Biharmonic Maps and Laguerre Minimal Surfaces

Abstract

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.

1. Introduction

Surfaces in ℝ³ 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 [1–3]. Of the many minimal surfaces, the Laguerre minimal surfaces are widely used.

Laguerre minimal surfaces were introduced by Weingarten in 1888 [3–6] and later studied in detail by Blaschke in a series of papers dating from 1924 [4–6].

A Laguerre minimal (L-minimal) surface 𝑆 is an ℝ³ surface that minimizes the geometric energy 𝑊=_𝑆𝐻²−𝐾𝐾𝑑𝐴, (1) where 𝐻 is the mean curvature and 𝐾 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 𝑊 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 ℝ³ [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 [7–13] 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 the sequel, the Laguerre surface (L-surface) is denoted by Φ∶=Φ(𝑎(𝑥,𝑦),𝑏(𝑥,𝑦),𝑐(𝑥,𝑦)), 𝑥,𝑦 real, Ψ=(1+𝑥²+𝑦²)Φ is the associated L-surface, and Φ^𝑖∶𝑧=𝑓(𝑥,𝑦) is the corresponding isotropic graph. The surface Ψ is called an associated L-minimal surface when Φ is L-minimal. Note that Ψ itself need not be L-minimal.

In Section 3, we write Ψ in the form Ψ=Ψ(𝑤(𝑢,𝑢),𝐶), (2) where 𝑢=𝑥+𝑖𝑦 lies in the unit disk 𝕌. Assuming that Φ is a L-minimal surface, it is shown in Lemma 3 that the projection 𝑤 and 𝐶 of Ψ are biharmonic. Additionally, if the isotropic map Φ^𝑖∶𝑧=𝑓(𝑥,𝑦) is harmonic, then 𝐶 is harmonic and 𝑤 takes the form 𝑤(𝑢,𝑢)=−𝑢𝐺(𝑢)+𝐻(𝑢), (3) with 𝐺 analytic and 𝐻 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 𝑤 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 𝐾 (isotropic sense) of an L-minimal surface when the function 𝐺 in (3) is analytic univalent satisfying 𝐺(0)=0 and 𝐺^′(0)=1.

Section 4 considers the case when Ψ is a graph; that is, it is a nonparametric surface. When the function 𝑤(𝑢,𝑢) 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 𝐺(𝑢)=𝐹(𝑢²) and 𝐹 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 𝐹 is harmonic [11] if Δ𝐹=0, and 𝐹 is biharmonic if Δ(Δ𝐹)=0, where 𝜕Δ=4²𝜕𝑧𝜕𝑧 (4) is the Laplacian operator. It is easy to show that a mapping 𝐹 is biharmonic in a simply connected domain 𝒟 if and only if 𝐹 has the representation 𝐹=𝑟²𝐺+𝐾,𝑟𝑒^𝑖𝜃∈𝒟, (5) where 𝐺 and 𝐾 are complex-valued harmonic functions in 𝒟, with 𝐺=𝑔₁+𝑔₂, (6)𝑔₁, 𝑔₂ being analytic in 𝒟 (for details see [7, 8, 10–12]). The Jacobian of a map 𝑊 is given by 𝐽_𝑊=𝑊_𝑧²−𝑊_𝑧². (7)

2. 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 𝑖-curvature of a regular surface 𝜙 given by the function 𝑧=𝑓(𝑥,𝑦) is the curvature along unit vectors in the 𝑥𝑦-plane. It is known that the principle 𝑖-curvatures 𝑚 and 𝑀 at a point on the surface are the eigenvalues of the Hessian matrix ∇²𝑓 given by ∇²𝑓=^{𝑓𝑥𝑥𝑓𝑥𝑦}_{𝑓𝑦𝑥𝑓𝑦𝑦}. Hence the 𝑖-mean curvature 𝐻 is given by 𝐻=𝑚+𝑀2=Δ𝑓2, (8) where Δ𝑓 is the Laplacian of 𝑓, while the 𝑖-Gaussian curvature 𝐾 is 𝐾=𝑚𝑀=det∇²𝑓. (9) These curvatures are much easier to deal with compared to the Euclidean curvatures.

2.2. Duality between Surfaces of Graphs

Let 𝜙^∗ be the dual of 𝜙∶𝑧=𝑓(𝑥,𝑦) given by the components of the tangent plane, specifically, 𝑥^∗=𝑓_𝑥𝑥,𝑦,𝑦^∗=𝑓_𝑦𝑥,𝑦,𝑧^∗=𝑥𝑓_𝑥+𝑦𝑓_𝑦−𝑓. (10) If 𝜎 is the corresponding map between 𝜙 and 𝜙^∗, then 𝜎 has an inverse 𝜎^∗. Hence, if 𝐻^∗ and 𝐾^∗ are the corresponding 𝑖-mean and 𝑖-Gaussian curvatures of 𝜙^∗, then [2, 6] 𝐻^∗=𝐻𝐾,𝐾^∗=1𝐾, (11) where 𝐻,𝐾 are, respectively, the 𝑖-mean and 𝑖-Gaussian curvatures of 𝜙.

2.3. Laguerre Geometry

In Laguerre geometry, a point on a surface in ℝ³ is represented by its oriented tangent plane. An oriented plane 𝑃 is given by 𝑛^𝑇⋅𝑥+ℎ=0, (12) where 𝑛 is the unit normal vector. An oriented sphere 𝑆, with center 𝑚 and signed radius 𝑅 (𝑅 can be negative), is tangent to an oriented plane 𝑃 if the signed distance from the center 𝑚 to 𝑃 equals 𝑅; that is, 𝑛^𝑇⋅𝑚+ℎ=𝑅. Points are viewed as oriented spheres with zero radius. The interested reader is referred to [2, 6] for additional details.

2.4. The Isotropic Image of an Oriented Plane

Let 𝑃∶𝑛₁𝑥+𝑛₂𝑦+𝑛₃𝑧+ℎ=0 be an oriented plane with unit normal vector 𝑛=(𝑛₁,𝑛₂,𝑛₃), and associate 𝑃 with the point (𝑛₁,𝑛₂,𝑛₃,ℎ)∈ℝ⁴. Next replace 𝑛 with its stereographic image 𝑛₁𝑛₃,𝑛+1𝑛₃+²1,0 (13) under the projection of the unit sphere 𝑆² from (0,0,−1) onto the plane 𝑧=0. Then the isotropic image 𝑃^𝑖 of 𝑃 is defined as 𝑃=𝑛₁,𝑛₂,𝑛₃,ℎ⟶𝑃^𝑖=𝑛₁𝑛₃,𝑛+1𝑛²₃,ℎ+1𝑛₃.+1 (14)

If we let 𝑢=𝑥+𝑖𝑦 and write 𝑛₁+𝑖𝑛₂=(2𝑥+𝑖2𝑦)/(1+𝑥²+𝑦²), then 𝑛₃=(1−𝑥²−𝑦²)/(1+𝑥²+𝑦²),𝑛₃+1=2/(1+𝑥²+𝑦²), and the unit vector 𝑛 in complex variables becomes 𝑛=𝑛₁,𝑛₂,𝑛₃=11+|𝑢|²2Re𝑢,2Im𝑢,1−|𝑢|². (15) In this case, (14) becomes 𝑃∶(𝑢,𝑢,ℎ)⟶𝑃^𝑖∶|𝑢|𝑥,𝑦,1+²ℎ2. (16)

2.5. Laguerre Surface

Let Φ be a Laguerre surface in ℝ³. Any regular point 𝑃 on Φ is thus represented as in (16). Denote the corresponding isotropic surface by Φ^𝑖 with 𝑃^𝑖 given by (16). By duality, their corresponding curvatures are related by 𝐻^∗=𝐻𝐾,𝐾^∗=1𝐾. (17) Blaschke [6] defined the middle tangent sphere to be the tangent to the tangent plane 𝑃 with radius 𝑅𝑅=¹+𝑅₂2, (18) where 𝑅₁=1/𝑚, 𝑅₂=1/𝑀, and 𝑚, 𝑀 are the principal curvatures of the L-surface Φ. Let Φ_𝑀 denote the middle surface consisting of centers of the middle spheres. It is shown in [6] that Φ_𝑀 is invariant under Laguerre transformations.

A surface Φ is an L-minimal surface when Φ_𝑀 minimizes the area functional 1Ω=4_Φ𝑅₁−𝑅₂²𝐾𝑑𝐴=_Φ𝐻²−𝐾𝐾𝑑𝐴=_Φ𝑖𝐻²_𝑖−𝐾_𝑖𝑑𝐴_𝑖 (19) (see (8) and (9)). This is also invariant under L-transforms.

If, in (16), 𝑃 on Φ is given by ℎ=ℎ(𝑢,𝑢), then [5, 6] 𝑖Ω=2ℎ_𝑢𝑢ℎ_𝑢𝑢𝑑𝑢𝑑𝑢, (20) (𝑑𝑢𝑑𝑢=−2𝑖𝑑𝑥𝑑𝑦), and when Ω is minimal, then ℎ_{𝑢𝑢𝑢𝑢}=0. (21) The latter implies that ℎ is biharmonic. Assume now that Φ^𝑖 is given by the function 𝑧=𝑓(𝑥,𝑦). Since 𝑧=ℎ/2, it follows from (20) that 𝑧=𝑓(𝑥,𝑦) is also biharmonic.

This leads to the following result.

Theorem 1 (see [2, 3]). Let Φ be a Laguerre surface and Φ^𝑖 its corresponding isotropic surface related as in (16). Suppose Φ^𝑖 is given by the function 𝑧=𝑓(𝑥,𝑦). Then Φ is minimal if and only if 𝑓 is biharmonic.

3. Projection of L-Minimal Surface onto a Plane

In (16), the Laguerre surface Φ is expressed in terms of the Laguerre coordinates (𝑢,𝑢,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^𝑖 be the graph of the biharmonic function 𝑧=𝑓(𝑥,𝑦). Then the parametric equations of the corresponding L-minimal surface Φ are given by 1𝑎=1+𝑥²+𝑦²𝑥²−𝑦²−1𝑓_𝑥+2𝑥𝑦𝑓_𝑦1−2𝑥𝑓𝑏=1+𝑥²+𝑦²𝑦²−𝑥²−1𝑓_𝑦+2𝑥𝑦𝑓_𝑥1−2𝑦𝑓𝑐=1+𝑥²+𝑦²2𝑥𝑓_𝑥+2𝑦𝑓_𝑦−2𝑓. (22)

Throughout this section, it is assumed that the L-minimal surface is parametrized by 𝑢=𝑥+𝑖𝑦, with 𝑢 in the unit disk 𝕌.

Equations (22) will first be written in terms of complex variables. For this purpose, let 𝑣=𝑎+𝑖𝑏,𝑤=1+|𝑢|²𝑣,𝐶=1+|𝑢|²𝑐. (23) Then from (22), the projection of the surface 𝑣=𝑣(𝑢) and the height 𝑐 becomes 2𝑢𝑣=𝑢𝑓_𝑢−𝑓−𝑓_𝑢1+|𝑢|²,2𝑐=𝑢𝑓_𝑢−𝑓+𝑢𝑓_𝑢1+|𝑢|², (24) and the coordinates of Ψ (see (2)) are 𝑤(𝑢)=2𝑢𝑢𝑓_𝑢−𝑓−𝑓_𝑢,𝐶=2𝑢𝑓_𝑢−𝑓+𝑢𝑓_𝑢. (25) Evidently, 𝑤_𝑢𝑢=2𝑢𝑓_𝑢𝑢+2𝑢²𝑓_𝑢𝑢𝑢−2𝑓_𝑢𝑢𝑢−2𝑓_𝑢,𝐶_𝑢𝑢=2𝑓_𝑢𝑢+2𝑢𝑓_𝑢𝑢𝑢+2𝑢𝑓_𝑢𝑢𝑢. (26) The relations (25) and (26) yield the following general lemma.

Lemma 3. (a) If 𝑓 is biharmonic in 𝕌, then 𝑤 and 𝐶 are biharmonic.
(b) If 𝑓=(𝐺+𝐺)/2 is harmonic and 𝐺 analytic, then 𝐶 is harmonic and 𝑤=−𝑢𝐺+𝐻, where the harmonic function 𝐻 is given by 𝐻(𝑢)=𝑢²𝐺^′(𝑢)−𝑢𝐺(𝑢)−𝐺^′(𝑢). (27)

Proof. (a) If 𝑓 is biharmonic in 𝕌, then 𝑓(𝑢)=|𝑢|²𝑔(𝑢)+ℎ(𝑢), (28) where ℎ, 𝑔 are harmonic. It follows directly by differentiation that 𝑓_𝑢=𝑢𝑔+|𝑢|²𝑔_𝑢+ℎ_𝑢,𝑢𝑓_𝑢𝑢=𝑢𝑔+𝑢𝑔_𝑢+𝑢𝑔_𝑢+ℎ_𝑢𝑢+𝑢𝑢𝑔_𝑢𝑢. (29) Then 𝑢𝑓_𝑢𝑢−𝑓_𝑢=𝑢²𝑔_𝑢+𝑢ℎ_𝑢𝑢+𝑢²𝑢𝑔_𝑢𝑢−ℎ_𝑢=𝑢²𝑔_𝑢−ℎ_𝑢 (30) in light that 𝑔 and ℎ are harmonic. Since 𝑓 is biharmonic, (26) shows that 𝑤_𝑢𝑢 is harmonic. That 𝑓 is biharmonic also implies that the leading three terms in the right-hand side of the second equation of (26) are harmonic. Hence 𝐶_𝑢𝑢 is harmonic, which proves part (a).
(b) Substituting 𝑓=(𝐺+𝐺)/2 into (25) yields 𝑤(𝑢)=𝑢²𝐺^′−𝑢𝐺−𝑢𝐺−𝐺^′. (31) Also the second equation of (25) shows that 𝐶 is harmonic.

Lemma 3 gives a structural connection between an L-minimal surface with its projection map, in other words, a connection between the surface Ψ∶(Re𝑤,Im𝑤,𝐶) and 𝑤 (see (2), (22), and (25)).

Corollary 4. (a) If Ψ is an associated L-minimal surface parametrized by the unit disk |𝑢|<1, then the projections 𝑤 and 𝐶 are biharmonic.
(b) If  Ψ is an associated L-minimal surface parametrized by the unit disk |𝑢|<1 and the corresponding isotropic surface is given by a harmonic function 𝑧=𝑓(𝑢), then 𝑤=−𝑢𝐺+𝐻, where 𝐺 is analytic, 𝐻 harmonic satisfying (27), and 𝐶 is harmonic.

Moving in the opposite direction is the following lemma.

Lemma 5. If 𝑤(𝑢)=−𝑢𝐺+𝐻, 𝑢∈𝕌, where 𝐺 is analytic and 𝐻 harmonic is given by (27), then the equation 𝑤 in (25) has a harmonic solution 𝑓 satisfying 𝑓=Re𝐺.

Proof. Comparing 𝑤 as given above and 𝑤 in (25), it follows that −𝑢𝐺+𝐻=2𝑢𝑢𝑓_𝑢−𝑓−𝑓_𝑢. (32)
Assume that the solution of (26) is harmonic. Differentiating both sides of the above equation leads to 𝑤_𝑢𝑢=−𝐺^′=2𝑢𝑓_𝑢𝑢+2𝑢²𝑓_𝑢𝑢𝑢−2𝑓_𝑢𝑢𝑢−2𝑓_𝑢=−2𝑓_𝑢. (33) The result now follows directly by integration, and the resulting 𝑓 clearly satisfies the conclusion.

The following corollary is obtained from Lemma 3 and (25).

Corollary 6. If 𝑤 is given by Lemma 3(b), then 𝐶(𝑢)=−2Re^𝑢₀𝑢𝐺^′′(𝑢)𝑑𝑢. (34)

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^𝑖 if and only if  Ψ is given by Ψ∶(Re𝑤(𝑢),Im𝑤(𝑢),𝐶), (35) where 𝑤 is a biharmonic map given by (27), and 𝐶 by (34) is harmonic in 𝕌.

The 𝑖-Gauss curvature of an L-minimal surface in terms of the projection map can be obtained from Lemma 3.

Corollary 8. Let 𝑤 be given as in Lemma 3(b). Then the i-Gauss curvature of the L-minimal surface is 2𝐾=−𝐺^′′(𝑢)²2|𝑢|=−²𝐶_𝑢². (36)

Proof. By (9), 𝐾_𝑖=𝑓_𝑥𝑥𝑓_𝑦𝑦−(𝑓_𝑥𝑦)²=−(𝑓²_𝑥𝑥+𝑓²_𝑥𝑦)=−2(𝑓_𝑢𝑢𝑓_𝑢𝑢)=−|𝐺^′′(𝑢)|²/2. Now 𝐾=1/𝐾_𝑖 and Corollary 4 gives both equalities.

We conclude this section with an estimate for 𝐾 when 𝐺 belongs to the class 𝒮 consisting of univalent analytic functions 𝐹 in 𝕌 normalized by 𝐹(0)=0 and 𝐹^′(0)=1.

Proposition 9. Let 𝐺∈𝒮 and the corresponding associated L-minimal surface Ψ be given as in Theorem 7. Then (𝑢)(|𝑢−∞≤𝐾≤−1−|)⁸2(2+|𝑢|)²,(0)1−∞≤𝐾≤−8. (37)

Proof. It is known [16, p. 21] that for 𝐺∈𝒮, 𝑢𝐺^′′(𝑢)𝐺^′(𝑢)≤4|𝑢|+2|𝑢|²1−|𝑢|²,𝐺^′(𝑢≤|𝑢|)1+(1−|𝑢|)³. (38) Thus 𝐺^′′(𝑢≤2(|𝑢)2+|)(1−|𝑢|)⁴, (39) which leads to the desired inequalities.

4. 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 𝑤.

Figure 1 makes this relationship evident and gives rise to the following theorem.

Figure 1 

Projection map of a Laguerre surface.

Theorem 10. An associated L-surface parametrized by the unit disk |𝑢|<1 is a graph if and only if 𝑤 is a univalent biharmonic map.

The following lemma gives the derivatives of the function 𝐶=𝐶(𝑤) of the graph surface.

Lemma 11. Let 𝑤 and 𝑓 be given as in Lemma 5. If 𝑤 is univalent with Jacobian 𝐽_𝑤(𝑢)=|−𝐺+𝐻_𝑢|²−|−𝑢𝐺^′+𝐻_𝑢|²>0 for all 𝑢, then 𝐶_𝑤=𝑤_𝑢𝐶_𝑢−𝑤_𝑢𝐶_𝑢𝐽_𝑤(𝑢)𝐶_𝑤=𝑤_𝑢𝐶_𝑢−𝑤_𝑢𝐶_𝑢𝐽_𝑤(𝑢). (40)

Proof. Differentiating 𝐶=𝐶(𝑤) leads to 𝐶_𝑢=𝐶_𝑤𝑤_𝑢+𝐶_𝑤𝑤_𝑢,𝐶_𝑢=𝐶_𝑤𝑤_𝑢+𝐶_𝑤𝑤_𝑢. (41) Since 𝑤_𝑢=𝑤_𝑢 and 𝑤_𝑢=𝑤_𝑢, it follows that 𝐶_𝑢=𝐶_𝑤𝑤_𝑢+𝐶_𝑤𝑤_𝑢𝐶_𝑢=𝐶_𝑤𝑤_𝑢+𝐶_𝑤𝑤_𝑢. (42) Solving the linear system gives the desired results.

Now (27) implies that 𝑤(𝑢)=𝑢²𝐺^′−𝑢𝐺−𝑢𝐺−𝐺^′, (43) and subsequently, 𝑤_𝑢=−𝐺^′′−𝑢𝐺^′𝑤_𝑢=2𝑢𝐺^′+𝑢²𝐺^′′−𝐺−𝑢𝐺^′−𝐺=𝑢²𝐺^′′+𝑢𝐺^′−2Re𝐺. (44)

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 𝑤(𝑢)=−𝑢𝐺+𝐻, 𝑢∈𝕌, where 𝐺 is analytic and 𝐻 harmonic is given by (27). If |𝐺^′(𝑢)| is bounded by a constant 𝑀, 𝑤(0)=0 and 𝐺^′′(0)=1, then there are uniform constants 𝜌(𝑀)>0 and 𝑅(𝑀)>0 so that 𝑤 and 𝑣(𝑢)=𝑤(𝑢)/(1+|𝑢|²) are univalent on the disk |𝑢|<𝜌, and the image of this disk contains a disk |𝑤|<𝑅 on which the surfaces are graphs.

Proof. The Jacobian of 𝑤 is given by 𝐽_𝑤(𝑢)=𝑤_𝑢(𝑢)²−𝑤_𝑢(𝑢)². (45) It follows from (44) that 𝐽_𝑤(0)=−|𝐺^′′(0)|²=−1. If |𝐺^′(𝑢)| is bounded, then |𝐺(𝑢)| and consequently |𝑤(𝑢)| is bounded. It can now be deduced from Theorem  1 in [7, 9, 12] that there are uniform constants 𝜌₁(𝑀)>0 and 𝑅₁(𝑀)>0 so that 𝑤 is univalent on the disk |𝑢|<𝜌₁ whose image contains the disk |𝑤|<𝑅₁. Consequently the surface Ψ is a graph above such a disk.
Clearly 𝑣 is univalent on each circle |𝑢|=𝑟<𝜌₁. Suppose now that there are 𝑢₁ and 𝑢₂ with |𝑢₁|<|𝑢₂|<𝜌₁ so that 𝑣(𝑢₁)=𝑣(𝑢₂). Then 𝑤𝑢₁−𝑤𝑢₂𝑢₁−𝑢₂=𝑤𝑢₂𝑢₂²−𝑢₁²𝑢₁−𝑢₂1+𝑢₂²≤2𝑤𝑢₂𝑢₂1+𝑢₂²≤𝑤𝑢₂. (46) But it was shown in the proof of Theorem  1 in [7, 9, 12] that 𝑤𝑢₁−𝑤𝑢₂𝑢₁−𝑢₂≥𝜋𝜌4𝑀−𝑜₁. (47) Hence 𝑤𝑢₂≥𝜋𝜌4𝑀−𝑜₁. (48) As 𝑤(0)=0, choose 𝜌<𝜌₁ so that |𝑤(𝑢₂)|<𝜋/8𝑀 and the result follows.

Corollary 13. Let 𝑤,𝐺,𝜌 be given as in Theorem 12. Then 𝑤(𝜌𝑢) is univalent in 𝕌 and the corresponding surface Ψ is a graph.

Interestingly when 𝐺(𝑢)=𝐹(𝑢²),𝐹∈𝒮, a similar result is obtained without imposing the boundedness condition. Recall that 𝒮 is the class of univalent analytic functions 𝐹 normalized by 𝐹(0)=0 and 𝐹^′(0)=1.

Theorem 14. Let 𝐺(𝑢)=𝐹(𝑢²), 𝐹∈𝒮, 𝐶₁(𝜌), 𝐶₂(𝜌) polynomials with positive coefficients of degree 2 and 3, respectively (see (51) and (53)), and 𝑅₂(𝜌₂) described by (56). Then there are two uniform radii 𝜌₂>0 and 𝑅₂(𝜌₂) satisfying 𝜌²₂𝐶₁𝜌²₂+𝐶₂𝜌²₂1−𝜌²₂⁴=2, (49) so that the corresponding 𝑤 and 𝑣 are univalent in |𝑢|<𝜌₂, and the image of 𝑤 contains a disk |𝑤+2|<𝑅₂. In this case, the surfaces Ψ and Φ are graphs on |𝑤+2|<𝑅₂.

Proof. Now 𝐺^′(𝑢)=2𝑢𝐹^′(𝑢²) and 𝐺^′′(𝑢)=2𝐹^′(𝑢²)+4𝑢²𝐹^′′(𝑢²). It follows from (44) that 𝑤_𝑢(0)=−2, 𝑤_𝑢(0)=0, and 𝐽_𝑤(0)=−4. The distortion estimates for the class 𝒮 [16, p. 21] are |𝐹(𝑧≤|𝑧|)|(1−|𝑧|)²,𝐹^′(𝑧≤|𝑧|)1+(1−|𝑧|)³,𝐹^′′(𝑧≤𝑐)(1−|𝑧|)⁴. (50) From (44), these inequalities imply that 𝑤_𝑢(𝑢≤|𝑢|)²𝐶₁|𝑢|²1−|𝑢|²⁴, (51) where 𝐶₁(𝑥) can be chosen as a polynomial of degree 2 with positive coefficients and 𝐶₁(0)=6.
The distortion inequality also implies that 𝐹^′(𝑧)≤4|𝑧||𝑧|−1−3²+|𝑧|³(1−|𝑧|)³. (52) The latter inequality together with the distortion inequalities imply that 𝑤_𝑢(𝑢)+2=−2𝐹^′𝑢²−1−4𝑢²𝐹^′′𝑢²−2𝑢²𝐹^′𝑢²≤|𝑢|²𝐶₂|𝑢|²1−|𝑢|²⁴, (53) where 𝐶₂(𝑥) is taken to be a polynomial with positive coefficients of degree 3 and 𝐶₂(0)=4𝑐+10.
Let 𝑢₁,𝑢₂ be two points near 0 in a disk |𝑢|<𝜌. It follows from (51) and (53) that 𝑤𝑢₁−𝑤𝑢₂=^𝑢2_𝑢1𝑤_𝑢(𝑢)𝑑𝑢+𝑤_𝑢(𝑢)𝑑𝑢=^𝑢2_𝑢1𝑤_𝑢(0)𝑑𝑢+𝑤_𝑢(0)𝑑𝑢+^𝑢2_𝑢1𝑤_𝑢(𝑢)−𝑤_𝑢(0+)𝑑𝑢_𝑢2𝑢1𝑤_𝑢(𝑢)−𝑤_𝑢(0)𝑑𝑢≥2𝑢₁−𝑢₂−^𝑢2_𝑢1𝑤_𝑢(𝑢)|𝑑𝑢|−^𝑢2_𝑢1𝑤_𝑢(𝑢)+2|𝑑𝑢|≥2𝑢₁−𝑢₂−𝑢₁−𝑢₂𝜌²𝐶₁𝜌²1−𝜌²⁴−𝑢₁−𝑢₂𝜌²𝐶₂𝜌²1−𝜌²⁴≥𝑢₁−𝑢₂𝜌2−²𝐶₁𝜌²+𝐶₂𝜌²1−𝜌²⁴. (54) Now choose 𝜌₂ so that 𝜌²(𝐶₁(𝜌²)+𝐶₂(𝜌²))/|1−𝜌²|⁴=2 to deduce that 𝑤 is univalent in |𝑢|<𝜌₂.
Let 𝛿>0 satisfy 𝛿²𝐶₁𝛿²+𝐶₂𝛿²1−𝛿²⁴=1, (55) and let 𝛾_𝜌=𝑤(|𝑢|=𝜌). Then the distance 𝑑(𝑤(0),𝜕𝛾_𝜌2)≥𝑑(𝑤(0),𝜕𝛾_𝛿). If we choose 𝑢₂=0 and 𝑢₁∈𝛾_𝛿 in (54), then 𝑑𝑤(0),𝜕𝛾_𝜌2≥𝑑𝑤(0),𝜕𝛾_𝛿≥𝛿. (56) Thus choose 𝑅₂(𝜌₂)=𝛿.
An argument similar to the proof of Theorem  1 in [7, 9, 12] gives the result for 𝑣 and consequently for Ψ and Φ.

Corollary 15. Let 𝐹∈𝒮 and 𝜌₂ be given by Theorem 14. If 𝐺(𝑢)=𝐹((𝜌₂𝑢)²), then the corresponding associated L-minimal surface Ψ is a graph.

Remark 16. (1) A result similar to Theorem 14 can be obtained for the L-surface Φ=Ψ/(1+𝑢²).
(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 𝐺∈𝒮.

Proposition 17. Let ϝ be the set of all convex univalent functions 𝐺 given by 𝐺(𝑢)=^𝑢₀𝑑𝑧(1−𝑥𝑧)^2𝑡1−𝑦𝑧^2(1−𝑡), (57) where 0≤𝑡≤1, |𝑥|=1 and |𝑦|=1, and let 𝑤 be the corresponding biharmonic map given by (27).(a)There is no uniform disk centered at 0 where 𝐽_𝑤(𝑢)≤0.(b)There is no uniform disk on which 𝑤 is univalent and, consequently, no uniform disk on which Ψ is a graph.

Proof. For 𝐺∈ϝ, 𝑢𝐺^′′𝐺^′+1=𝑡1+𝑥𝑢+()1−𝑥𝑢1−𝑡1+𝑦𝑢.1−𝑦𝑢 (58) First we show that 𝑤_𝑢(𝑢)=0, for any 𝑢∈𝕌⧵{0}, and for the choices 𝑡=1/2, 𝑥,𝑦 satisfying 𝑦𝑢=𝑥𝑢 with 𝑥𝑢,𝑦𝑢 being the intersection points between the circles |𝑧|=|𝑢| and |1−𝑧|=1. These conditions imply that |1−𝑥𝑢|=|1−𝑦𝑢|=1. In this case, (58) gives 𝑢𝐺^′′𝐺^′+|𝑢|²=121+𝑥𝑢+11−𝑥𝑢21+𝑦𝑢−|𝑢|1−𝑦𝑢1−²,Re𝑢𝐺^′′𝐺^′+|𝑢|²=121−|𝑢|²|1−𝑥𝑢|²+121−|𝑢|²1−𝑦𝑢²−1−|𝑢|²=121−|𝑢|²+121−|𝑢|²−1−|𝑢|²=0,Im𝑢𝐺^′′𝐺^′+|𝑢|²=0. (59) Hence (44) becomes 𝑤_𝑢(𝑢)=−𝐺^′′−𝑢𝐺^′=−𝐺^′𝑢𝑢𝐺^′′𝐺^′+|𝑢|²=0. (60) For the above choices of 𝑢,𝑥,𝑦, and 𝑡, we next show that 𝑤_𝑢≠0. From (44) 𝑤_𝑢(𝑢)=𝑢²𝐺^′′+𝑢𝐺^′−2Re𝐺=𝑢𝐺^′𝑢𝐺^′′𝐺^′+1−2Re𝐺. (61) However, (57) and (58) give 𝐺^′(𝑢)=1()=11−𝑥𝑢1−𝑦𝑢|1−𝑥𝑢|²,𝑢𝐺^′′𝐺^′+1=Re1+𝑥𝑢,(𝑢)1−𝑥𝑢Re𝐺=Re^𝑢₀𝑑𝑧()=1−𝑥𝑧1−𝑦𝑧¹₀Re𝑢𝑑𝑡|1−𝑥𝑡𝑢|². (62) Since |1−𝑥𝑢|=1, it is geometrically clear that |1−𝑥𝑡𝑢|≤1 and consequently Re𝐺/Re𝑢>1. Hence (61) becomes 𝑤_𝑢(𝑢)|𝑢|=𝑢1−²|1−𝑥𝑢|⁴−¹₀2Re𝑢𝑑𝑡|1−𝑥𝑡𝑢|²=𝑢1−|𝑢|²−¹₀2Re𝑢𝑑𝑡|1−𝑥𝑡𝑢|². (63) If the last expression is zero, then 𝑢 should be chosen real satisfying 𝑢=Re𝑢 and 1−|𝑢|²=∫¹₀(2𝑑𝑡/|1−𝑥𝑡𝑢|²)>2. Since this is impossible, we conclude, for arbitrary 𝑢≠0 and with the above choices of 𝑥,𝑦 and 𝑡=1/2, that 𝑤_𝑢(𝑢)≠0 and consequently 𝐽_𝑤(𝑢)>0. In general, it follows from (44) that 𝐽_𝑤(0)=−|𝐺^′′(0)|², and this is negative for certain choices of 𝐺, especially for 𝐺(𝑢)=𝑢/(1−𝑢). Hence there is no uniform disk for the family on which 𝐽_𝑤(𝑢)<0. This completes the proof of part (a).
For the proof of part (b), choose 𝑢=−𝑖|𝑢|,𝑥=1,𝑦=−1, and 𝑡=1/2. From (57), and with the present choices, we conclude that 𝐺^′(|𝑢=1−𝑖|)(|𝑢|𝑖|𝑢|𝑖)=11+)(1−1+|𝑢|²,Re𝐺(−𝑖|𝑢|)=Re₀^{−|𝑢|𝑖}𝑑𝑧(1+𝑧)(1−𝑧)=Re¹₀−|𝑢|𝑖𝑑𝑡(1+𝑡|𝑢|𝑖)(1−𝑡|𝑢|𝑖)=0. (64) It is clear from (27) that 𝑤(−|𝑢|𝑖)=−|𝑢|²1+|𝑢|²−11+|𝑢|²=𝑤(|𝑢|𝑖). (65) Hence 𝑤 is not univalent near 0.

We conclude our exposition with several examples.

Example 18. Let 𝐺(𝑢)=𝑢. Then 𝑤(𝑢)=−1−|𝑢|² and |𝑤_𝑢/𝑤_𝑢|=1. Hence the surface Ψ degenerates.

Example 19. Choose 𝐺(𝑢)=𝑢². Then 𝑤(𝑢)=𝑢³−|𝑢|²𝑢−2𝑢. From (44), 𝑤_𝑢𝑤_𝑢=3𝑢²−𝑢²−2−2|𝑢|²=3|𝑢|²21+|𝑢|²11−3𝑢²𝑢²<1, (66) and thus the corresponding 𝑤 is locally one-to-one in 𝕌.

When 𝑢=𝑒^𝑖𝑡,𝑤(𝑢)=𝑢³−3𝑢=𝑒^3𝑖𝑡−3𝑒^−𝑖𝑡, Im𝑑𝑤/𝑑𝑡𝑤(𝑡)𝑒=3Re^3𝑖𝑡+𝑒^−𝑖𝑡𝑒^3𝑖𝑡−3𝑒^−𝑖𝑡𝑒=3Re^4𝑖𝑡+1𝑒^4𝑖𝑡()−3=−61+cos4𝑡3−𝑒^4𝑖𝑡2≤0. (67) Hence 𝑤 is univalent on |𝑢|=1, and since 𝐽_𝑤(𝑢)≠0 in 𝕌, it must be univalent in 𝕌. Figures 2 and 3 show that the associated L-surface Ψ and the corresponding L-minimal surface Φ are total graphs.

Figure 2 

The associated L-surface Ψ when 𝐺(𝑢)=𝑢2.

Figure 3 

The L-minimal surface Φ when 𝐺(𝑢)=𝑢2.

Example 20. Let 𝐺(𝑢)=𝑢/(1−𝑥𝑢),|𝑥|=1. It follows from (44) that 𝑤_𝑢=2𝑥𝑢²(1−𝑥𝑢)³+𝑢(1−𝑥𝑢)²𝑢−2Re,1−𝑥𝑢𝑤_𝑢=−2𝑥(1−𝑥𝑢)³−𝑢(1−𝑥𝑢)². (68) Consequently 𝑤_𝑢𝑤_𝑢≤|𝑢|²+|𝑢|+2|𝑢|(1+|𝑢|)²(|𝑢|𝑢.1−|)(2+|) (69)

The value of this expression ranges between 0 at 0 and ∞ at |𝑢|=1. Hence placing it less than 1 and solving for |𝑢| give a uniform disk |𝑢|<0.32471 for all 𝑥. The corresponding 𝑤 is then locally univalent, with 𝐽_𝑤(𝑢)<0 in |𝑢|<0.32471.

Note that in the case 𝐺(𝑢)=𝑢/(1+𝑢),𝑤_𝑢→−1,𝑤_𝑢→0 when 𝑢→1. Hence 𝐽_𝑤→1. This implies that 𝑤 may not be locally univalent in all of 𝕌. Figures 4 and 5 show that neither the associated L-surface Ψ nor the corresponding L-surface Φ is a total graph.

Figure 4 

The associated L-surface Ψ when 𝐺(𝑢)=𝑢/(1−𝑢).

Figure 5 

The L-surface Φ when 𝐺(𝑢)=𝑢/(1−𝑢).

Acknowledgments

The work presented here was supported in parts by research grants from the American University of Sharjah and Universiti Sains Malaysia.