Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 67-76.

Parallel and Minimal Surfaces in Heisenberg Space^{1}

Mohamed Belkhelfa^{2}

Department of Mathematics, K U Leuven Celistijnlaan 200B, Heverlee, Belgium e-mail: mohamed.belkhelfa@wis.kuleuven.ac.be

Abstract. After a brief historical review on minimal surfaces in Euclidean spaces 𝔼^{3} and in Heisenberg spaces H_{3}, ℝ^{3}(−3), the Beltrami formula and parallel surfaces are investigated in these Heisenberg spaces respectively. Keywords: Heisenberg spaces, minimal surfaces, parallel surfaces

1. Minimal surface in 𝔼^{3}

The generalization from the straight lines, as 1-dimensional objects, being the lines of shortest length in the Euclidean plane 𝔼^{2}, to 2-dimensional objects, i.e. surfaces, in the 3-dimensional Euclidean space 𝔼^{3}, yields the notion of the so-called “minimal” surfaces. More precisely, a surface M in 𝔼^{3} 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 [12], i.e. when M satisfies the problem of J. A. F. Plateau [25]. The study of the minimal graphs as surfaces in 𝔼^{3} was historically one of the first applications of the variational problem for double integrals [25]. As a result, J. L. Lagrange in 1760 obtained the following non-linear elliptic partial differential equation: A surface as graph of a function z = f(x, y) is minimal if and only if

f_{xx}[1 + (f_{y})^{2}] − 2f_{xy}f_{x}f_{y} + f_{yy}[1 + (f_{x})^{2}] = 0,

where f_{x} = \frac{∂f}{∂x}, f_{y} = \frac{∂f}{∂y} and f_{xx} = \frac{∂^{2}f}{∂x^{2}} , f_{yy} = \frac{∂^{2}f}{∂y^{2}} , f_{xy} = \frac{∂^{2}f}{∂x∂y}.

1This is an original research article and no version has been submitted for publication elsewhere.

2The author thanks the organizing committee of the Summer school for the financial support, the PADGE members of KU Leuven for multiple help and support and Verstraelen family for the generous hospitality.

J. B. Meusnier in 1775 showed that the condition of minimality of a surface in 𝔼^{3} is equivalent with the vanishing of its mean curvature function, H = 0.

Examples. The first non-trivial examples of minimal surfaces in 𝔼^{3}, – i.e. non-planar minimal surfaces – were found by L. Euler in 1744, considering the class of surfaces of revolution. As a result, he found that the only (non-trivial) minimal surfaces of revolution are the catenoids. Euler called the catenoid as alysseid. Around 1775 Meusnier found a second class of non-trivial minimal surfaces, namely the helicoids, and E. Catalan in 1842 proved that the helicoids are the only non-trivial minimal ruled surfaces. H. F. Scherk in 1835 found other examples, the so-called minimal translation surfaces of Scherk, as graph of functions z(x, y) = g(x) + h(y) with separated variables.

2. Minimal surfaces in H_{3}

In non-Euclidean spaces, in particular in Heisenberg spaces the problem of minimal surfaces was recently studied in [2], [3], [10], [27], [15], for surfaces of revolution with constant mean curvature or Gaussian curvature are studied by R. Caddeo, P. Piu and A. Ratto [7], [8] and P. Tomter [24].

First I would like to mention different metrics that will be used in this paper. We shall denote ℝ^{3} endowed with the following metrics:

1. ds_{1}^{2} = dx^{2} + dy^{2} + (dz + \frac{y}{2}dx − \frac{x}{2}dy)^{2},

2. ds_{2}^{2} = dx^{2} + dy^{2} + (dz + xdy)^{2},

3. ds_{3}^{2} = \frac{1}{4}(dx^{2} + dy^{2}) + (dz − ydx)^{2} by H_{3}, ℝ^{3}(−\frac{3}{4}), and ℝ^{3}(−3), respectively. On the 3-dimensional Riemannian manifold H_{3} is called the Heisenberg 3-space. We can define the Lie group structure on ℝ^{3} by a multiplication law as follows:

(x̄, ȳ, z̄) ∗ (x, y, z) = (x̄ + x, ȳ + y, z̄ + z + \frac{yx̄}{2} − \frac{xȳ}{2}) .

Then the metric ds_{1}^{2} is left invariant metric on H_{3}. We can take the following left invariant orthonormal frame:

e_{1} = \frac{∂}{∂x} − \frac{y}{2} · \frac{∂}{∂z} , e_{2} = \frac{∂}{∂y} + \frac{x}{2} · \frac{∂}{∂z} , e_{3} = \frac{∂}{∂z} .

This orthonormal frame satisfies

[e_{1}, e_{2}] = e_{3}, [e_{1}, e_{3}] = 0, [e_{2}, e_{3}] = 0.

The sectional curvatures are given by:

K(e_{1}, e_{2}) = \frac{−3}{4}, K(e_{1}, e_{3}) = \frac{1}{4}, K(e_{2}, e_{3}) = \frac{1}{4}.

The identity component I_{0}(H_{3}, ds_{1}^{2}) of the isometry group is given by the following ([2], [14], [11]):

( \begin{matrix}x \\ y \\ z \end{matrix} ) ⟶ (\begin{matrix}cos θ & − sin θ & 0 \\ sin θ & cos θ & 0 \\ A & B & 1 \end{matrix}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) + ( \begin{matrix}a \\ b \\ c \end{matrix} ) ,  (2.1)

where θ, a, b, and c are real numbers and

A = \frac{1}{2}(a sin θ − b cos θ), B = \frac{1}{2}(a cos θ + b sin θ).

The identity component I_{0}(H_{3}, ds_{1}^{2}) of the isometry group contains the rotations (θ, 0, 0, 0) of ℝ^{3} around the z-axis and the left translations (0,a,b,c). The Lie group I_{0}(H_{3}, ds_{1}^{2}) is the semi-direct product of SO(2) and H_{3}, SO(2)αH_{3}. In [14], the authors define an imbedding of H_{3} into the real general linear group GL(3; ℝ); i : H_{3} → GL(3; ℝ);

i(x, y, z) = (\begin{matrix}1 & x & t \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) ,

where t is defined by t(x, y, z) = z + \frac{1}{2}xy. With respect to this imbedding, the left invariant metric ds_{1}^{2} is expressed as dx^{2} + dy^{2} + (dz − \frac{1}{2}xdy)^{2}. It easy to see that i is an injective Lie group homomorphism. Hence the Heisenberg group H_{3} is identified with the following closed Lie subgroup N of GL(3; ℝ) via i:

N = \{(\begin{matrix}1 & x & t \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) | x, y, t ∈ ℝ\} .

The Lie algebra n is naturally identified with the tangent space of N at e = (0, 0, 0);

T_{e}N = \{(\begin{matrix}0 & x & u \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix}) | x, y, u ∈ ℝ\} .

The exponential mapping exp : T_{e}N → N is given explicitly

exp (\begin{matrix}0 & u & w \\ 0 & 0 & v \\ 0 & 0 & 0 \end{matrix}) = (\begin{matrix}1 & u & w + \frac{1}{2}uv \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix}) .

In particular the exponential mapping is a diffeomorphism onto N. See also [11]. The left invariant orthonormal frame e_{1}, e_{2}, e_{3} corresponds to the basis A_{1}, A_{2}, A_{3} of T_{e}N given by

A_{1} = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , A_{2} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , A_{3} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

In 1991, Bekkar [2] found the equation of minimal surfaces as graphs of functions z = f(x, y):

f_{xx}[1 + (f_{y} − \frac{x}{2})^{2}] − 2f_{xy}(f_{y} − \frac{x}{2})(f_{x} + \frac{y}{2}) + f_{yy}[1 + (f_{x} + \frac{y}{2})^{2}] = 0.

In 1992, M. Bekkar and T. Sari [3] classified the ruled surfaces in H_{3}.

Theorem 2.1. The minimal ruled surfaces of H_{3} ruled by straight lines are (up to isometry of H_{3}) parts of:

1. planes;

2. hyperbolic paraboloids z = \frac{xy}{2};

3. helicoids;

4. surfaces defined by the equation:

z = \frac{λ}{2} [x\sqrt{(1 + x^{2})} + log (x + \sqrt{(1 + x^{2})})] − \frac{xy}{2} , λ ∈ ℝ \ \{0\}

5. surfaces with parametrisation:

(x(t, s), y(t, s), z(t, s)) = (t + su(t), s, a(t) − \frac{st}{2})

where u and a are solutions of the system:

(1 + u^{2} + t^{2})ü − (1 + 2u̇ȧ)tu̇ = 0,
(1 + u^{2} + t^{2})ä − (1 + 2u̇ȧ)(tȧ − u) = 0.

Remark 2.2. [3] The parts of helicoids are ruled only by geodesic straight lines, but parts of planes and parts of hyperbolic paraboloids are possibly ruled by straight line geodesics and by straight lines which are not necessary geodesics.

Example 2.3. The surface as graph of the function

z = \frac{1}{2} [x\sqrt{(1 + x^{2})} + log (x + \sqrt{(1 + x^{2})})] − \frac{xy}{2}

is a minimal surface in H_{3}. This surface is ruled by straight lines L_{t} spanned by the vectors (0, 1, −t) passing through the points

(t, 0, \frac{1}{2} [x\sqrt{(1 + x^{2})} + log (x + \sqrt{(1 + x^{2})})]),

which are not geodesics.

Remark 2.4. [15, 13] Let G be a Lie group with left invariant metric and X, Y be two vectors in the Lie algebra g of G such that |X| = |Y | and X ⊥ Y. If the metric of G is biinvariant then the mapping r : ℝ^{2} → G defined by

r(u, v) = exp(uX) exp(vY)

is a flat minimal surface in G. If we choose G = 𝔼^{3} and X = (1, 0, 0) and Y = (0, 1, 0), then this surface is the xy-plane in 𝔼^{3}. If the metric of G is only left invariant the mapping r is not necessarily immersion or harmonic map. In the case G = H_{3}, if we choose X = A_{1} and e_{2} = A_{2} or e_{1} = A_{2} and e_{2} = A_{1}, then

r_{+}(x, y) = exp(xA_{1}) ∗ exp(yA_{2}),

and

r_{−}(x, y) = exp(xA_{2}) ∗ exp(yA_{1}),

are (nonflat) minimal surfaces in H_{3}. The minimal surfaces r_{±} are explicitly given by r_{+}(x, y) = +\frac{1}{2}xy and r_{−}(x, y) = −\frac{1}{2}xy. These surfaces r_{+} and r_{−} may be considered as ∗-analogues in H_{3} of xy-plane in 𝔼^{3}.

Example 2.5. (Translation surfaces [15]) It is well-known that only the minimal translation surfaces in E^{3} are planes or Scherk’s minimal surfaces. In H_{3}, we can consider ∗-translation minimal surfaces of nonparametric form:

r(x, y) = (x, 0, u(x)) ∗ (0, y, v(y)) = (x, y, u(x) + v(y) + \frac{xy}{2} ).

The minimal surface equation for the function f(x, y) = u(x) + v(y) + \frac{xy}{2} is

u_{xx}(1 + v_{y}^{2}) − (u_{x} + y)v_{y} + v_{yy}{1 + (u_{x} + y)^{2}} = 0.  (2.2)

We shall solve this differential equation under the assumption that either u or v is constant.

(1) v(y) =constant: In this case (2.2) becomes u_{xx} = 0. Hence we get

z = \frac{xy}{2} + ax + b, a, b ∈ ℝ  (2.3)

(2) u(x) =constant: In this case (2.2) becomes

\frac{d^{2}v}{dy^{2}} (1 + y^{2}) − \frac{dv}{dy} y = 0.  (2.4)

Solving this equation we get

z = \frac{xy}{2} + c \{y\sqrt{1 + y^{2}} + log (y + \sqrt{1 + y^{2}})\} + d, c, d ∈ ℝ.  (2.5)

The minimal surfaces defined by (2.3) and (2.5) are rewritten in the following forms:

r(x, y) = (x, 0, u(x)) ∗ (0, y, 0) = (0, y, 0) ∗ (x, 0, u(x)), u(x) = ax + b.  (2.6)

\begin{aligned}r(x, y) & = (x, 0, 0) ∗ (0, y, v(y)) = (0, y, v(y)) ∗ (x, 0, 0)v(y) \\ & = c \{y\sqrt{1 + y^{2}} + log (y + \sqrt{1 + y^{2}})\} + d, c, d ∈ ℝ \end{aligned}  (2.7)

These formulae imply that the minimal surface (2.6) (resp.(2.7)) is a cylinder over a curve in the xz- (resp. yz-) plane.

Proposition 2.6. [15]

(1) Let r(x, y) be a cylinder over a curve (x, 0, u(x)) in the xz-plane. Then r is minimal if and only if

u(x) = ax + b, a, b ∈ ℝ

(2) Let r(x, y) be a cylinder over a curve (0, y, v(y)) in the yz-plane. Then r is minimal if and only if

v(y) = c \{y\sqrt{1 + y^{2}} + log (y + \sqrt{1 + y^{2}})\} + d, c, d ∈ ℝ

(3) Let (x(u), y(u)) be a curve in the xy-plane, parameterized by the arclength u. Then the cylinder over (x(u), y(u)) is

r(u, v) = (x(u), y(u), v).

Its first and second fundamental forms are

I = (1 + ω(r_{u})^{2}) du^{2} + 2ω(r_{u})dudv + dv^{2}
II = {ẍ(u)ẏ(u) − ẋ(u)ÿ(u) + ω(r_{u})}du^{2} + dudv,
n = ẏ(u)e_{1} − ẋ(u)e_{2}.

The mean curvature H of the cylinder is

H = \frac{1}{2} \{ẍ(u)ẏ(u) − ẋ(u)ÿ(u)\} = −\frac{κ(u)}{2}.

Here κ is the curvature of the curve (x(u), y(u)).

One can easily check that the base curve of r is a line in the xy-plane. Hence r is a plane parallel to the z-axis [20]. (More generally constant mean curvature cylinder is a circular cylinder. Kokubu [17] obtained corresponding results for SL(2, ℝ). Sanini characterised these cylinders in terms of the harmonicity of their tangential Gauss maps. See Propsition 3 in [20].

Remark 2.7. [10] For a surface with parametrization r(u, v) = (x(u, v), y(u, v), z(u, v)) is minimal if and only if satisfy the equation

(r_{1}G − 2s_{1}F + t_{1}E)δ_{23} + (r_{2}G − 2s_{2}F + t_{2}E)δ_{31} + (r_{3}G − 2s_{3}F + t_{3}E)δ_{12} = 0;

where

δ_{12} = (p_{1}q_{2} − p_{2}q_{1}), δ_{23} = p_{2}q_{3} − p_{3}q_{2}, δ_{31} = p_{3}q_{1} − p_{1}q_{3},

x_{u} = \frac{∂x}{∂u} = p_{1}, x_{v} = \frac{∂x}{∂v} = q_{1}, x_{uu} = \frac{∂^{2}x}{∂u^{2} = r_{1}} , x_{vv} = \frac{∂^{2}x}{∂v^{2} = t_{1}} , x_{uv} = \frac{∂^{2}x}{∂u∂v} = s_{1};

for p_{2}, q_{2}, r_{2}, s_{2}, t_{2} and p_{3}, q_{3}, r_{3}, s_{3}, t_{3}, we change x by y and z respectively.

Theorem 2.8. [10] Let S be a surface in Heisenberg space with paramerization r(u, v) = (x(u, v), y(u, v), z(u, v)) such that

x(u, v) = α_{1}u^{2} + β_{1}v^{2} + 2λ_{1}uv,
y(u, v) = α_{2}u^{2} + β_{2}v^{2} + 2λ_{2}uv,
z(u, v) = α_{3}u^{2} + β_{3}v^{2} + 2λ_{3}uv.

If S satisfies (δ_{12})^{2} + (δ_{23})^{2} + (δ_{31})^{2} > 0 and one of the following conditions:

a) (δ_{12})(δ_{23})(δ_{31}) = 0,

b) two colons of the determinant δ = det(α, β, λ) are proportional, where

α_{i}, β_{i}, λ_{i} ∈ ℝ, α^{t} = (α_{1}, α_{2}, α_{3}), β^{t} = (β_{1}, beta_{2}, beta_{3}), λ^{t} = (λ_{1}, λ_{2}, λ_{3}). Then the surface S is regular and minimal.

3. Minimal surfaces in ℝ^{3}(−3)

We consider ℝ^{3} with coordinates (x, y, z) and its usual contact form η = \frac{1}{2}(dz − ydx), the metric ds_{3}^{2}, the tensor field φ and the caracteristic vector field ξ given by

φ = (\begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & y & 0 \end{matrix}) , ξ = 2\frac{∂}{∂z} .

ℝ^{3}, ds_{3}^{2}, η, ξ is a Sasakian manifold. The vector field e_{1} = 2\frac{∂}{∂y}, e_{2} = 2(\frac{∂}{∂x} + y \frac{∂}{∂z}) and ξ = 2\frac{∂}{∂z} is an orthonormal basis called φ-basis. The sectional curvature of any plane section spanned by a vector X and φX is equal -3, for this reason the Sasakian space ℝ^{3}, ds_{3}^{2}, η, ξ is called Sasakian space form denoted by ℝ^{3}(−3) see [4]). G. Zafindratafa in 1997 obtained other examples of Sasakian space form ℝ^{3}(−3) as following:

The graphs of the functions z = axy + P (x) + Q(y), where P, Q ∈ ℝ[X], are minimal if and only if

1. z = ax + b,

2. z = xy + ay + b,

3. z = \frac{1}{2}xy + ax + by + c,

4. z = \frac{1}{2}xy + \frac{ε}{4}x^{2} − \frac{ε}{4}y^{2} + εax + ay + b, a, b ∈ ℝ, ε = ±1,

5. z = \frac{1+ε\sqrt{1−16a^{2}}}{2} xy + ax^{2} − ay^{2} + b, ε = ±1, b ∈ ℝ, a ∈ [−\frac{1}{4}, \frac{1}{4}].

4. Beltrami formula in ℝ^{3}(−\frac{3}{4})

Let M be a surface of a Euclidean space 𝔼^{3}, with x as its position vector field in 𝔼^{3}. Then the relation between the Laplacian of x and the mean curvature vector field is given by the following Bertrami formula:

△x = −2\overrightarrow{H}

where △ is the Laplacian of the surface and \overrightarrow{H} is the mean curvature vector field of M.

A Beltrami formula in ℝ^{3}(\frac{−3}{4}) is given by the following:

△x = −(2H + Hξ_{3} + N_{3})E_{3} − (Hξ_{2} + N_{2})E_{2} − (Hξ_{1} + N_{1})E_{1},

where E_{1}, E_{2}, E_{3} are adapted frame and F_{i}^{j} are coefficients of E_{i} with respect to the orthonormal basis e_{j} of ℝ^{3}(\frac{−3}{4}).

N_{1} = −F_{1}^{3}C − F_{2}^{3}F_{3}^{3}, N_{2} = −F_{2}^{3}C + F_{1}^{3}F_{3}^{3},
N_{3} = −\frac{1}{2}g(E_{3}, x) − F_{3}^{3}C + F_{3}^{3} \frac{x^{3}}{4} ,
C = F_{1}^{3}g(E_{1}, x) + F_{2}^{3}g(E_{2}, x) − x^{3}(\frac{1}{4} + (F_{1}^{3})^{2} + (F_{2}^{3})^{2},
ξ_{1} = (F_{1}^{2}F_{3}^{3} + F_{1}^{3}F_{3}^{2})x^{1} − (F_{1}^{1}F_{3}^{3} + F_{1}^{3}F_{3}^{1})x^{2} + (−F_{1}^{1}F_{3}^{2} + F_{1}^{2}F_{3}^{1})x^{3},
ξ_{2} = (F_{2}^{2}F_{3}^{3} + F_{2}^{3}F_{3}^{2})x^{1} − (F_{2}^{1}F_{3}^{3} + F_{2}^{3}F_{3}^{1})x^{2} + (−F_{2}^{1}F_{3}^{2} + F_{2}^{2}F_{3}^{1})x^{3}
ξ_{3} = 2F_{3}^{2}F_{3}^{3})x^{1} − 2F_{3}^{1}F_{3}^{3})x^{2}.

An application of the Bertrami formula

The minimal surfaces as graphs of functions z = f(x, y) in ℝ^{3}(\frac{−3}{4}) are not always harmonic:

△x ≠ 0.

Remark 4.1. J. Inoguchi et al. [15] and G. Zafindratafa [27] studied harmonic maps in N^{3} and vector valued harmonic functions on ℝ^{3}(−3). A vector valued harmonic function on M into ℝ^{3}(−3) is a smooth map φ from M to ℝ^{3}(−3) such that △φ = 0. Here △ is the Laplacian of M. A harmonic map φ from M into ℝ^{3}(−3) is a smooth map whose tension field vanishes. For the definition of tension field, see [15].

Definition 4.2. A submanifold M of a Riemannian manifold N is called parallel when ( ¯∇_{X}h)(Y, Z) = 0 where h is the second fundamental form and

( ¯∇_{X}h)(Y, Z) = (∇_{X}^{⊥}h)(Y, Z) − h(∇_{X}Y, Z) − h(Y, ∇_{X}Z)

where ¯∇, ∇, ∇^{⊥} are the connection of van der Waerden-Bortolotti, the Levi-Civita connection of M, the normal connection of M in N respectively.

In 1948 V. F. Kagan [16] investigated the parallel surfaces in Euclidean space 𝔼^{3}. U. Simon and A. Weinstein [21] generalized Kagan’s work to hypersurfaces in 𝔼^{n} in 1969. J. Vilms [26] studied parallel general submanifolds in Euclidean spaces in 1972. D. Ferus [9] and W. Strübing [22] after showing that the parallel submanifolds of Euclidean space ℝ^{n} are related to their extrinsic symmetry, classified them. E. Backes and H. Reckziegel [1] and M. Takeushi [23] investigated parallel submanifolds in spaces of constant curvature. G. Pitis [19] studied parallel submanifolds in the Sasakian space form of dimension 2n + 1, n ≥ 2. D. E. Blair and C. Baikoussis [5, 6] studied C-parallel submanifolds, i.e., submanifolds such that ( ¯∇h ∥ ξ) in Sasakian space forms. We refer the reader to Ü. Lumiste’s paper [18]. We classify parallel surfaces in the Heisenberg 3-space H_{3}. Before that we recall some definitions and properties of surfaces in Heisenberg space H_{3}.

Theorem 4.3. [16] A surface of the Euclidean space 𝔼^{3} is parallel if and only if it is (a part of) a plane R^{2}, a sphere S^{2} or a round cylinder S^{1} × R^{1}.

Theorem 4.4. [20] The Heisenberg group H_{3} does not admit totally umbilical surfaces, in particularly totally geodesic ones.

Our main theorem is the following:

Theorem 4.5. The only parallel surfaces in the Heisenberg group H_{3} are the vertical planes (planes parallel to the z-axis of revolution of H_{3}). (See Proposition 2.6.)

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