Biharmonic Maps and Laguerre Minimal Surfaces

A Laguerre surface is known to be minimal if and only if its corresponding isotropic map is biharmonic. For every Laguerre surface Φ is its associated surface Ψ = (1+|𝑢| 2 )Φ , where 𝑢 lies in the unit disk. In this paper, the projection of the surface Ψ associated to a Laguerre minimal surface is shown to be biharmonic. A complete characterization of Ψ is obtained under the assumption that the corresponding isotropic map of the Laguerre minimal surface is harmonic. A sufficient and necessary condition is also derived for Ψ to be a graph. Estimates of the Gaussian curvature to the Laguerre minimal surface are obtained, and several illustrative examples are given.


Introduction
Surfaces in R 3 that minimize geometric energies are of great interest to architects because of their stability over other surfaces.These surfaces are used in the design and construction process of certain discrete meshed surfaces such as surfaces covered by special quadrilateral meshes with planar faces and conical meshes [1][2][3].Of the many minimal surfaces, the Laguerre minimal surfaces are widely used.
A Laguerre minimal (L-minimal) surface  is an R 3 surface that minimizes the geometric energy where  is the mean curvature and  the Gaussian curvature in the isotropic sense.This will be given more light in Section 2. Of interest to Weingarten and Blaschke was the fact that  is invariant under the group of Laguerre transformations.These are transformations on the space of oriented spheres which preserve oriented contact of spheres and take planes into planes in R 3 [3,6].Section 2 will give a brief description of the Laguerre geometry used in this paper.
Our keen interest in the geometric aspect of biharmonic maps [7][8][9][10][11][12][13] moved us to study L-minimal surfaces.The link between the two comes from the fact that isotropic models of L-minimal surfaces are described by biharmonic functions [3,6].
In Section 3, we write Ψ in the form where  = + lies in the unit disk U. Assuming that Φ is a Lminimal surface, it is shown in Lemma 3 that the projection  and  of Ψ are biharmonic.Additionally, if the isotropic map Φ  :  = (, ) is harmonic, then  is harmonic and  takes the form  (, ) = − () +  () , with  analytic and  harmonic.In Theorem 7, the associated L-minimal surface Ψ is completely characterized when the isotropic map is harmonic.It is shown that Ψ is an associated L-minimal surface if and only if the projection map  is a biharmonic map of the form (3). The associated surface Ψ is given more emphasis than the L-minimal surface because the coordinates of Ψ are either biharmonic or harmonic, and therefore are much easier to handle.We also give in Proposition 9 an estimate for the Gaussian curvature  (isotropic sense) of an L-minimal surface when the function  in (3) is analytic univalent satisfying (0) = 0 and   (0) = 1.
Section 4 considers the case when Ψ is a graph; that is, it is a nonparametric surface.When the function (, ) is univalent and biharmonic, it is shown in Theorem 10 that Ψ is an L-minimal graph.In Theorem 12, Landau's theorem for biharmonic maps [7,9,12] is used to find a uniform disk centered at 0 over which Ψ is locally a graph.In Theorem 14, a universal disk is obtained over which Ψ is a graph when () = ( 2 ) and  a normalized analytic univalent function.Neither one of the uniform disks described in Theorems 12 and 14 is sharp.Theorem 14 does not hold though over the entire class of normalized analytic univalent functions.Finally, three examples of graphs and local graphs are given to illustrate the results obtained.
Recall that a function  is harmonic [11] if Δ = 0, and  is biharmonic if Δ(Δ) = 0, where is the Laplacian operator.It is easy to show that a mapping  is biharmonic in a simply connected domain D if and only if  has the representation where  and  are complex-valued harmonic functions in D, with 1 ,  2 being analytic in D (for details see [7,8,[10][11][12]).The Jacobian of a map  is given by

Laguerre Geometry
For the sake of completeness, the basic essentials of Laguerre geometry is presented in this section.Additional details may be obtained from the works of [1-6, 14, 15].

2.1.
Isotropic Curvature for Graphs.The -curvature of a regular surface  given by the function  = (, ) is the curvature along unit vectors in the -plane.It is known that the principle -curvatures  and  at a point on the surface are the eigenvalues of the Hessian matrix ∇ 2  given by ∇ 2  = ( Hence the -mean curvature  is given by where Δ is the Laplacian of , while the -Gaussian curvature  is These curvatures are much easier to deal with compared to the Euclidean curvatures.

Duality between Surfaces of Graphs.
Let  * be the dual of  :  = (, ) given by the components of the tangent plane, specifically, If  is the corresponding map between  and  * , then  has an inverse  * .Hence, if  * and  * are the corresponding mean and -Gaussian curvatures of  * , then [2,6] where ,  are, respectively, the -mean and -Gaussian curvatures of .

Laguerre Geometry.
In Laguerre geometry, a point on a surface in R 3 is represented by its oriented tangent plane.An oriented plane  is given by where  is the unit normal vector.An oriented sphere , with center  and signed radius  ( can be negative), is tangent to an oriented plane  if the signed distance from the center  to  equals ; that is,   ⋅  + ℎ = .Points are viewed as oriented spheres with zero radius.The interested reader is referred to [2,6] for additional details.

The Isotropic Image of an Oriented Plane.
Let  :  1  +  2 + 3 +ℎ = 0 be an oriented plane with unit normal vector  = ( 1 ,  2 ,  3 ), and associate  with the point ( 1 ,  2 ,  3 , ℎ) ∈ R 4 .Next replace  with its stereographic image under the projection of the unit sphere  2 from (0, 0, −1) onto the plane  = 0. Then the isotropic image   of  is defined as If we let  = + and write and the unit vector  in complex variables becomes In this case, (14) becomes 2.5.Laguerre Surface.Let Φ be a Laguerre surface in R 3 .Any regular point  on Φ is thus represented as in (16).Denote the corresponding isotropic surface by Φ  with   given by (16).By duality, their corresponding curvatures are related by Blaschke [6] defined the middle tangent sphere to be the tangent to the tangent plane  with radius where  1 = 1/,  2 = 1/, and ,  are the principal curvatures of the L-surface Φ.Let Φ  denote the middle surface consisting of centers of the middle spheres.It is shown in [6] that Φ  is invariant under Laguerre transformations.A surface Φ is an L-minimal surface when Φ  minimizes the area functional (see (8) and ( 9)).This is also invariant under L-transforms.If, in (16),  on Φ is given by ℎ = ℎ(, ), then [5,6] ( = −2), and when Ω is minimal, then The latter implies that ℎ is biharmonic.Assume now that Φ  is given by the function  = (, ).Since  = ℎ/2, it follows from (20) that  = (, ) is also biharmonic.This leads to the following result.
Theorem 1 (see [2,3]).Let Φ be a Laguerre surface and Φ  its corresponding isotropic surface related as in (16).Suppose Φ  is given by the function  = (, ).Then Φ is minimal if and only if  is biharmonic.

Projection of L-Minimal Surface onto a Plane
In ( 16), the Laguerre surface Φ is expressed in terms of the Laguerre coordinates (, , ℎ).In this section, the Euclidean coordinates are used instead.Simple calculations from ( 16) and use of Theorem 1 lead to the following known result.
Theorem 2 (see [3]).Let Φ  be the graph of the biharmonic function  = (, ).Then the parametric equations of the corresponding L-minimal surface Φ are given by Throughout this section, it is assumed that the L-minimal surface is parametrized by  =  + , with  in the unit disk U.
Equations (22) will first be written in terms of complex variables.For this purpose, let Then from ( 22), the projection of the surface V = V() and the height  becomes and the coordinates of Ψ (see (2)) are Evidently, The relations (25) and (26) yield the following general lemma.
where ℎ,  are harmonic.It follows directly by differentiation that Then in light that  and ℎ are harmonic.Since  is biharmonic, (26) shows that   is harmonic.That  is biharmonic also implies that the leading three terms in the right-hand side of the second equation of (26) are harmonic.Hence   is harmonic, which proves part (a).(b) Substituting  = ( + )/2 into (25) yields Also the second equation of (25) shows that  is harmonic.
Lemma 3 gives a structural connection between an Lminimal surface with its projection map, in other words, a connection between the surface Ψ : (Re , Im , ) and  (see ( 2), ( 22), and (25)).Proof.Comparing  as given above and  in (25), it follows that Assume that the solution of (26) is harmonic.Differentiating both sides of the above equation leads to The result now follows directly by integration, and the resulting  clearly satisfies the conclusion.
The following corollary is obtained from Lemma 3 and (25).
Combining the above lemmas and corollaries results in the following characterization of minimal surfaces with harmonic isotropic maps.Theorem 7. Let Φ be a Laguerre surface.A surface Ψ is an associated L-minimal surface with a harmonic isotropic map Φ  if and only if Ψ is given by where  is a biharmonic map given by (27), and  by (34) is harmonic in U.
The -Gauss curvature of an L-minimal surface in terms of the projection map can be obtained from Lemma 3.

Corollary 8.
Let  be given as in Lemma 3(b).Then the i-Gauss curvature of the L-minimal surface is Proof.By (9), (37) which leads to the desired inequalities.

The Associated L-Surface Is a Graph
This section looks at the case when Ψ is a graph; that is, when Ψ is a nonparametric surface.Interestingly, the graph of the associated L-minimal surface is closely connected to its corresponding projection map .
Proof.Differentiating  = () leads to Since   =   and   =   , it follows that Solving the linear system gives the desired results.

Now (27) implies that
and subsequently, We next present a theorem about the surfaces Ψ and Φ which is a consequence of Landau's theorem for biharmonic maps.This was first proved in [7] and the universal constant was later sharpened in [9,12].This theorem will also help provide examples of graphs for L-surfaces.Theorem 12. Let Ψ be a surface given by () = − + ,  ∈ U, where  is analytic and  harmonic is given by (27).If |  ()| is bounded by a constant , (0) = 0 and   (0) = 1, then there are uniform constants () > 0 and () > 0 so that  and V() = ()/( 1 As (0) = 0, choose  <  1 so that |( 2 )| < /8 and the result follows.
Corollary 13.Let , ,  be given as in Theorem 12. Then () is univalent in U and the corresponding surface Ψ is a graph.
The distortion inequality also implies that The latter inequality together with the distortion inequalities imply that where  2 () is taken to be a polynomial with positive coefficients of degree 3 and  2 (0) = 4 + 10.
An argument similar to the proof of Theorem 1 in [7,9,12] gives the result for V and consequently for Ψ and Φ.
(2) Theorem 14 is not true for the class S. The following proposition shows that there is no uniform disk on which the surface Ψ is a graph for all  ∈ S.
(a) There is no uniform disk centered at 0 where   () ≤ 0.
(b) There is no uniform disk on which  is univalent and, consequently, no uniform disk on which Ψ is a graph.
We conclude our exposition with several examples.
The value of this expression ranges between 0 at 0 and ∞ at || = 1.Hence placing it less than 1 and solving for || give a uniform disk || < 0.32471 for all .The corresponding  is then locally univalent, with   () < 0 in || < 0.32471.
Note that in the case () = /(1+),   → −1,   → 0 when  → 1. Hence   → 1.This implies that  may not be locally univalent in all of U. Figures 4 and 5 show that neither the associated L-surface Ψ nor the corresponding Lsurface Φ is a total graph.

Lemma 3 .
(a) If  is biharmonic in U, then  and  are biharmonic.(b) If  = ( + )/2 is harmonic and  analytic, then  is harmonic and  = − + , where the harmonic function  is given by

Corollary 4 .Lemma 5 .
(a) If Ψ is an associated L-minimal surface parametrized by the unit disk || < 1, then the projections  and  are biharmonic.(b) If Ψ is an associated L-minimal surface parametrized by the unit disk || < 1 and the corresponding isotropic surface is given by a harmonic function  = (), then  = − + , where  is analytic,  harmonic satisfying (27), and  is harmonic.Moving in the opposite direction is the following lemma.If () = − + ,  ∈ U, where  is analytic and  harmonic is given by (27), then the equation  in (25) has a harmonic solution  satisfying  = Re .

2 .Proposition 9 .
Now  = 1/  and Corollary 4 gives both equalities.We conclude this section with an estimate for  when  belongs to the class S consisting of univalent analytic functions  in U normalized by (0) = 0 and   (0) = 1.Let  ∈ S and the corresponding associated L-minimal surface Ψ be given as in Theorem 7. Then −∞ ≤  ()

Figure 1 Figure 1 :
Figure 1: Projection map of a Laguerre surface.