Minimal surfaces in the 3-dimensional Heisenberg group

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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)]).
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Minimal surfaces in the 3-dimensional Heisenberg group

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