal surface (see [13]), parametrizedby ψ ( u, θ ) = (cid:16)(cid:112) u + 1 cos θ, (cid:112) u + 1 sin θ, u + θ + arctan u (cid:17) . Also, as θ ( u, t ) = t − arctan u + √ (cid:16) u √ (cid:17) , we have that the parametrization Ψ( u, t ) = ψ ( u, θ ( u, t )) = (cid:18) cos (cid:16) t + √ (cid:16) u √ (cid:17)(cid:17) + u sin (cid:16) t + √ (cid:16) u √ (cid:17)(cid:17) , sin (cid:16) t + √ (cid:16) u √ (cid:17)(cid:17) − u sin (cid:16) t + √ (cid:16) u √ (cid:17)(cid:17) , (cid:16) u + t + √ (cid:16) u √ (cid:17)(cid:17)(cid:19) represents a natural parametrization of the helicoidal catenoid with g Ψ = du + U ( u ) dt . Now, if we start from a = 1 / and in the equations (3.16) we consider all the decreasing values of a inthe interval [0 , / , we obtain an isometric deformation of the helicoidal catenoid into a rotationalsurface (obtained for a = 0 ), through helicoidal surfaces parametrized by natural parameters. ���� �� ���� �� a = 1 / a = 1 / ���� �� ���� �� a = 1 / a = 0 Figure 1.
Isometric deformation of the helicodal catenoid into a rotation surfacein H . The surfaces in the picture are obtained for a = 1 / , / , / and a = 0 ,respectively; only that with angular pitch a = 1 / is a minimal surface (see Re-mark 4.7). orollary 3.4. The rotation surfaces given by ψ [ U,n, ( u, t ) , with n (cid:54) = 0 , give rise in the BCV-space N κ,τ to a -parameter family of isometric surfaces that are also isometric to the helicoidal surfaces ψ [ U,m,a ] ( u, t ) . This family is determined by the formulas: (3.17) ξ ( u ) = 2 n U (cid:113) √ ∆) − κ n U ,ξ ( u ) = (cid:90) (cid:115) (1 + √ ∆) √ ∆) − κ n U − n (1 + √ ∆) U ∆ [2(1 + √ ∆) − κ n U ] du,θ ( u, t ) = tn + 2 τ (cid:90) (cid:115) √ ∆) − κ n U − n (1 + √ ∆) U (cid:48) ∆ [2(1 + √ ∆) − κ n U ] du, where ∆ = 1 + (4 τ − κ ) n U . In particular, (1) if κ = 4 τ (cid:54) = 0 , i.e. N κ,τ ∼ = S ( τ ) , the equations (3.17) become ξ ( u ) = n U √ − τ n U ,ξ ( u ) = (cid:90) (cid:112) − n ( τ U + U (cid:48) )1 − τ n U du,θ ( u, t ) = tn + τ (cid:90) (cid:112) − n ( τ U + U (cid:48) )1 − τ n U du, (2) if τ = 0 , i.e. N κ,τ ∼ = R , N κ,τ ∼ = S ( κ ) × R or N κ,τ ∼ = H ( κ ) × R , the equations (3.17) become ξ ( u ) = 2 n U √ − κ n U ,ξ ( u ) = (cid:90) (cid:114) − n ( κ U + U (cid:48) )1 − κ n U du,θ ( u, t ) = tn , (3) if κ = 0 and τ (cid:54) = 0 , i.e. N κ,τ ∼ = H , the equations (3.17) become ξ ( u ) = √ n U (cid:112) √ ∆ , ∆ = 1 + 4 τ n U ,ξ ( u ) = (cid:90) √ ∆2 (cid:115)
21 + √ ∆ − n U (cid:48) ∆ du,θ ( u, t ) = tn + τ (cid:90) (cid:115)
21 + √ ∆ − n U (cid:48) ∆ du, (4) if κ (cid:54) = 0 and τ (cid:54) = 0 , the equations (3.17) give the family of rotation surfaces in the cases N κ,τ ∼ = SU(2) and N κ,τ ∼ = (cid:94) SL(2 , R ) . From the formula for the Gaussian curvature of an invariant surface obtained in [18], it followsthat the helicoidal surfaces of the Bour’s family in the BCV-space N κ,τ have all the same Gaussiancurvature given by K ( u ) = − U (cid:48)(cid:48) ( u ) U ( u ) . ith regard to the mean curvature H of these surfaces, in next section we shall see that differentvalues of a and m can give rise to different values of H .4. Helicoidal surfaces of constant mean curvature
In this section, we will describe the helicoidal surfaces in the BCV-spaces that have the sameconstant mean curvature. We start by computing the mean curvature of a helicoidal surface (2.6).It turns out that the mean curvature of an invariant immersion is tightly related to the geodesiccurvature of the profile curve, as shown by the remarkable following theorem. But first we recallthat if on a three-dimensional connected Riemannian manifold ( N , g ) we consider the -parametersubgroup G X of isometries generated by X , an orbit G ( p ) of p ∈ N is called principal if there existsan open neighborhood U ⊂ N of p such that all orbits G ( q ) , q ∈ U , are of the same type as G ( p ) (i.e. the isotropy subgroups G q and G p are coniugated). This implies that G ( q ) is diffeomorphic to G ( p ) . We denote with N r the regular part of N , that is, the subset consisting of points belongingto principal orbits [1]. Then we have Theorem 4.1 (Reduction Theorem [2]) . Let H be the mean curvature of a G X -invariant surface M r ⊂ N r and k g the geodesic curvature of the profile curve M r /G X ⊂ B r . Then H ( x ) = k g ( π ( x )) − D n ln ω ( π ( x )) , x ∈ M r , where n is the unit normal of the profile curve and ω = (cid:112) g ( X, X ) is the volume function of theprincipal orbit. Let now ˜ γ ( u ) = ( ξ ( u ) , ξ ( u )) be a curve in B r , parametrized by arc-length, that under the actionof G X generates the helicoidal surface. From (2.5), it follows that(4.1) ξ (cid:48) = B cos σ, ξ (cid:48) = (cid:112) ξ + ( a B − τ ξ ) sin σξ and the geodesic curvature of ˜ γ takes the expression(4.2) k g = ( (cid:101) g ) ξ ξ (cid:48) − ( (cid:101) g ) ξ ξ (cid:48) (cid:112)(cid:101) g (cid:101) g + σ (cid:48) = B [ ξ + ( a B − τ ξ ) ] ( (cid:101) g ) ξ ξ sin σ + σ (cid:48) , where σ is the angle that ˜ γ makes with the ∂∂ξ direction. Also, as n = (cid:18) − B sin σ, (cid:112) ξ + ( a B − τ ξ ) ξ cos σ (cid:19) , the normal derivative is given by D n ln ω = 4 ξ [8 aτ − ξ ( κ + 2 aκ τ − τ )] sin σ ξ + [ a (4 + κ ξ ) − τ ξ ] and thus we obtain that the mean curvature is given by(4.3) H = σ (cid:48) + (cid:18) ξ − κ ξ (cid:19) sin σ. Proposition 4.2.
A helicoidal surface Ψ( u, t ) := ψ [ U,m,a ] ( u, t ) has constant mean curvature H ifand only if U ( u ) satisfies the differential equation (4.4) H (cid:115) m U − a )(1 + √ ∆) − τ m U − m B U U (cid:48) ∆ = 2 − B − m B (cid:16) U U (cid:48) √ ∆ (cid:19) (cid:48) , here (4.5) ∆ = (1 − a τ ) + ( m U − a )(4 τ − κ ) and B = 2 1 − aτ + √ ∆(1 + √ ∆) − τ m U . Proof.
If we consider a helicoidal surface Ψ( u, t ) of the Bour’s family, from (4.1) we can writeequation (4.3) as H = − (cid:0) ξ (cid:48) B (cid:1) (cid:48) (cid:113) − (cid:0) ξ (cid:48) B (cid:1) + (cid:18) ξ − κ ξ (cid:19) (cid:114) − (cid:16) ξ (cid:48) B (cid:17) and therefore, using (3.12) we get(4.6) − (cid:16) ξ (cid:48) B (cid:17) (cid:48) + (cid:18) ξ − κ ξ (cid:19) (cid:104) − (cid:16) ξ (cid:48) B (cid:17) (cid:105) = H (cid:114) − (cid:16) ξ (cid:48) B (cid:17) = Hξ (cid:114) ξ − m B U U (cid:48) ∆ . Then,(4.7) H (cid:114) ξ − m B U U (cid:48) ∆= ξ (cid:20) − (cid:16) ξ (cid:48) B (cid:17) (cid:48) + (cid:18) ξ − κ ξ (cid:19) (cid:16) − (cid:16) ξ (cid:48) B (cid:17) (cid:17)(cid:21) = 1 − (cid:16) ξ (cid:48) B (cid:17) − ξ ξ (cid:48)(cid:48) B + 3 κ ξ (cid:16) ξ (cid:48) B (cid:17) − k ξ . Now, deriving (3.11) we get ξ ξ (cid:48)(cid:48) = m B √ ∆ ( U (cid:48) + U U (cid:48)(cid:48) ) + m B ∆ (cid:18) κ B + κ − τ √ ∆ (cid:19) U U (cid:48) − ξ (cid:48) and hence, taking into account (3.10), (3.11) and (3.12), we can write equation (4.7) as H (cid:114) ξ − m B U U (cid:48) ∆= 2 − B − m B √ ∆ ( U (cid:48) + U U (cid:48)(cid:48) ) + m B ∆ (cid:18) τ − κ √ ∆ − κ B B − Bξ (cid:19) U U (cid:48) = 2 − B − m B √ ∆ ( U (cid:48) + U U (cid:48)(cid:48) ) + (4 τ − κ ) m B ∆ / U U (cid:48) . (cid:3) The solution of the mean curvature equation.
Next, we will give a description of thehelicoidal surfaces in N κ,τ with constant mean curvature H . For this purpose, we assume that thehelicoidal surfaces are parametrized by natural coordinates ( u, t ) and we determine explicitly theexpression of the function U ( u ) , that gives the metric, by integrating (4.4). Theorem 4.3.
In the BCV-space N κ,τ the helicoidal surface Ψ( u, t ) := ( ξ ( u ) , θ ( u, t ) , ξ ( u ) + a θ ( u, t )) with ξ ( u ) , ξ ( u ) and θ ( u, t ) given by (3.1) and (3.2) , has constant mean curvature H if and only if U ( u ) is given by: (1) if κ = τ = H = 0 , U ( u ) = u + a + c / m ; if κ = 4 τ (cid:54) = − H , U ( u ) = c + (cid:112) c + c ( H + 4 τ ) sin( √ τ + H u ) m ( H + 4 τ ) ; (3) if − H = κ (cid:54) = 4 τ , U ( u ) = (cid:16) b u + b (cid:17) + b m (4 τ + H ) ; (4) if − H < κ (cid:54) = 4 τ , U ( u ) = ( H + κ ) b + (cid:104) b + (cid:112) b + b ( H + κ ) sin( √ H + κ u ) (cid:105) m (4 τ − κ )( H + κ ) ; (5) if − H > κ (cid:54) = 4 τ , U ( u ) = ( H + κ ) b + (cid:104) b − (cid:112) − b − b ( H + κ ) sinh( (cid:112) − ( H + κ ) u ) (cid:105) m (4 τ − κ )( H + κ ) , b + b ( H + κ ) < , ( H + κ ) b + (cid:104) b − (cid:112) b + b ( H + κ ) cosh( (cid:112) − ( H + κ ) u ) (cid:105) m (4 τ − κ )( H + κ ) , b + b ( H + κ ) > , where b, b i , c i ∈ R , are the constants given by b = (1 − aτ ) [ κ (1 + 2 aτ ) − τ ] − c , b = 4 τ − aκτ − c H,b = − b b , b = 4 aτ − a κ − ,c = 1 + (1 − a τ ) − c H, c = − c − a (1 − a τ ) . These expressions define a -parameter family { U c ( u ) } of functions U ( u ) such that the helicoidalsurface ψ [ U c ,m,a ] ( u, t ) has constant mean curvature and equal to H .Proof. Using the transformation of coordinates given by(4.8) x ( u ) = m U ( u ) ,y ( u ) = (cid:115) ( x ( u ) − a ) [(1 + (cid:112) ∆( u )) − τ x ( u )][1 − aτ + (cid:112) ∆( u )] − x ( u ) x (cid:48) ( u ) ∆( u ) , where ∆( u ) = (1 − a τ ) + (4 τ − κ )( x ( u ) − a ) , equation (4.4) becomes(4.9) H y ( u ) = 2 B ( u ) − − (cid:32) x ( u ) x (cid:48) ( u ) (cid:112) ∆( u ) (cid:33) (cid:48) , with B ( u ) = 2 1 − aτ + (cid:112) ∆( u )(1 + (cid:112) ∆( u )) − τ x ( u ) . Therefore, y (cid:48) ( u ) = x ( u ) x (cid:48) ( u ) y ( u ) (cid:112) ∆( u ) (cid:20) B ( u ) − − (cid:32) x ( u ) x (cid:48) ( u ) (cid:112) ∆( u ) (cid:33) (cid:48) (cid:21) = H x ( u ) x (cid:48) ( u ) (cid:112) ∆( u ) nd thus(4.10) y ( u ) = H x ( u ) + c (cid:112) ∆( u ) , τ − κ = 0 ,H (cid:112) ∆( u ) + c τ − κ , τ − κ (cid:54) = 0 , where c is an arbitrary constant. Consequently, we have the following cases: Case τ − κ = 0 . From (4.8) and (4.10), if we suppose that − a τ > , then we have (cid:0) x ( u ) x (cid:48) ( u ) (cid:1) = 4 (cid:2) (1 − a τ ) − τ ( x ( u ) − a ) (cid:3) ( x ( u ) − a ) − ( H x ( u ) + c ) . Putting z = x the above expression is transformed into the following(4.11) z (cid:48) ( u ) = (cid:112) − ( H + 4 τ ) z ( u ) + 2 c z ( u ) + c , where c = 1 + (1 − a τ ) − c H, c = − c − a (1 − a τ ) . Consequently,i) If H + 4 τ = 0 , we have that the BCV-space is R and c = 2 , c = − c − a . Thus,adjusting the origin of u , we get (cid:114) z + c u and since z = m U we have(4.12) U ( u ) = u + a + c / m . We observe that when c = 0 , from (3.15) it follows that the helicoidal surface ψ [ U,m,a ] is ahelicoid.ii) If H + 4 τ (cid:54) = 0 , by integrating (4.11) we have that u = 1 √ H + 4 τ sin − (cid:18) ( H + 4 τ ) z − c (cid:112) c + c ( H + 4 τ ) (cid:19) , up to a constant. Thus, as z = m U , it follows that(4.13) U ( u ) = c + (cid:112) c + c ( H + 4 τ ) sin( √ τ + H u ) m ( H + 4 τ )= 1 m ( H + 4 τ ) (cid:20) − a τ ) − c H + 2 (cid:112) (1 − aτ ) − H (1 − aτ )( Ha + c ) − τ ( Ha + c ) sin( (cid:112) H + 4 τ u ) (cid:21) . Case τ − κ (cid:54) = 0 . In this case, as x ( u ) = 1 + a κ − a τ − ∆( u ) κ − τ ,x ( u ) x (cid:48) ( u ) (cid:112) ∆( u ) = ( (cid:112) ∆( u )) (cid:48) τ − κ , rom (4.8) and (4.10) we get ( H (cid:112) ∆( u ) + c ) = (4 τ − κ ) y ( u )= (1 − aτ − (cid:112) ∆( u )) ( κ + κ (cid:112) ∆( u ) + 2 aκτ − τ ) − (cid:16) ( (cid:112) ∆( u )) (cid:48) (cid:17) . Thus(4.14) ( (cid:112) ∆( u )) (cid:48) = (cid:113) − ( H + κ ) ∆( u ) + 2 b (cid:112) ∆( u ) + b, where b = (1 − aτ ) ( κ (1 + 2 aτ ) − τ ) − c , b = 4 τ − aκτ − c H. In the sequel we integrate equation (4.14), up to a change of the origin of u , considering the followingpossibilities:i) If H + κ = 0 , then we obtain that (cid:112) ∆( u ) = b u + b , with b = 2 aτ H + 4 τ − c H, b = − b b . Then, substituting in the first equation of (4.5), we get(4.15) U ( u ) = 1 m (cid:18) ∆( u ) − (1 − aτ ) τ + H + a (cid:19) = (cid:16) b u + b (cid:17) + a H + 4 aτ − m (4 τ + H ) . ii) If H + κ > , then the integration of (4.14) gives (cid:112) ∆( u ) = 1 H + κ (cid:104) b + (cid:113) b + b ( H + κ ) sin( (cid:112) H + κ u ) (cid:105) . Therefore, substituting in the first equation of (4.5), we obtain(4.16) U ( u ) = ( H + κ ) (4 aτ − a κ −
1) + (cid:104) b + (cid:112) b + b ( H + κ ) sin( √ H + κ u ) (cid:105) m (4 τ − κ )( H + κ ) . iii) If H + κ < , then the integration of (4.14) gives (cid:112) ∆( u ) = H + κ (cid:104) b − (cid:113) − b − b ( H + κ ) sinh( (cid:112) − ( H + κ ) u ) (cid:105) , b + b ( H + κ ) < , H + κ (cid:104) b − (cid:113) b + b ( H + κ ) cosh( (cid:112) − ( H + κ ) u ) (cid:105) , b + b ( H + κ ) > . Therefore, substituting in the first equation of (4.5), we obtain(4.17) U ( u ) = ( H + κ ) b + (cid:104) b − (cid:112) − b − b ( H + κ ) sinh( (cid:112) − ( H + κ ) u ) (cid:105) m (4 τ − κ )( H + κ ) , b + b ( H + κ ) < , ( H + κ ) b + (cid:104) b − (cid:112) b + b ( H + κ ) cosh( (cid:112) − ( H + κ ) u ) (cid:105) m (4 τ − κ )( H + κ ) , b + b ( H + κ ) > . (cid:3) emark . In particular, if we consider m = 1 and a = a in the expressions of U ( u ) obtainedin Theorem 4.3, we see that an arbitrary helicoidal surface in N κ,τ has constant mean curvature H if and only if the functions ξ ( u ) , ξ ( u ) and θ ( u, t ) are given by (3.1) and (3.2) by substituting thecorresponding function U ( u ) . Remark . Putting κ = τ = 0 in the equations (4.12) and (4.13) we obtain the following expres-sions: U ( u ) = u + a + c / m , if H = 0 , − c H + 2 √ − c H − a H sin ( H u ) m H , if H (cid:54) = 0 , the second of which was given by Do Carmo and Dajczer in [12].In regard to the helicoidal minimal surfaces we have the following result: Corollary 4.6.
In each BCV-space N κ,τ , the helicoidal minimal surfaces Ψ( u, t ) := ( ξ ( u ) , θ ( u, t ) , ξ ( u )+ a θ ( u, t )) , where ξ ( u ) , ξ ( u ) and θ ( u, t ) are given by (3.1) and (3.2) , are determined by the followingfunctions U ( u ) : (1) If κ = τ = 0 , then N κ,τ ∼ = R and (4.18) U ( u ) = u + a + c / m ; (2) if κ = 4 τ (cid:54) = 0 , then N κ,τ ∼ = S ( τ ) and U ( u ) = 1 − a τ + (cid:112) (1 − aτ ) − τ c sin(2 τ u )4 m τ , with | c | < | /τ − a | ; (3) if κ > and τ = 0 , then N κ,τ ∼ = S ( κ ) × R and U ( u ) = κ ( a κ + 1) + ( c − κ ) sin ( √ κ u ) m κ , | c | < √ κ ; (4) if κ < and τ = 0 , then N κ,τ ∼ = H ( κ ) × R and U ( u ) = κ ( a κ + 1) + ( c − κ ) cosh ( √− κ u ) m κ ; (5) if κ = 0 and τ (cid:54) = 0 , then N κ,τ ∼ = H and (4.19) U ( u ) = (cid:16) τ u + 1 − aτ + c τ (cid:17) + 4 a τ − m τ , (6) if κ > and τ (cid:54) = 0 , then N κ,τ ∼ = SU(2) and U ( u ) = κ (4 aτ − a κ −
1) + (cid:104) τ − aκτ + (cid:112) (4 τ − κ ) − c κ sin( √ κ u ) (cid:105) m κ (4 τ − κ ) , with | c | < | τ − κ | / √ k ; (7) if κ < and τ (cid:54) = 0 , then N κ,τ ∼ = (cid:94) SL(2 , R ) and U ( u ) = κ (4 aτ − a κ −
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Università degli Studi di Cagliari, Dipartimento di Matematica e Informatica, Via Ospedale 72,09124 Cagliari, Italia
Email address : [email protected] Università degli Studi di Cagliari, Dipartimento di Matematica e Informatica, Via Ospedale 72,09124 Cagliari
Email address : [email protected] Università degli Studi di Cagliari, Dipartimento di Matematica e Informatica, Via Ospedale 72,09124 Cagliari
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