International Journal of Analysis and Applications Volume 18, Number 5 (2020), 838-848 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-838 SURFACES AS GRAPHS OF FINITE TYPE IN H2 ×R AHMED AZZI1,∗, ZOUBIR HANIFI2, MOHAMMED BEKKAR1 1Department of Mathematics, Faculty of Sciences, University of Oran 1, Ahmed Benbella Algeria 2Ecole Nationale Polytechnique d’Oran, Département de Mathématiques, et Informatique B.P 1523 El M’Naour, Oran, Algérie ∗Corresponding author: azzi.mat@hotmail.fr Abstract. In this paper, we prove that ∆X = 2H where ∆ is the Laplacian operator, r = (x,y,z) the position vector field and H is the mean curvature vector field of a surface S in H2 ×R and we study surfaces as graphs in H2 × R which has finite type immersion. 1. Introduction The H2 ×R geometry is one of eight homogeneous Thurston 3-geometries E3, S3, H3, S2 ×R, H2 ×R, ˜SL(2,R), Nil, Sol. The Riemannian manifold (M,g) is called homogeneous if for any x, y ∈ M there exists an isometry φ : M → M such that y = φ(x). The two and three-dimensional homogeneous geometries are discussed in detail in [6] . A Euclidean submanifold is said to be of finite Chen-type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian [3]. B. Y. Chen posed the problem of classifying the finite type surfaces in the Received May 24th, 2020; accepted July 1st, 2020; published July 27th, 2020. 2010 Mathematics Subject Classification. 53B05, 53B21, 53C30. Key words and phrases. Laplacian operator; H2 × R geometry; surfaces of coordinate finite type; minimal surfaces. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 838 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-838 Int. J. Anal. Appl. 18 (5) (2020) 839 3-dimensional Euclidean space E3. Further, the notion of finite type can be extended to any smooth function on a submanifold of a Euclidean space or a pseudo-Euclidean space. Let S be a 2-dimensional surface of the Euclidean 3-space E3. If we denote by r, H and ∆ the position vector field, the mean curvature vector field and the Laplace operator of S respectively, then it is well-known that [3] (1.1) ∆r = −2H. A well-known result due to Takahashi states that minimal surfaces and spheres are the only surfaces in E3 satisfying the condition ∆r = λr for a real constant λ. From (1.1), we know that minimal surfaces and spheres also verify the condition (1.2) ∆H = λH, λ ∈ R. Equation (1.1) shows that S is a minimal surface of E3 if and only if its coordinate functions are harmonic. In [9], D. W. Yoon studied surfaces invariant under the 1-parameter subgroup in Sol3. In 2012, M. Bekkar and B. Senoussi [1] studied the translation surfaces in the 3-dimensional Euclidean and Lorentz-Minkowski space under the condition ∆IIIri = µiri, µi ∈ R, where ∆III denotes the Laplacian of the surface with respect to the third fundamental form III. A surface S in the Euclidean 3-space E3 is called minimal when locally each point on the surface has a neighborhood which is the surface of least area with respect to its boundary [5]. In 1775, J. B. Meusnier showed that the condition of minimality of a surface in E3 is equivalent with the vanishing of its mean curvature function, H = 0. Let z = f(x,y) define a graph S in the Euclidean 3-space E3. If S is minimal, the function f satisfies (1 + (fy) 2)fxx − 2fxyfxfy + (1 + (fx)2)fyy = 0, which was obtained by J. L. Lagrange in 1760. In 1835, H. F. Scherk studied translation surfaces in E3 and proved that, besides the planes, the only minimal translation surfaces are given by z(x, y) = 1 λ log |cos(λx)|− 1 λ log |cos(λy)| , where λ is a non-zero constant. In 1991, F. Dillen, L. Verstraelen and G. Zafindratafa. [4] generalized this result to higher-dimensional Euclidean space. Int. J. Anal. Appl. 18 (5) (2020) 840 In 2015, D. W. Yoon [8] studied translation surfaces in the product space H2×R and classified translation surfaces with zero Gaussian curvature in H2 ×R. In 2019, B. Senoussi, M. Bekkar [7] studied translation surfaces of finite type in H3 and Sol3 and the authors gived some theorems. A surface S(γ1, γ2) in H2 ×R is a surface parametrized by S : Ω ⊆ R2 → H2 ×R, X(s, t) = γ1(s) ∗γ2(t) or X(s, t) = γ2(t) ∗γ1(s), where γ1 and γ2 are any generating curves in R3. Since the multiplication ∗ is not commutative. In this work we study the surfaces as graphs of functions ϕ = f(s, t)) in H2 ×R satisfy the condition (1.3) ∆xi = λixi, λi ∈ R. 2. Preliminaries Let H2 be represented by the upper half-plane model {(x, y) ∈ R | y > 0} equipped with the metric gH = (dx2 + dy2) y2 . The space H2, with the group structure derived by the composition of proper affine maps, is a Lie group and the metric gH is left invariant. Therefore, the product space H2 ×R is a Lie group with the left invariant product metric g = (dx2 + dy2) y2 + dz2, we can define the multiplication law on H2 ×R as follows (x,y,z) ∗ (x̄, ȳ, z̄) = (yx̄ + x,yȳ,z + z̄). The left identity is (0, 1, 0) and the inverse of (x,y,z) is (− x y , 1 y ,−z), on H2 ×R a left-invariant metric ds2 = (ω1)2 + (ω2)2 + (ω3)2, where ω1 = dx y , ω2 = dy y , ω3 = dz, is the orthonormal coframe associated with the orthonormal frame e1 = y ∂ ∂x , e2 = y ∂ ∂y , e3 = ∂ ∂z , The corresponding Lie brackets are [e1, e2] = −e1, [ei, ei] = [e3, e1] = [e2, e3] = 0,∀i = 1, 2, 3. Int. J. Anal. Appl. 18 (5) (2020) 841 The Levi-Civita connection ∇ of H2 ×R is given by  ∇e1e1 ∇e1e2 ∇e1e3   =   0 1 0 −1 0 0 0 0 0     e1 e2 e3   , ∇e2ei = ∇e3ei = 0, ∀i = 1, 2, 3. Let S be an immersed surface in H2×R given as the graph of the function z = f(x,y). Hence, the position vector is described by r(x,y) = (x,y,f(x,y)) and the tangent vectors rx = ∂r ∂x and ry = ∂r ∂y in terms of the orthonormal frame (e1,e2,e3) are described by rx = ∂ ∂x + fr ∂ ∂z = 1 y e1 + fxe3,(2.1) ry = ∂ ∂y + fy ∂ ∂z = 1 y e2 + fye3.(2.2) Definition 2.1. [3] The immersion (S,r) is said to be of finite Chen-type k if the position vector X admits the following spectral decomposition r = r0 + k∑ i=1 ri, where ri are E3-valued eigenfunctions of the Laplacian of (S,r) : ∆ri = λiri, λi ∈ R, i = 1, 2, ..,k. If λi are different, then S is said to be of k-type. For the matrix G = (gij) consisting of the components of the induced metric on S, we denote by G−1 = (gij) (resp. D = det(gij)) the inverse matrix (resp. the determinant) of the matrix (gij). The Laplacian ∆ on S is, in turn, given by (2.3) ∆ = −1√ |D| ∑ ij ∂ ∂ri (√ |D|gij ∂ ∂rj ) . If r = r(x,y) = (r1 = r1(x,y),r2 = r2(x,y),r3 = r3(x,y)) is a function of class C 2 then we set ∆r = (∆r1, ∆r2, ∆r3). 3. Surfaces as graphs of finite type in H2 ×R Let S be a graph of a smooth function f : Ω ⊂ R2 → R. We consider the following parametrization of S r(x,y) = (x,y, f(x, y)), (x,y) ∈ Ω. Int. J. Anal. Appl. 18 (5) (2020) 842 Theorem 3.1. A Beltrami formula in H2 ×R is given by the following: ∆r = 2H, where ∆ is the Laplacian of the surface and H is the mean curvature vector field of S. Proof. A basis of the tangent space TpS associated to this parametrization is given by rx = ∂ ∂x + fx ∂ ∂z = 1 y e1 + fxe3, ry = ∂ ∂y + fy ∂ ∂z = 1 y e2 + fye3, The coefficients of the first fundamental form of S are given by E = g(rx, rx) = 1 y2 + f2x, F = g(rx, ry) = fxfy, G = g(ry, ry) = 1 y2 + f2y . The unit normal vector field N on S is given by N = 1 W ( − 1 y fxe1 − 1 y fye2 + 1 y2 e3 ) , where W = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . To compute the second fundamental form of S, we have to calculate the following rxx = ∇rxrx = 1 y2 e2 + fxxe3, rxy = ∇rxry = ∇ryrx = − 1 y2 e1 + fxye3,(3.1) ryy = ∇ryry = − 1 y2 e2 + fyye3. So, the coefficients of the second fundamental form of S are given by L = g(∇rxrx, N) = 1 Wy2 ( fxx − 1 y fy ) , M = g(∇rxry, N) = 1 Wy2 ( fxy + 1 y fx ) , N = g(∇ryry, N) = 1 Wy2 ( fyy + 1 y fy ) , where W = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . Thus, the mean curvature H of S is given by H = EN − 2FM + GL 2W2 . H = 1 2W3y2 [ 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ] . Int. J. Anal. Appl. 18 (5) (2020) 843 By (2.3), the Laplacian operator ∆ of r can be expressed as (3.2) ∆ = − 1 W4 [ W2 ( G ∂2 ∂x2 − 2F ∂2 ∂x∂y + E ∂2 ∂y2 ) + ∆1 ∂ ∂x + ∆2 ∂ ∂y ] , where ∆1 = 2 y2 fyf 2 xfxy − 1 y4 fxfxx − 1 y2 fxf 2 y fxx − 1 y4 fxfyy − 1 y2 f3xfyy − 2 y5 fxfy − 1 y3 f3xfy − 1 y3 fxf 3 y , and ∆2 = 2 y2 fxf 2 y fxy − 1 y4 fyfyy − 1 y2 f2xfyfyy − 1 y4 fyfxx − 1 y2 f3y fxx − 1 y5 f2y + 1 y5 f2x + 1 y3 f4x + 1 y3 f2xf 2 y . By a straightforward computation, the Laplacian operator ∆ of r with the help of (3.1) and (3.2) turns out to be ∆r = − 1 W4   ( 2 y3 f2xfyfxy − 1 y5 fxfxx − 1 y3 fxf 2 y fxx − 1 y5 fxfyy − 1 y3 f3xfyy + 1 y4 f3xfy + 1 y4 fxf 3 y ) e1 + ( 2 y3 fxf 2 y fxy − 1 y5 fyfyy − 1 y3 f2xfyfyy − 1 y5 fyfxx − 1 y3 f3y fxx + 1 y4 f2xf 2 y + 1 y4 f4y ) e2 + ( − 2 y4 fxfyfxy − 1 y5 f2xfy − 1 y5 f3y + 1 y6 fxx + 1 y4 f2y fxx + 1 y6 fyy + 1 y4 f2xfyy ) e3   , ∆r =   ( −fx Wy ) 1 W3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e1 + ( −fy Wy ) 1 W3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e2 + ( 1 Wy2 ) 1 W3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e3   , ∆r = 1 W3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy )   ( −fx Wy ) e1 + ( −fy Wy ) e2 + ( 1 Wy2 ) e3   , thus we get ∆r = 2HN,(3.3) = 2H, Int. J. Anal. Appl. 18 (5) (2020) 844 where H is the mean curvature vector field of S. S is a minimal surfaces in H2 ×R if and only if its coordinate functions are harmonic . � 4. Surfaces as graphs in H2 ×R satisfying 4xi = λixi Let S be an immersed surface in H2 × R given as the graph of function z = f(x,y). Hence, the vector position is described by r(x,y) = (x,y,f(x,y)). We have rx = 1 y e1 + fxe3, ry = 1 y e2 + fye3, where rx = ∂r ∂x , ry = ∂r ∂x , and fx = ∂f ∂x , fy = ∂f ∂y . From an earlier results the mean curvature H of S and the unit normal vector field N on S are given by H = 1 2W3y2 [ 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ] , and (4.1) N = 1 W ( − 1 y fxe1 − 1 y fye2 + 1 y2 e3 ) , where W = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . If the vector position on the tangent space TpS is described by r = (x,y,f(x,y)) r(x,y) = x ∂ ∂x + y ∂ ∂y + f(x,y) ∂ ∂z , then (4.2) r(x,y) = x y e1 + e2 + f(x,y)e3. The equation (1.3) by means of (3.3), (4.1) and (4.2) gives rise to the following system of ordinary differential equations ( 2H W ) fx = −λ1x,(4.3) ( 2H W ) fy = −λ2y,(4.4) 2H W = λ3y 2f.(4.5) Int. J. Anal. Appl. 18 (5) (2020) 845 Therefore, the problem of classifying the surfaces S of (1.3) is reduced to the integration of this system of ordinary differential equations. Next we study it according to the constants λ1, λ2 and λ3. Case 1. Let λ3 = 0. In this case the system (4.3), (4.4) and (4.5) is reduced equivalently to( 2H W ) fx = −λ1x,(4.6) ( 2H W ) fy = −λ2y,(4.7) 2H W = 0.(4.8) The equation (4.8) implies that the mean curvature H is identically zero. Thus, the surface S is minimal; and we get also λ1 = λ2 = 0. Case 2. Let λ3 6= 0. in this case we study the general system (4.3), (4.4) and (4.5). 2-i): If λ1 = λ2 = 0, then H = 0. From (4.5) we obtain λ3 = 0, so we get a contradiction. 2-ii): If λ1 = 0 and λ2 6= 0., from (4.3) we obtain Hfx = 0. 2-ii-a: If H = 0 (4.4), (4.5) implies that λ2 = λ3 = 0. So we get a contradiction. 2-ii-b: if fx = 0, then f(x,y) = ϕ(y), where ϕ is smooth function of y. The mean curvature H turns to (4.9) H = 1 2Wy3 ( 1 y ϕ′′ −ϕ′3 ) , where ϕ′ = dϕ dy . Using (4.4) and (4.5) we obtain ϕ′ = −λ2 λ3yϕ , which leads to, λ3ϕ ′ϕ = −λ2 y . After integrating with respect to y, we obtain Int. J. Anal. Appl. 18 (5) (2020) 846 λ3 2 ϕ2(y) = −λ2 ln y + φ(x); y > 0, where φ is smooth function of x, and hence f(x,y) = ϕ(y) = ± √ λ2 λ3 ln 1 y2 + φ(x). Using the condition fx = 0 we get φ(x) = a, a ∈ R. Thus, f(x,y) = ϕ(y) = ± √ λ2 λ3 ln 1 y2 + c; c = 2 λ3 a, in this subcase, the surfaces S are given by r(x,y) = ( x,y,± √ λ2 λ3 ln 1 y2 + c ) ; λ2 6= 0, λ3 6= 0, c ∈ R. 2-iii): If λ1 6= 0 and λ2 = 0., from (4.4) we obtain Hfy = 0. 2-iii-a: If H = 0, (4.3) and (4.5) implies that λ2 = λ3 = 0. So we get a contradiction. 2-iii-b: If fy = 0,then f(x,y) = ψ(x), where ψ is smooth function of x. The mean curvature H turns to (4.10) H = 1 2Wy4 ψ′′, where ψ′ = dψ dx . Using (4.3) and (4.5) we get ψ′ = −λ1x λ3y2ψ , so we can write (4.11) λ3y 2 + λ1 x ψψ′ = 0, A differentiation with respect to y gives λ3y = 0, this implies that λ3 = 0 and from (4.8) we get the mean curvature H is identically zero. From (4.6) and (4.7) we obtain λ1 = λ2 = 0, which leads to a contradiction. Int. J. Anal. Appl. 18 (5) (2020) 847 2-iv): If λ1 6= 0 and λ2 6= 0 From (4.3), we have (4.12) 2H W = − λ1x ψ′ . Substituting (4.12) into (4.5), we get − λ1x ψ′ = λ3y 2ψ, A differentiation with respect to x gives −λ1 ( ψ −xψ′′ ψ′2 ) = λ3y 2ψ′, this equation gives (4.13) λ1 ( ψ′ −xψ′′ ψ′3 ) + λ3y 2 = 0. A differentiation with respect to y gives λ3y = 0, this implies that λ3 = 0 and from (4.8) we get the mean curvature H is identically zero. From (4.6) and (4.7) we obtain λ1 = λ2 = 0, which leads to a contradiction. Therefore, we have the following theorem, Theorem 4.1. Let S be a surface as graph of function parametrized by r(x,y) = (x,y,f(x,y)) in H2 × R Then, S satisfies the equation ∆ri = λiri, λi ∈ R if and only if S is minimal surfaces or parametrized as S : r(x,y) = ( x,y,± √ λ2 λ3 ln 1 y2 + c ) ; λ2 6= 0, λ3 6= 0, c ∈ R. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] M. Bekkar, B. Senoussi, Translation surfaces in the 3-dimensional space satisfying ∆IIIri = µiri, J. Geom. 103 (2012), 367-374. [2] M. Bekkar, H, Zoubir, Surfaces of revolution in the 3-dimensional Lorentz Minkowski space satisfying ∆jri = µiri, Int. J. Contemp. Math. Sci. 3 (2008), 1173-1185. [3] B-Y. Chen, Total mean curvature and submanifolds of finite type, (2nd edition), World Scientific Publisher, Singapore, 1984. Int. J. Anal. Appl. 18 (5) (2020) 848 [4] F. Dillen, L. Verstraelen, G. 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