Research Article | Open Access
Yusuf Abu Muhanna, Rosihan M. Ali, "Biharmonic Maps and Laguerre Minimal Surfaces", Abstract and Applied Analysis, vol. 2013, Article ID 843156, 9 pages, 2013. https://doi.org/10.1155/2013/843156
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 is its associated surface , where lies in the unit disk. In this paper, the projection of the surface associated to a Laguerre minimal surface is shown to be biharmonic. A complete characterization of 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 to be a graph. Estimates of the Gaussian curvature to the Laguerre minimal surface are obtained, and several illustrative examples are given.
Surfaces in 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.
A Laguerre minimal (L-minimal) surface is an 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 [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 , real, is the associated L-surface, and is the corresponding isotropic graph. The surface is called an associated L-minimal surface when is L-minimal. Note that itself need not be L-minimal.
In Section 3, we write in the form where lies in the unit disk . Assuming that is a L-minimal surface, it is shown in Lemma 3 that the projection and of are biharmonic. Additionally, if the isotropic map is harmonic, then is harmonic and takes the form with analytic and harmonic.
In Theorem 7, the associated L-minimal surface is completely characterized when the isotropic map is harmonic. It is shown that is an associated L-minimal surface if and only if the projection map is a biharmonic map of the form (3). The associated surface is given more emphasis than the L-minimal surface because the coordinates of 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 and .
Section 4 considers the case when is a graph; that is, it is a nonparametric surface. When the function is univalent and biharmonic, it is shown in Theorem 10 that 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 over which is locally a graph. In Theorem 14, a universal disk is obtained over which is a graph when 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  if , and is biharmonic if , where is the Laplacian operator. It is easy to show that a mapping is biharmonic in a simply connected domain if and only if has the representation where and are complex-valued harmonic functions in , with , being analytic in (for details see [7, 8, 10–12]). The Jacobian of a map is given by
2. Laguerre Geometry
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 given by . Hence the -mean curvature is given by where is the Laplacian of , while the -Gaussian curvature is These curvatures are much easier to deal with compared to the Euclidean curvatures.
2.2. 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 .
2.3. Laguerre Geometry
In Laguerre geometry, a point on a surface in 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, . 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 be an oriented plane with unit normal vector , and associate with the point . Next replace with its stereographic image under the projection of the unit sphere from onto the plane . Then the isotropic image of is defined as
If we let and write , then , and the unit vector in complex variables becomes In this case, (14) becomes
2.5. Laguerre Surface
Let be a Laguerre surface in . Any regular point on is thus represented as in (16). Denote the corresponding isotropic surface by with given by (16). By duality, their corresponding curvatures are related by Blaschke  defined the middle tangent sphere to be the tangent to the tangent plane with radius where , , and , are the principal curvatures of the L-surface . Let denote the middle surface consisting of centers of the middle spheres. It is shown in  that is invariant under Laguerre transformations.
If, in (16), on is given by , then [5, 6] (), and when is minimal, then The latter implies that is biharmonic. Assume now that is given by the function . Since , it follows from (20) that is also biharmonic.
This leads to the following result.
3. Projection of L-Minimal Surface onto a Plane
In (16), the Laguerre surface is expressed in terms of the Laguerre coordinates . 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 ). Let be the graph of the biharmonic function . Then the parametric equations of the corresponding L-minimal surface are given by
Throughout this section, it is assumed that the L-minimal surface is parametrized by , with in the unit disk .
Equations (22) will first be written in terms of complex variables. For this purpose, let Then from (22), the projection of the surface and the height becomes and the coordinates of (see (2)) are Evidently, The relations (25) and (26) yield the following general lemma.
Lemma 3. (a) If is biharmonic in , then and are biharmonic.
(b) If is harmonic and analytic, then is harmonic and , where the harmonic function is given by
Proof. (a) If is biharmonic in , then
where , are harmonic. It follows directly by differentiation that
in light that and 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 into (25) yields Also the second equation of (25) shows that is harmonic.
Corollary 4. (a) If is an associated L-minimal surface parametrized by the unit disk , then the projections and are biharmonic.
(b) If is an associated L-minimal surface parametrized by the unit disk 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.
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.
Corollary 6. If is given by Lemma 3(b), then
Combining the above lemmas and corollaries results in the following characterization of minimal surfaces with harmonic isotropic maps.
Theorem 7. Let be a Laguerre surface. A surface is an associated L-minimal surface with a harmonic isotropic map if and only if is given by where is a biharmonic map given by (27), and by (34) is harmonic in .
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
We conclude this section with an estimate for when belongs to the class consisting of univalent analytic functions in normalized by and .
Proposition 9. Let and the corresponding associated L-minimal surface be given as in Theorem 7. Then
Proof. It is known [16, p. 21] that for , Thus which leads to the desired inequalities.
4. The Associated L-Surface Is a Graph
This section looks at the case when is a graph; that is, when is a nonparametric surface. Interestingly, the graph of the associated L-minimal surface is closely connected to its corresponding projection map .
Figure 1 makes this relationship evident and gives rise to the following theorem.
Theorem 10. An associated L-surface parametrized by the unit disk is a graph if and only if is a univalent biharmonic map.
The following lemma gives the derivatives of the function of the graph surface.
Lemma 11. Let and be given as in Lemma 5. If is univalent with Jacobian for all , then
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 and which is a consequence of Landau’s theorem for biharmonic maps. This was first proved in  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 be a surface given by , , where is analytic and harmonic is given by (27). If is bounded by a constant , and , then there are uniform constants and so that and are univalent on the disk , and the image of this disk contains a disk on which the surfaces are graphs.
Proof. The Jacobian of is given by
It follows from (44) that . If is bounded, then and consequently is bounded. It can now be deduced from Theorem 1 in [7, 9, 12] that there are uniform constants and so that is univalent on the disk whose image contains the disk . Consequently the surface is a graph above such a disk.
Clearly is univalent on each circle . Suppose now that there are and with so that . Then But it was shown in the proof of Theorem 1 in [7, 9, 12] that Hence As , choose so that and the result follows.
Corollary 13. Let be given as in Theorem 12. Then is univalent in and the corresponding surface is a graph.
Interestingly when , a similar result is obtained without imposing the boundedness condition. Recall that is the class of univalent analytic functions normalized by and .
Theorem 14. Let , , , polynomials with positive coefficients of degree 2 and 3, respectively (see (51) and (53)), and described by (56). Then there are two uniform radii and satisfying so that the corresponding and are univalent in , and the image of contains a disk . In this case, the surfaces and are graphs on .
Proof. Now and . It follows from (44) that , , and . The distortion estimates for the class [16, p. 21] are
From (44), these inequalities imply that
where can be chosen as a polynomial of degree with positive coefficients and .
The distortion inequality also implies that The latter inequality together with the distortion inequalities imply that where is taken to be a polynomial with positive coefficients of degree and .
Let be two points near in a disk . It follows from (51) and (53) that Now choose so that to deduce that is univalent in .
Let satisfy and let . Then the distance . If we choose and in (54), then Thus choose .
An argument similar to the proof of Theorem 1 in [7, 9, 12] gives the result for and consequently for and .
Corollary 15. Let and be given by Theorem 14. If , then the corresponding associated L-minimal surface is a graph.
Remark 16. (1) A result similar to Theorem 14 can be obtained for the L-surface .
(2) Theorem 14 is not true for the class . The following proposition shows that there is no uniform disk on which the surface is a graph for all .
Proposition 17. Let be the set of all convex univalent functions given by where , and , and let be the corresponding biharmonic map given by (27).(a)There is no uniform disk centered at where .(b)There is no uniform disk on which is univalent and, consequently, no uniform disk on which is a graph.
Proof. For ,
First we show that , for any , and for the choices , satisfying with being the intersection points between the circles and . These conditions imply that . In this case, (58) gives
Hence (44) becomes
For the above choices of , and , we next show that . From (44)
However, (57) and (58) give
Since , it is geometrically clear that and consequently . Hence (61) becomes
If the last expression is zero, then should be chosen real satisfying and . Since this is impossible, we conclude, for arbitrary and with the above choices of and , that and consequently . In general, it follows from (44) that , and this is negative for certain choices of , especially for . Hence there is no uniform disk for the family on which . This completes the proof of part (a).
For the proof of part (b), choose , and . From (57), and with the present choices, we conclude that It is clear from (27) that Hence is not univalent near .
We conclude our exposition with several examples.
Example 18. Let . Then and . Hence the surface degenerates.
Example 19. Choose . Then . From (44), and thus the corresponding is locally one-to-one in .
Example 20. Let . It follows from (44) that Consequently
The value of this expression ranges between at and at . Hence placing it less than and solving for give a uniform disk for all . The corresponding is then locally univalent, with in .
Note that in the case when . Hence . This implies that may not be locally univalent in all of . Figures 4 and 5 show that neither the associated L-surface nor the corresponding L-surface is a total graph.
The work presented here was supported in parts by research grants from the American University of Sharjah and Universiti Sains Malaysia.
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