# Convex Combinations of Minimal Graphs

Authors: Michael Dorff; Ryan Viertel; Magdalena Wołoszkiewicz
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Given a collection of minimal graphs,M1,M2,…,Mn, with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions on M1,M2,…,Mn 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.


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## Body

## 1. Introduction

Consider a surfaceM in ℝ3.Definition 1.1.
Thenormal curvature at a point p∈M in the w direction is
(1.1)k(w)=α′′⋅n,
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.
Aminimal surface is a surface M with mean curvature
(1.2)H=k1+k22=0,
at all points p∈M, where k1 and k2 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 inℝ3 has a local isothermal parametric representation of the following form:
(1.3)X(z)=(Re∫φ1(z)dz,Re∫φ2(z)dz,Re∫φ3(z)dz),
where each φk is analytic, ϕ2=φ12+φ22+φ32=0, and |ϕ|2=|φ1|2+|φ2|2+|φ3|2≠0, and is finite.A common way to use the Weierstrass representation is in terms of the Gauss map,G, and height differential, dh. 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
(1.4)X(z)=Re∫(12(1G-G),i2(1G+G),1)dh.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 functionf(x,y)=u(x,y)+iv(x,y) defined in a domain D⊂ℂ 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,𝔻={z:|z|<1}.Theorem 1.5 (see [4]).
Iff=u+iv is harmonic in 𝔻, then f can be written as f=h+g¯, where h and g are analytic.Definition 1.6.
Thedilatation of f=h+g¯ is ω(z)=g′(z)/h′(z).Theorem 1.7 (see [5]).
The harmonic functionf=h+g¯ is locally univalent and orientation preserving in 𝔻 if and only if |w(z)|<1, for all z∈𝔻.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 x1x2-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]).
Letf=h+g¯ be an orientation-preserving harmonic univalent mapping of a domain 𝔻 onto some domain Ω with dilatation, ω, that is, the square of an analytic function in 𝔻. Then
(1.5)X(z)=(Re{h(z)+g(z)},Im⁡{h(z)-g(z)},2Im⁡{∫0zg′(ζ)h′(ζ)dζ})
gives an isothermal parametrization of a minimal graph whose projection onto the complex plane is f(𝔻). Conversely, if a minimal graph is parameterized by orientation-preserving isothermal parameters z=x+iy∈𝔻, then the projection onto its base plane defines a harmonic univalent mapping f(z)=Re{h(z)+g(z)}+iIm{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
(1.6)G=-ih′g′=-iω,dh=-2ig′h′dz.


## 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 Theorem1.8 and some established results concerning the univalence of planar harmonic mappings. We will first need some background information.Definition 2.1.
A domainΩ is convex in the directioneiα if for every a∈ℂ the set
(2.1)Ω∩{a+teiα:t∈R}
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 functionf=h+g¯ locally univalent in 𝔻 is a univalent mapping of 𝔻 onto a domain convex in the eiα direction if and only if ψ=h-e2iαg is a analytic univalent mapping of 𝔻 onto a domain convex in the eiα direction.We will also need the following from Hengartner and Schober [7].Condition 1.
Letψ be a nonconstant analytic function in 𝔻, and there exist sequences zn′, zn′′ converging to z=1, z=-1, respectively, such that
(2.2)lim⁡n→∞Re{ψ(zn′)}=sup⁡|z|<1Re{ψ(z)},lim⁡n→∞Re{ψ(zn′′)}=inf⁡|z|<1Re{ψ(z)}.Theorem 2.3 (see [7]).
Suppose thatψ is analytic and nonconstant in 𝔻. Then
(2.3)Re{(1-z2)ψ′(z)}≥0,z∈D
if and only if ψ is univalent in 𝔻, ψ(𝔻) 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.
LetM1,…,Mn:𝔻→ℝ3 be minimal graphs with isothermal parametrizations ϕk=Re(ϕk1,ϕk2,ϕk3)=Re∫((1/2)(1/Gk-Gk),(i/2)(1/Gk+Gk),1)dhk, where Gk is the Gauss map and dhk is the height differential (k=1,…,n). Let (1)
Gk=G1 for each k,(2)
the projection ofMk on to the x1x2-plane,Ωk, be CID,(3)
Condition1 holds for each ϕk1, fork=1,…,n.If (t1ϕ11+⋯+tnϕn1)′≠0, then M=t1M1+⋯+tnMn is a minimal graph for all 0≤tk≤1, where t1+⋯+tn=1 with G=G1 and dh=t1dh1+⋯+tndhn.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 Theorem1.8, the projection of each minimal graph, Mk, onto the x1x2-plane defines a univalent harmonic mapping fk=hk+gk¯ with dilatation ωk=gk′/hk′. Let
(3.1)f=h+g¯=(t1h1+⋯+tnhn)+(t1g1+⋯+tngn)¯.
We will show that f is a univalent harmonic mapping of 𝔻 onto a domain convex in the imaginary direction. Since G1=Gk, we see from (1.6) that ω1=ωk for all k=2,…,n. Also, ω=g′/h′ equals ω1 because
(3.2)ω=t1g1′+⋯+tngn′t1h1′+⋯+tnhn′=t1h1′ω1+⋯+tnhn′ωnt1h1′+⋯+tnhn′=ω1.
Hence, f is locally univalent since |ω(z)|=|ω1(z)|<1 for every z∈𝔻. We now will show that h+g is a univalent analytic mapping of 𝔻 onto a domain convex in the imaginary direction, so we can apply the shearing theorem. By Theorem 2.2, we know that each hk+gk is univalent and CID. Also, hk+gk satisfies Condition 1 since Re{hk+gk}=Re{ϕk1}. Applying Theorem 2.3, we have
(3.3)Re{(1-z2)(hk′(z)+gk′(z))}≥0
for every k∈{1,2,…,n}. Then
(3.4)Re{(1-z2)(h′(z)+g′(z))}=Re{(1-z2)[t1(h1′(z)+g1′(z))+⋯+tn(hn′(z)+gn′(z))]}=t1Re{(1-z2)(h1′(z)+g1′(z))}+⋯+tnRe{(1-z2)(hn′(z)+gn′(z))}≥0.
Since h′+g′=(t1ϕ11+⋯+tnϕn1)′≠0, by applying Theorem 2.3 in the other direction, we have that h+g is a conformal univalent mapping of 𝔻 onto a CID domain. Thus, by Theorem 2.2, f is a harmonic univalent mapping with f(𝔻) 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
(3.5)u=Re{h+g}u=Re{(t1h1+t1g1)+⋯+(tnhn+tngn)}u=t1Re{ϕ11}+⋯+tnRe{ϕn1}.
Similarly, v=Im{h-g}=t1Re{ϕ12}+⋯+tnRe{ϕn2}. Finally,
(3.6)F(u,v)=2Im⁡{∫0z‍(t1g1′(ζ)+⋯+tngn′(ζ))(t1h1′(ζ)+⋯+tnhn′(ζ))dζ}F(u,v)=2Im⁡{∫0z‍(t1ω1(ζ)h1′(ζ)+⋯+tnωn(ζ)hn′(ζ))(t1h1′(ζ)+⋯+tnhn′(ζ))dζ}F(u,v)=2Im⁡{∫0z‍ω1(ζ)(t1h1′(ζ)+⋯+tnhn′(ζ))dζ}F(u,v)=2Im⁡{∫0z(t1g1′(ζ)h1′(ζ)+⋯+tngn′(ζ)hn′(ζ))dζ}F(u,v)=t1Re{ϕ13}+⋯+tnRe{ϕn3}.
Thus, M~=t1M1+⋯+tnMn=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 catenoidG1=-1/z and dh1=z/(1-z2)2dz, where z∈𝔻. Using (1.6), we get
(3.7)h1=14log⁡⁡(1+z1-z)+12z1-z2,g1=14log⁡(1+z1-z)-12z1-z2.
Notice that ϕ11=h1+g1=(1/2)log((1+z)/(1-z)) and Re{(1-z2)((ϕ11)′)}=1≥0. So by Theorem 2.3, ϕ11(𝔻) is convex in the imaginary direction and ϕ11 satisfies Condition 1. Since ω1=-z2, the harmonic map f1=h1+g1¯ lifts to a minimal graph by Theorem 1.8.
Similarly, the Weierstrass dataG2=-1/z and dh2=(z/(z4-1))dz, where z∈𝔻, results in a graph of Scherk's doubly periodic surface with ω2=-z2, and
(3.8)h2=14log⁡(1+z1-z)+i4log⁡(i+zi-z),g2=-14log⁡(1+z1-z)+i4log⁡(i+zi-z).
Now ϕ21=h2+g2=(i/2)log((i+z)/(i-z)) and
(3.9)Re{(1-z2)((ϕ21)′)}=Re{1-z21+z2}≥0.
Again, by Theorem 2.3, ϕ21(𝔻) is convex in the imaginary direction and ϕ21 satisfies Condition 1.Since both parametrizations satisfy the hypotheses of Theorem3.1, the harmonic map f=t(h1+g1¯)+(1-t)(h2+g2¯) will lift to a minimal graph over 𝔻 with Weierstrass data G=-1/z and dh=(t)dh1+(1-t)dh2 for all 0≤t≤1 (see Figure 1).Figure 1
Images of concentric circles underf and corresponding minimal surfaces for various values of t in Example 3.3.Example 3.4.
The Weierstrass dataG1=-i/z and dh1=-iz gives a parametrization of Enneper's surface. The functions G2=-i/z and dh2=z/(z4-1) give a different paramaterization of Scherk's doubly periodic surface than in Example 3.3. Notice that ϕ11=z+(1/3)z3 and ϕ21=-(i/2)log((1+z)/(1-z)). By Theorem 2.3, ϕ11 and ϕ21 satisfy Condition 1. Thus, both surfaces satisfy the hypotheses of Theorem 3.1, and the function f=t(h1+g1¯)+(1-t)(h2+g2¯) will lift to a minimal graph for all 0≤t≤1 (see Figure 2).Figure 2
Images of concentric circles underf and corresponding minimal surfaces for various values of t in Example 3.4.Using this method, we were able to show that the combinations shown in Figure3 are also minimal graphs.Various combinations of 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 periodicRemark 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 Theorem3.1.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.



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