Minimal surfaces in the 3-dimensional Heisenberg group

Minimal surfaces in the 3-dimensionalHeisenberg groupJun-ichi InoguchiAbstract. An integral representation formula in terms of the normalGauss map for minimal surfaces in the 3-dimensional Heisenberg groupwith canonical left invariant metric is obtained.M.S.C. 2000: 53A10, 53C15, 53C30.Key words: minimal surfaces, Heisenberg group, harmonic map.IntroductionIn our previous papers ([3], [5]), an integral representation formula for minimal surfacesin the model space Sol 3 of the solvegeometry (in the sense of Thurston [9]).In [4], an integral representation formula for minimal surfaces in the Heisenberggroup Nil 3 (the model space of nilgeometry in the sense of Thurston) is obtained.F. Mercuri, S. Montaldo and P. Piu [7] independently obtained such an integral representationformula for minimal surfaces in Nil 3 .D. A. Berdinskiĭ and I. A. Taĭmanov [1] obtained a representation formula forminimal surfaces in Nil 3 in terms of spinors and Dirac operators.In this paper, we study normal Gauss maps for minimal surfaces in Nil 3 and reformulatethe integral representation formula due to [4] and [7] in terms of the normalGauss map. Via the reformulation in terms of normal Gauss map, the geometricmeaning of the data for the integral representation formula is clarified. In fact, weshall show that every minimal surface in Nil 3 other than vertical plane is determinedby a harmonic map into the Poincáre disc.1 Nilpotent Lie groupsIn this paper we study minimal surfaces in the simply connected nilpotent Lie groupG(λ).We define a 1-parameter family {G(λ)} λ∈R of 3-dimensional nilpotent Lie groupbyG(λ) = (R 3 (x 1 , x 2 , x 3 ), ·)Differential Geometry - Dynamical Systems, Vol.10, 2008, pp. 163-169.c○ Balkan Society of Geometers, Geometry Balkan Press 2008.

164 Jun-ichi Inoguchiwith multiplication:(x 1 , x 2 , x 3 ) · (˜x 1 , ˜x 2 , ˜x 3 ) = (x 1 + ˜x 1 , x 2 + ˜x 2 , x 3 + ˜x 3 + λ 2 (x1˜x 2 − ˜x 1 x 2 ) ).The unit element of G(λ) is ⃗0 = (0, 0, 0). The inverse element of (x 1 , x 2 , x 3 ) is−(x 1 , x 2 , x 3 ). Obviously, G(0) is the abelian group (R 3 , +).The Lie algebra g(λ) of G(λ) is R 3 with commutation relations:(1) [E 1 , E 2 ] = λ E 3 , [E 2 , E 3 ] = [E 3 , E 1 ] = ⃗0with respect to the natural basis E 1 = (1, 0, 0), E 2 = (0, 1, 0), E 3 = (0, 0, 1). Theformulae (1) imply that g(λ) is nilpotent. The left translated vector fields of E 1 , E 2 ,E 3 aree 1 = ∂∂x − λy ∂2 ∂z , e 2 = ∂ ∂y + λx ∂2 ∂z , e 3 = ∂ ∂z ,respectively.We equip an inner product 〈·, ·〉 on g(λ) so that {E 1 , E 2 , E 3 } is orthonormal withrespect to it. Then the resulting left invariant Riemannian metric g = g λ on G(λ) is(2) g λ = (dx 1 ) 2 + (dx 2 ) 2 + ω ⊗ ω,where(3) ω = dx 3 + λ 2 (x2 dx 1 − x 1 dx 2 ).The one-form ω satisfies dω ∧ ω = −λdx 1 ∧ dx 2 ∧ dx 3 . Thus ω is a contact form onG(λ) if and only if λ ≠ 0.The homogeneous Riemannian 3-manifold (G(λ), g λ ) is called the 3-dimensionalHeisenberg group if λ ≠ 0. Note that (G(0), g 0 ) is the Euclidean 3-space. Thehomogeneous Riemannian 3-manifold (G(1), g 1 ) is frequently referred as the modelspace Nil 3 of the nilgeometry in the sense of Thurston [9].2 Matrix group model of G(λ)The Lie group G(λ) is realised as a closed subgroup of the general linear group GL 4 R.In fact, G(λ) is imbedded in GL 4 R by ι : G(λ) → GL 4 R;⎛⎞e x1 0 0 0ι(x 1 , x 2 , x 3 ) = ⎜ 0 1 λx 1 x 3 + λ 2⎝x1 x 2⎟0 0 1 x 2 ⎠ .0 0 0 1Clearly ι is an injective Lie group homomorphism. Thus G(λ) is identified with⎧⎛⎞⎫e ⎪⎨x1 0 0 0⎜ 0 1 λx 1 x 3 + λ 2⎝x1 x 2⎪⎬⎟0 0 1 x ⎪⎩2 ⎠ ∣ x1 , x 2 , x 3 ∈ R .⎪⎭0 0 0 1

Minimal surfaces in the 3-dimensional Heisenberg group 165The Lie algebra g(λ) corresponds to⎧⎛⎪⎨⎜⎝⎪⎩u 1 0 0 00 0 λu 1 u 30 0 1 u 20 0 0 0The orthonormal basis {E 1 , E 2 , E 3 } is identified withE 1 =⎛⎜⎝1 0 0 00 0 λ 00 0 0 00 0 0 0⎞⎟⎠ , E 2 =⎛⎜⎝⎞⎫⎪⎬⎟⎠ ∣ u1 , u 2 , u 3 ∈ R .⎪⎭1 0 0 00 0 0 00 0 0 10 0 0 0The Levi-Civita connection ∇ if g is given by⎞⎟⎠ , E 3 =⎛⎜⎝1 0 0 00 0 0 10 0 0 00 0 0 0⎞⎟⎠ .∇ E1 E 1 = 0, ∇ E1 E 2 = λ 2 E 3, ∇ E1 E 3 = − λ 2 E 2,(1) ∇ E2 E 1 = − λ 2 E 3, ∇ E2 E 2 = 0, ∇ E2 E 3 = λ 2 E 1,∇ E3 E 1 = − λ 2 E 2, ∇ E3 E 2 = λ 2 E 1, ∇ E3 E 3 = 0.The Riemannian curvature tensor R defined by R(X, Y ) = [∇ X , ∇ Y ]−∇ [X,Y ] is givenby(2) R 1 212 = − 3λ24 , R1 313 = R 2 323 = λ24 .The Ricci tensor field Ric is given by(3) R 11 = R 22 = − λ22 , R 33 = λ22 .The scalar curvature ρ of G is ρ = −λ 2 /2. The natural-reducibility obstruction Udefined by2g(U(X, Y ), Z) = g(X, [Z, Y ]) + g(Y, [Z, X]),X, Y, Z ∈ g(λ)is given by(4) U(E 1 , E 3 ) = − λ 2 E 3, U(E 2 , E 3 ) = λ 2 E 1.Note that U measures the non right-invariance of the metric. In fact U = 0 if andonly if g is right invariant (and hence biinvariant). The formulae (4) implies that g isbiinvariant if and only if λ = 0.The following formula was obtained in [4] (see also [7, (9)]).

166 Jun-ichi InoguchiProposition 2.1 ([4]-II) Let {ω 1 , ω 2 , ω 3 } be a solution to(5)(6)(7)¯∂ω 1 =λ2 (ω2 ∧ ω 3 − ω 2 ∧ ω 3 );¯∂ω 2 = − λ 2 (ω1 ∧ ω 3 − ω 1 ∧ ω 3 );¯∂ω 3 = − λ 2 (ω1 ∧ ω 2 + ω 1 ∧ ω 2 )on a simply connected coordinate region D ⊂ C. Then∫ z(ϕ(z, ¯z) = 2 Re ω 1 , ω 2 , ω 3 − λ )z 02 (x2 · ω 1 − x 1 · ω 2 )is a harmonic map into Nil 3 . Moreover if in addition, {ω 1 , ω 2 , ω 3 } satisfiesandω 1 ⊗ ω 1 + ω 2 ⊗ ω 2 + ω 3 ⊗ ω 3 = 0,ω 1 ⊗ ω 1 + ω 2 ⊗ ω 2 + ω 3 ⊗ ω 3 ≠ 0.Then ϕ is an minimal immersion.Conversely, any harmonic map of D into Nil 3 can be represented in this form.3 The normal Gauss mapLet ϕ : M → G(λ) be a conformal immersion. Take a unit normal vector field Nalong ϕ. Then, by the left translation we obtain the following smooth map:ψ := ϕ −1 · N : M → S 2 ⊂ g(λ).The resulting map ψ takes value in the unit sphere in the Lie algebra g(λ). Here,via the orthonormal basis {E 1 , E 2 , E 3 }, we identify g(λ) with Euclidean 3-spaceE 3 (u 1 , u 2 , u 3 ).The smooth map ψ is called the normal Gauss map of ϕ ([6], [3, p. 370]).Express the data as ω i = φ i dz. First we consider the case “φ 3 ≡ 0”. In this case,we have(x 1 z) 2 + (x 2 z) 2 = 0, |x 1 z| 2 + |x 2 z| 2 > 0,x 3 z + λ 2 (x1 zx 2 − x 1 zx 2 ) = 0.These equation imply that the minimal surface determined by the condition φ 3 ≡ 0is an integral surface of the distribution ω = 0. The distribution ω = 0 is integrableif and only if λ = 0. When λ = 0, then the surface is a vertical plane x 3 = constant.Hereafter we assume that φ 3 ≢ 0 and introduce the mapping f and g by(1) f := φ 1 − √ −1φ 2 , g :=φ 3φ 1 − √ −1φ 2 .By definition, f and g take values in the extended complex plane C = C∪{∞}. Usingthese functions f and g, we obtain the following formula.

Minimal surfaces in the 3-dimensional Heisenberg group 167Theorem 3.1 Let f and g be solutions to the system:√ √ −1λ−1λ(2) f¯z = − |f| 2 ḡ(1 − |g| 2 ), g¯z = − ¯f(1 − |g| 2 ) 224over a simply connected coordinate region D ⊂ C. Then the mapping(3) ϕ(z, ¯z) = ( ϕ 1 (z, ¯z), ϕ 2 (z, ¯z), ϕ 3 (z, ¯z) ) : D → G(λ),defined by∫ z( 1ϕ 1 (z, ¯z) = 2 Re))z 02 f(1 − g2 dz,∫ z(√ )−1(4) ϕ 2 (z, ¯z) = 2 Rez 02 f(1 + g2 ) dz,∫ z[ {ϕ 3 (z, ¯z) = 2 Re f g + λ (√−1ϕ 1 (1 + g 2 ) − ϕ 2 (1 − g 2 ) )}] dz.z 04is a weakly conformal harmonic map into G(λ).Conversely, every weakly conformal harmonic map ϕ : D → G(λ) (other thanvertical plane when λ = 0) is represented in this form.Proof. By the assumption φ 3 ≢ 0. Hence the harmonicity together with integrability(5)–(7) for ϕ areφ 1¯z + λ 2 (φ2 φ 3 + φ 2 φ 3 ) = 0,φ 2¯z − λ 2 (φ1 φ 3 + φ 1 φ 3 ) = 0,φ 3¯z − λ 2 (φ1 φ 2 − φ 1 φ 2 ) = 0.This system is equivalent to√ √ −1λ−1λf¯z = − |f| 2 ḡ(1 − |g| 2 ), g¯z = ¯f(1 − |g| 2 ) 2 .24Thus we obtain the required result. ✷Remark 3.1 In [7], Mercuri, Montaldo and Piu introduced the following auxiliaryfunctions:for Nil 3 = G(1). Then we haveG 2 = f 2 , H = g · Gφ 1 = G 2 − H 2 , φ 2 = √ −1(G 2 + H 2 ), φ 3 = 2GH.These functions are solutions to the system:2 √ −1G¯z = (|G| 2 − |H| 2 )H, 2 √ −1H¯z = (|G| 2 − |H| 2 )G.

168 Jun-ichi InoguchiThe integral representation formula is rewritten as ([7, Theorem 4.3])∫ zϕ 1 (z, ¯z) = 2 Re ( G 2 − H 2) dz,z 0ϕ 2 (z, ¯z) = 2ϕ 3 (z, ¯z) = 2∫ zRe {√ −1 ( G 2 + H 2)} dz,z 0[Re 2GH + 1 {ϕ 2 (G 2 − H 2 ) − √ −1ϕ 1 (G 2 + H 2 ) }] dz. □z 02∫ zThe normal Gauss map of (3-4) is computed as(5) ψ(z, ¯z) =11 + |g| 2 (2Re (g)E1 + 2Im (g)E 2 + (|g| 2 − 1)E 3).Under the stereographic projection π : S 2 \ {∞} ⊂ g(λ) → C := RE 1 + RE 2 ,the map ψ is identified with the C-valued function g. Based on this fundamentalobservation, we may call the function g the normal Gauss map of ϕ.When λ ≠ 0, from (2), we can deduce the following partial differential equationfor g:(6) (1 − |g| 2 )g z¯z + 2ḡg z g¯z = 0.If |g| = 1, then (6) implies that g is constant. In such a case, ψ has the formψ = cos θE 1 + sin θE 2 , g = cos θ + √ −1 sin θ.Thus the corresponding minimal surface is a vertical plane, i.e., a plane parallel tothe x 3 -axis ([4, II, Example 1.11], [8, p. 91]).From (5), one can see that the third component of ψ is positive if and only if|g| 2 < 1.The equation (6) means that if g satisfies |g| 2 < 1, then g is a harmonic map intothe unit disc D in C = RE 1 + RE 2 ;D = {w = uE 1 + √ −1vE 2 |u, v ∈ R}equipped with the Poincaré metric 4dwd ¯w/(1 − |w| 2 ).Corollary 3.1 Let D be a simply connected region in C and g : D → D be a harmonicmap into the unit disc. Let f be a function defined by f = −4 √ −1λḡ z /(1 − |g| 2 ) 2 .Then the mapping ϕ defined by (3)-(4) is a weakly conformal harmonic map into theHeisenberg group G(λ) (λ ≠ 0).Acknowledgement The author was partially supported by Kakenhi 16740029,18540068.

Minimal surfaces in the 3-dimensional Heisenberg group 169References[1] D. A. Berdinskiĭ and I. A. Taĭmanov, Surfaces in three-dimensional Lie groups(Russian), Sibirsk. Mat. Zh. 46 (2005), no. 6, 1248–1264. English Translation:Siberian Math. J. 46 (2005), no. 6, 1005–1019.[2] J. Inoguchi, Minimal surfaces in 3-dimensional solvable Lie groups, Chinese Ann.Math. B. 24 (2003), 73–84.[3] J. Inoguchi, Minimal surfaces in 3-dimensional solvable Lie groups II, Bull. Austral.Math. Soc. 73 (2006), 365–374.[4] J. Inoguchi, T. Kumamoto, N. Ohsugi and Y. Suyama, Differential geometry ofcurves and surfaces in 3-dimensional homogeneous spaces I–IV, Fukuoka Univ.Sci. Rep. 29 (1999), 155–182, 30 (2000), 17–47, 131–160, 161–168.[5] J. Inoguchi and S. Lee, A Weierstrass repesenttaion for minimal surfaces in Sol,Proc. Amer. Math. Soc., 136 (2008), 2209–2216.[6] M. Kokubu, Weierstrass representation for minimal surfaces in hyperbolic space,Tôhoku Math. J. 49 (1997), 367–377.[7] F. Mercuri, S. Montaldo and P. Piu, A Weierstrass representation formula forminimal surfaces in H 3 and H 2 × R, Acta Math. Sin. Engl. Ser. 22 (2006), no. 6,1603–1612[8] A. Sanini, Gauss map of a surface of the Heisenberg group, Boll. Un. Mat. Ital.B (7), 11-B suppl. facs. 2 (1997), 79–93.[9] W. M. Thurston, Three-dimensional Geometry and Topology I, Princeton Math.Series. 35 (Princeton University Press, Princeton N. J., 1997).Author’s address:Jun-ichi InoguchiDepartment of Mathematics Education,Faculty of Education, Utsunomiya University,Utsunomiya, 321-8505, Japan.E-mail: inoguchi@cc.ustunomiya-u.ac.jp