Abstract
Given a collection of minimal graphs, , with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions on so that a convex combination of them will be a minimal graph. We will then provide two examples, taking a convex combination of Scherk's doubly periodic surface with the catenoid and Enneper's surface, respectively.
1. Introduction
Consider a surface in .
Definition 1.1. The normal curvature at a point in the direction is where is the unit normal at , is a tangent vector of at , and is an arclength parametrization of the curve created by the intersection of with the the plane containing and .
Definition 1.2. A minimal surface is a surface with mean curvature at all points , where and are the maximum and minimum normal curvature values at .
The standard tool for representing minimal surfaces is the Weierstrass representation as the following theorem demonstrates.
Theorem 1.3 (Weierstrass representation). Every regular minimal surface in has a local isothermal parametric representation of the following form: where each is analytic, , and , and is finite.
A common way to use the Weierstrass representation is in terms of the Gauss map, , and height differential, . These are analytic functions that provide information about the geometry of the surface (see [1, 2]). When represented in these terms, the Weierstrass representation becomes
Another way to represent minimal surfaces is in terms of planar harmonic mappings. Planar harmonic mappings have been studied independently of minimal surfaces and results about them can be used to establish results about minimal surfaces (see [3]). The following definitions and theorems will be useful in this discussion.
Definition 1.4. A continuous function defined in a domain is a planar harmonic mapping or harmonic function in if and are real harmonic functions in .
In this paper, we will only consider harmonic functions defined on the unit disk, .
Theorem 1.5 (see [4]). If is harmonic in , then f can be written as , where and are analytic.
Definition 1.6. The dilatation of is .
Theorem 1.7 (see [5]). The harmonic function is locally univalent and orientation preserving in if and only if , for all .
Notice that the first and second coordinates of the Weierstrass representation (1.3) are the real part of analytic functions and are thus harmonic. The projection of a minimal surface onto the -plane can then be viewed as the image of a planar harmonic mapping in the complex plane. This gives rise to another Weierstrass representation in terms of the planar harmonic mapping . One advantage of this representation is that the univalence of the harmonic mapping guarantees that the corresponding minimal surface will be a graph over the image of and will thus be embedded. These ideas are summarized in the following theorem.
Theorem 1.8 (Weierstrass representation , see [6]). Let be an orientation-preserving harmonic univalent mapping of a domain onto some domain with dilatation, , that is, the square of an analytic function in . Then gives an isothermal parametrization of a minimal graph whose projection onto the complex plane is . Conversely, if a minimal graph is parameterized by orientation-preserving isothermal parameters , then the projection onto its base plane defines a harmonic univalent mapping whose dilatation is the square of an analytic function.
It can be derived from (1.4) and (1.5) that the Gauss map and height differential are related to the harmonic mapping by
2. Harmonic Univalent Functions
We wish to establish conditions on a collection of minimal graphs to guarantee that a convex combination of them will be a minimal graph. To do this, we will make use of Theorem 1.8 and some established results concerning the univalence of planar harmonic mappings. We will first need some background information.
Definition 2.1. A domain is convex in the direction if for every the set is either connected or empty. In particular, a domain is convex in the imaginary direction (CID) if every line parallel to the imaginary axis has a connected intersection with .
In general, it is difficult to establish the univalence of a planar harmonic mapping. The shearing technique of Clunie and Sheil-Small however provides one way to do this.
Theorem 2.2 (see [4]). A harmonic function locally univalent in is a univalent mapping of onto a domain convex in the direction if and only if is a analytic univalent mapping of onto a domain convex in the direction.
We will also need the following from Hengartner and Schober [7].
Condition 1. Let be a nonconstant analytic function in , and there exist sequences , converging to , , respectively, such that
Theorem 2.3 (see [7]). Suppose that is analytic and nonconstant in . Then if and only if is univalent in , is convex in the imaginary direction, and Condition 1 holds.
Note that the normalization in (2.2) can be thought of in some sense as if and are the right and left extremes in the image domain in the extended complex plane.
3. Convex Combinations of Minimal Graphs
We are now ready to prove our main result.
Theorem 3.1. Let be minimal graphs with isothermal parametrizations , where is the Gauss map and is the height differential . Let (1) for each ,(2)the projection of on to the -plane, be CID,(3)Condition 1 holds for each , for.If , then is a minimal graph for all , where with and .
Remark 3.2. This definition of the convex combinations of minimal graphs is very close to the definition of the sum of two complete minimal surfaces with finite total curvature given by Rosenberg and Toubiana in [8].
Proof. By Theorem 1.8, the projection of each minimal graph, , onto the -plane defines a univalent harmonic mapping with dilatation . Let We will show that is a univalent harmonic mapping of onto a domain convex in the imaginary direction. Since , we see from (1.6) that for all . Also, equals because Hence, is locally univalent since for every . We now will show that is a univalent analytic mapping of onto a domain convex in the imaginary direction, so we can apply the shearing theorem. By Theorem 2.2, we know that each is univalent and CID. Also, satisfies Condition 1 since . Applying Theorem 2.3, we have for every . Then Since , by applying Theorem 2.3 in the other direction, we have that is a conformal univalent mapping of onto a CID domain. Thus, by Theorem 2.2, is a harmonic univalent mapping with being convex in the imaginary direction. We can now apply the Weierstrass representation from Theorem 1.8 to lift to a minimal graph . Notice that Similarly, . Finally, Thus, .
Using this theorem we can take a convex combination of several classical minimal surfaces to produce new minimal graphs.
Example 3.3. Consider the Weierstrass data for the catenoid and , where . Using (1.6), we get
Notice that and . So by Theorem 2.3, is convex in the imaginary direction and satisfies Condition 1. Since , the harmonic map lifts to a minimal graph by Theorem 1.8.
Similarly, the Weierstrass data and , where , results in a graph of Scherk's doubly periodic surface with , and
Now and
Again, by Theorem 2.3, is convex in the imaginary direction and satisfies Condition 1.
Since both parametrizations satisfy the hypotheses of Theorem 3.1, the harmonic map will lift to a minimal graph over with Weierstrass data and for all (see Figure 1).
Example 3.4. The Weierstrass data and gives a parametrization of Enneper's surface. The functions and give a different paramaterization of Scherk's doubly periodic surface than in Example 3.3. Notice that and . By Theorem 2.3, and satisfy Condition 1. Thus, both surfaces satisfy the hypotheses of Theorem 3.1, and the function will lift to a minimal graph for all (see Figure 2).
Using this method, we were able to show that the combinations shown in Figure 3 are also minimal graphs.
(a) Enneper-Helicoid
(b) Enneper-Scherk's doubly periodic
(c) Enneper-Scherk's singly periodic
(d) Helicoid-Scherk's doubly periodic
(e) Helicoid-Scherk's doubly periodic 2
(f) Helicoid-Scherk's singly periodic
Remark 3.5. In the examples of this paper we have only taken convex combinations using two minimal graphs. It is possible, however, to take a convex combination of any finite number of minimal graphs using Theorem 3.1.
Area for Further Investigation
The condition that two minimal graphs share the same Gauss map does not seem to be necessary. It would be interesting to find an example of two minimal graphs with different Gauss maps such that a convex combination of them is a minimal graph.