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Michael Dorff 1 and Ryan Viertel 1 and Magdalena Woloszkiewicz 2

Recommended by Ilya M. Spitkovsky

1, Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
2, Department of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland

Received 11 May 2012; Accepted 14 July 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Consider a surface M in ^{... 3} .

Definition 1.1.

The normal curvature at a point p ∈ M in the w direction is [figure omitted; refer to PDF] where n is the unit normal at p , w is a tangent vector of M at p , and α is an arclength parametrization of the curve created by the intersection of M with the the plane containing w and n .

Definition 1.2.

A minimal surface is a surface M with mean curvature [figure omitted; refer to PDF] at all points p ∈ M , where _{k 1} and _{k 2} are the maximum and minimum normal curvature values at p .

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 ^{... 3} has a local isothermal parametric representation of the following form: [figure omitted; refer to PDF] where each _{[straight phi] k} is analytic, ^{[varphi] 2} = ^{[straight phi] 1 2} + ^{[straight phi] 2 2} + ^{[straight phi] 3 2} = 0 , and | [varphi] ^{| 2} = | _{[straight phi] 1} ^{| 2} + | _{[straight phi] 2} ^{| 2} + | _{[straight phi] 3} ^{| 2} ...0; 0 , and is finite.

A common way to use the Weierstrass representation is in terms of the Gauss map, G , and height differential, d h . 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 [figure omitted; refer to PDF]

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 f ( x , y ) = u ( x , y ) + i v ( x , y ) defined in a domain D ⊂ ... is a planar harmonic mapping or harmonic function in D if u and v are real harmonic functions in D .

In this paper, we will only consider harmonic functions defined on the unit disk, ... = { z : | z | < 1 } .

Theorem 1.5 (see [4]).

If f = u + i v is harmonic in ... , then f can be written as f = h + g ¯ , where h and g are analytic.

Definition 1.6.

The dilatation of f = h + g ¯ is ω ( z ) = ^{g [variant prime]} ( z ) / ^{h [variant prime]} ( z ) .

Theorem 1.7 (see [5]).

The harmonic function f = h + g ¯ is locally univalent and orientation preserving in ... if and only if | w ( z ) | < 1 , for all z ∈ ... .

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 _{x 1} _{x 2} -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 f = h + g ¯ . One advantage of this representation is that the univalence of the harmonic mapping f guarantees that the corresponding minimal surface will be a graph over the image of f and will thus be embedded. These ideas are summarized in the following theorem.

Theorem 1.8 (Weierstrass representation ( h , g ) , see [6]).

Let f = h + g ¯ be an orientation-preserving harmonic univalent mapping of a domain ... onto some domain Ω with dilatation, ω , that is, the square of an analytic function in ... . Then [figure omitted; refer to PDF] gives an isothermal parametrization of a minimal graph whose projection onto the complex plane is f ( ... ) . Conversely, if a minimal graph is parameterized by orientation-preserving isothermal parameters z = x + i y ∈ ... , then the projection onto its base plane defines a harmonic univalent mapping f ( z ) = Re { h ( z ) + g ( z ) } + i Im { h ( z ) - g ( z ) } 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 f = h + g ¯ by [figure omitted; refer to PDF]

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 Ω is convex in the direction ^{e i α} if for every a ∈ ... the set [figure omitted; refer to PDF] 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 Ω .

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 f = h + g ¯ locally univalent in ... is a univalent mapping of ... onto a domain convex in the ^{e i α} direction if and only if ψ = h - ^{e 2 i α} g is a analytic univalent mapping of ... onto a domain convex in the ^{e i α} direction.

We will also need the following from Hengartner and Schober [7].

Condition 1.

Let ψ be a nonconstant analytic function in ... , and there exist sequences ^{z n [variant prime]} , ^{z n [variant prime][variant prime]} converging to z = 1 , z = - 1 , respectively, such that [figure omitted; refer to PDF]

Theorem 2.3 (see [7]).

Suppose that ψ is analytic and nonconstant in ... . Then [figure omitted; refer to PDF] if and only if ψ is univalent in ... , ψ ( ... ) 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 ψ ( 1 ) and ψ ( - 1 ) 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 _{M 1} , ... , _{M n} : ... [arrow right] ^{... 3} be minimal graphs with isothermal parametrizations _{[varphi] k} = Re ( ^{[varphi] k 1} , ^{[varphi] k 2} , ^{[varphi] k 3} ) = Re ^∫ ( ( 1 / 2 ) ( 1 / _{G k} - _{G k} ) , ( i / 2 ) ( 1 / _{G k} + _{G k} ) , 1 ) d _{h k} , where _{G k} is the Gauss map and d _{h k} is the height differential ( k = 1 , ... , n ) . Let

(1) _{G k} = _{G 1} for each k ,

(2) the projection of _{M k} on to the _{x 1} _{x 2} -plane , _{Ω k} , be CID,

(3) Condition 1 holds for each ^{[varphi] k 1} , for k = 1 , ... , n .

If ⁽^{_{t 1}}^{[varphi] 1 1}^{+ ... +}^{_{t n}}^{[varphi] n 1}^{) [variant prime]} ...0; 0 , then M = _{t 1} _{M 1} + ... + _{t n} _{M n} is a minimal graph for all 0 ...4; _{t k} ...4; 1 , where _{t 1} + ... + _{t n} = 1 with G = _{G 1} and d h = _{t 1} d _{h 1} + ... + _{t n} d _{h n} .

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, _{M k} , onto the _{x 1} _{x 2} -plane defines a univalent harmonic mapping _{f k} = _{h k} + _{g k} ¯ with dilatation _{ω k} = ^{g k [variant prime]} / ^{h k [variant prime]} . Let [figure omitted; refer to PDF] We will show that f is a univalent harmonic mapping of ... onto a domain convex in the imaginary direction. Since _{G 1} = _{G k} , we see from (1.6) that _{ω 1} = _{ω k} for all k = 2 , ... , n . Also, ω = ^{g [variant prime]} / ^{h [variant prime]} equals _{ω 1} because [figure omitted; refer to PDF] Hence, f is locally univalent since | ω ( z ) | = | _{ω 1} ( z ) | < 1 for every z ∈ ... . We now will show that h + g 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 _{h k} + _{g k} is univalent and CID. Also, _{h k} + _{g k} satisfies Condition 1 since Re { _{h k} + _{g k} } = Re { ^{[varphi] k 1} } . Applying Theorem 2.3, we have [figure omitted; refer to PDF] for every k ∈ { 1,2 , ... , n } . Then [figure omitted; refer to PDF] Since ^{h [variant prime]} + ^{g [variant prime]} = ( _{t 1} ^{[varphi] 1 1} + ... + _{t n} ^{[varphi] n 1} ) [variant prime] ...0; 0 , by applying Theorem 2.3 in the other direction, we have that h + g is a conformal univalent mapping of ... onto a CID domain. Thus, by Theorem 2.2, f is a harmonic univalent mapping with f ( ... ) being convex in the imaginary direction. We can now apply the Weierstrass representation from Theorem 1.8 to lift f = h + g ¯ to a minimal graph M ~ = ( u , v , F ( u , v ) ) . Notice that [figure omitted; refer to PDF] Similarly, v = Im { h - g } = _{t 1} Re { ^{[varphi] 1 2} } + ... + _{t n} Re { ^{[varphi] n 2} } . Finally, [figure omitted; refer to PDF] Thus, M ~ = _{t 1} _{M 1} + ... + _{t n} _{M n} = M .

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 _{G 1} = - 1 / z and d _{h 1} = z / ( 1 - ^{z 2} ^{) 2} d z , where z ∈ ... . Using (1.6), we get [figure omitted; refer to PDF] Notice that ^{[varphi] 1 1} = _{h 1} + _{g 1} = ( 1 / 2 ) log ( ( 1 + z ) / ( 1 - z ) ) and Re { ( 1 - ^{z 2} ) ( ⁽^{[varphi] 1 1}^{) [variant prime]} ) } = 1 ...5; 0 . So by Theorem 2.3, ^{[varphi] 1 1} ( ... ) is convex in the imaginary direction and ^{[varphi] 1 1} satisfies Condition 1. Since _{ω 1} = - ^{z 2} , the harmonic map _{f 1} = _{h 1} + _{g 1} ¯ lifts to a minimal graph by Theorem 1.8.

Similarly, the Weierstrass data _{G 2} = - 1 / z and d _{h 2} = ( z / ( ^{z 4} - 1 ) ) d z , where z ∈ ... , results in a graph of Scherk's doubly periodic surface with _{ω 2} = - ^{z 2} , and [figure omitted; refer to PDF] Now ^{[varphi] 2 1} = _{h 2} + _{g 2} = ( i / 2 ) log ( ( i + z ) / ( i - z ) ) and [figure omitted; refer to PDF] Again, by Theorem 2.3, ^{[varphi] 2 1} ( ... ) is convex in the imaginary direction and ^{[varphi] 2 1} satisfies Condition 1.

Since both parametrizations satisfy the hypotheses of Theorem 3.1, the harmonic map f = t ( _{h 1} + _{g 1} ¯ ) + ( 1 - t ) ( _{h 2} + _{g 2} ¯ ) will lift to a minimal graph over ... with Weierstrass data G = - 1 / z and d h = ( t ) d _{h 1} + ( 1 - t ) d _{h 2} for all 0 ...4; t ...4; 1 (see Figure 1).

Figure 1: Images of concentric circles under f and corresponding minimal surfaces for various values of t in Example 3.3.