Math. J. Okayama Univ. 59 (2017), 117–130

A NOTE ON BALANCE EQUATIONS FOR DOUBLYPERIODIC MINIMAL SURFACESPeter Connor
Abstract. Most known examples of doubly periodic minimal surfacesin R3 with parallel ends limit as a foliation of R3 by horizontal noded
planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a corresponding three-parameter family of doubly periodic minimal surfaces. In this note we derive a differential equationthat is equivalent to the balance equations for doubly periodic minimalsurfaces. This allows for the generation of many more solutions to thebalance equations, enabling the construction of increasingly complicatedsurfaces.

1. Introduction
Many doubly periodic minimal surfaces in R3 with parallel ends limit asa foliation of parallel planes connected by tiny catenoid necks that shrink tonodes at the limit. This was the case with the first examples of genus one
constructed by Karcher [3] and Meeks and Rosenberg [4], and of genus twoconstructed by Wei [7]. It was also the case with the surfaces constructed in[2], in which the author and Weber proved that for any genus g ≥ 1 and any
even number N ≥ 2 there are three-parameter families of embedded doubly
periodic minimal surfaces of genus g and 2N parallel ends. Each family ofsurfaces is constructed in a neighborhood of a noded limit. Given a set ofpoints in the complex plane that satisfy a set of balance equations, theorem
2.1 in [2] provides a three-parameter family of surfaces that geometricallylook like parallel planes connected by periodically placed catenoid necks,
with the location of the necks given by the solutions to the balance equations.
See figure 1.1.
Solving the balance equations proved difficult due to the fact that there
are many equivalent solutions by permuting the locations of the nodes at agiven level. Employing techniques used by Traizet in [5, 6] to find balance
configurations for minimal surfaces with finite total curvature, the balance
equations can be combined into a differential equation that mitigates thisdifficulty. This note demonstrates how to do so with the doubly periodicbalance equations.
Mathematics Subject Classification. Primary 53C43; Secondary 53C45.
Key words and phrases. minimal surfaces, doubly periodic, balance equations.
117

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PETER CONNOR

Figure 1.1. Two views of a genus 8 doubly periodic minimal surface

In section 2, we discuss forces, balance equations, and the known balanced
configurations for doubly periodic minimal surfaces. In section 3, we provethat the balance equations are equivalent to a second order differential equation. In section 4, we examine configurations of type (2, n). In section 5,we examine configurations of type (3, 4), which is the smallest configuration
with no non-trivial symmetries.

2. Forces and Balance Equations
A doubly periodic minimal surface M in R3 is invariant under two linearly independent translations given by a two dimensional lattice Λ. Thereis a corresponding minimal surface M̃ in the quotient space R3 /Λ, fromwhich one can recover M . Assume that the generators of Λ are the vector (0, 2π, 0) and a non-horizontal vector and that the ends of the surface
have vertical limiting normal. Then, each level of the quotient surface has
domain C/(2πiZ). For convenience of calculations, this is identified withC∗ = C − {0} via the exponential map.
Consider N copies of C∗ , labeled C∗k for k = 1, . . . , N , which correspondto the different levels of the surface. The ends of the surface are placed at
0k = 0 and ∞k = ∞ in Ck . On each C∗k , place nk points pk,1 , . . . , pk,nk .
Extend this definition of pk,i for any integer k by making it periodic in thesense that pk+N,i = pk,i for k = 1, . . . , N and i = 1, . . . , nk , with nk+N = nk .
Each point p̃k,i = log pk,i corresponds to the location of a catenoid shaped
neck between the k − 1 an k levels of the surface.
Given a family of doubly periodic minimal surfaces that limits as a foliation of noded planes, the location of the nodes must satisfy a balancing
condition given in terms of the following force equations.

BALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

119Definition 2.1. The force exerted on pk,i by the other points in {pk,i } isdefined byFk,i :=Xnk+1
Xpk,i + pk+1,jpk,i + pk,j−22nk nk+1 (pk,i − pk+1,j )nk (pk,i − pk,j )j=1

j6=ink−1−X
j=1

pk,i + pk−1,j.
2nk nk−1 (pk,i − pk−1,j )The equations Fk,i = 0 are referred to as balance equations.
Definition 2.2. The configuration {pk,i } is called a balanced configuration
if Fk,i = 0 for k = 1, . . . , N and i = 1, . . . , nk . It is a balanced configurationof type (n1 , n2 , . . . , nN ).
Definition 2.3. A configuration {pk,i } is said to be non-degenerate if theN
XJacobian matrix ∂Fk,i /∂pj,h has complex rank m − 1, where m =nk .
k=1

The Jacobian matrix can’t have full rank 2m because
nk
N X
X

Fk,i = 0.

k=1 i=1

This holds whether or not the configuration {pk,i } is balanced.
Theorem 2.1 from [2] states that, given a non-degenerate balanced configuration {pk,i }, there exists a three-parameter family of embedded doubly
periodic minimal surfaces that limit as a foliation of R3 by horizontal noded
planes. Each quotient surface has genus
g =1+NX
k=1(nk − 1)and 2N ends asymptotic to flat cylinders, two at each of the N levels. Thereare nk catenoid necks joining the k − 1 and k levels, with the horizontal
position of the necks given by the terms p̃k,i = log pk,i , i = 1, 2, . . . , nk .
When the surfaces are viewed in R3 , there are infinitely many levels, withthe height of level N + k equal to the sum of the heights of level N andlevel k. Also, there are infinitely many periodically placed necks between
successive levels, with the horizontal locations p̃k,i of the necks periodic withrespect to the translation vector (0, 2π, 0).
Theorem 2.1 was proven by constructing the Weierstrass representation
for the desired surfaces in a neighborhood of a noded limit and solving theperiod problem on the noded limit. Part of solving the period problem is120

PETER CONNOR

having a balanced configuration. The configuration being non-degenerate
allows the use of the implicit function theorem to solve the period problem
in an open neighborhood of the noded limit.
In [2], non-degenerate balanced configurations were shown to exist when
N = 2, n1 = 1, and n2 = n for any n ∈ N. On each quotient surface, thisdata corresponds to two levels, each with two Scherk ends. Between thelevels, there are catenoid necks. From level one to level two, there are nnecks. From level two to level three (level one in the quotient), there is oneneck. We refer to these as (1, n) configurations, designating two levels with 1and n necks between successive levels. The surface in figure 1.1 corresponds
to a (1, 8) balanced configuration.
For each n ∈ N there is only one (1, n) balanced configuration. The
location of the nodes are p1,1 and p2,k , k = 1, 2, . . . , n, with p1,1 = 1 and thep2,k corresponding to roots of the polynomialpn (z) =n  2Xn
k=0kzk .

It was also proven that sequences of this type of configuration can beconcatenated to produce a new non-degenerate balanced configuration. If
there exist non-degenerate balanced configurations of type (1, nj ) for j =1, 2, . . . , m then they can be combined to create a non-degenerate balanced
configuration of type (1, n1 , 1, n2 , . . . , 1, nm ), with corresponding embedded,
doubly periodic minimal surface with 2m levels and the number of necks
between successive levels alternating between 1 and the integers nj .
Two (2, 3) balanced configurations were discovered, which led to the question of whether there are always balanced configurations of the form (m, n)with 1 ≤ m ≤ n. Numerical evidence indicates that the number of balanced
configurations of a fixed type (m, n) increases as m increases. The locationsof the necks of the surface in figure 2.1 are given by one of the seven balanced
configurations of type (2, 13).

3. An alternative to the balance equations
The balance equations corresponding to more complicated configurations
such as those of type (n1 , n2 ) with 1 < n1 < n2 are very difficult to solve
algebraically. In [5, 6], Traizet combined a set of balance equations for minimal surfaces in R3 with finite total curvature into one differential equation.
One solution of the differential equation corresponds to many equivalentbalanced configurations by permutation of the nodes at each level, and so it
is much easier to find balanced configurations by solving the correspondingBALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

121Figure 2.1. Surface corresponding to a (2, 13) balancedconfigurationdifferential equation. We use Traizet’s method to find a differential equation corresponding to the balance equations for doubly periodic minimalsurfaces.
Theorem 3.1. Let N be an even positive integer, n1 , n2 , . . . , nN ∈ N, andsuppose {pk,i } is a configuration such that the pk,i are distinct. LetnkNYY
Pk (z) =Pk (z)(z − pk,i ), P (z) =i=1

andQ(z) =N XzP 00 (z)P (z)k=1kn2k Pk (z)k=10(z)P (z)zPk0 (z)Pk+1P 0 (z)P (z)−+ k2
nk nk+1 Pk (z)Pk+1 (z)nk Pk (z)Then the configuration {pk,i } is balanced if and only if Q(z) ≡ 0.
Proof. An equivalent expression for the force Fk,i is given by
nk+1
2pk,i X
pk,i X
1
1Fk,i = 2−pk,i − pk,jnk nk+1
p − pk+1,jnk
j=1 k,ij6=ink−1

pk,i X1
1−+ 2.
nk−1 nk
pk,i − pk−1,jnk
j=1

Since the pk,i are distinct for each k,
Pk00 (pk,i ) X
2=,0Pk (pk,i )
pk,i − pk,j
j6=i.

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PETER CONNOR

nk+10
X
Pk+1(pk,i )1=,Pk+1 (pk,i )
pk,i − pk+1,j

j=1

nk−10
X
Pk−1
(pk,i )
1=,
Pk−1 (pk,i )
pk,i − pk−1,jj=1

and the force equations can be rewritten in terms of the polynomials Pk :Fk,i =00(pk,i )
pk,i Pk+1
pk,i Pk−1(pk,i )
pk,i Pk00 (pk,i )1−−+ 220
nk nk+1 Pk+1 (pk,i ) nk nk−1 Pk−1 (pk,i ) nk
nk Pk (pk,i )Substituting z for pk,i and multiplying by
Pk0 (z)P (z)Pk (z)
we get the polynomial
Qk (z) =0(z)P (z)zPk0 (z)Pk+1
zPk00 (z)P (z)−nk nk+1 Pk (z)Pk+1 (z)n2k Pk (z)00zPk−1 (z)Pk (z)P (z)Pk0 (z)P (z),−+ 2nk nk−1 Pk−1 (z)Pk (z)nk Pk (z)and for each i = 1, 2, . . . , nk , Fk,i = 0 if and only if Qk (pk,i ) = 0.
Then,N 0X(z)P (z)zPk0 (z)Pk+1
zPk00 (z)P (z)Pk0 (z)P (z)Q(z) =−+ 2
nk nk+1 Pk (z)Pk+1 (z)n2k Pk (z)nk Pk (z)k=1
P
is a polynomial with degree less than m =nk , and Q(pk,i ) = Qk (pk,i ) fori = 1, 2, . . . , nk and k = 1, 2, . . . , N .
If Q(z) ≡ 0 then Qk (pk,i ) = 0 and Fk,i = 0 for i = 1, 2, . . . , nk and k =1, 2, . . . , N , and so the configuration {pk,i } is balanced. If the configuration{pk,i } is balanced then Q(pk,i ) = Qk (pk,i ) = Fk,i = 0. Thus, Q(z) has degree
less than m and at least m distinct roots, and so Q(z) ≡ 0.
Note that if we re-express
Pk (z) =nk
Xak,i z ii=0

then the Q(z) ≡ 0 is a system of at most m − 1 equations with m variables
ak,i .

BALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

123

3.1. Configurations of type (n1 , n2 ). If N =2 then, after multiplying by
n21 n22 ,Q(z) =n22 zP100 (z)P2 (z) + n21 zP200 (z)P1 (z) − 2n1 n2 zP10 (z)P20 (z)+ n22 P10 (z)P2 (z) + n21 P20 (z)P1 (z).

With some extra assumptions, the non-degeneracy of configurations oftype (n1 , n2 ) is guaranteed.
Proposition 3.2. If pk,i ∈ R with p1,i > 0 for i = 1, 2, . . . , n1 and p2,i < 0for i = 1, 2, . . . , n2 then the configuration {pk,i } is non-degenerate.

Proof. If pk,i ∈ R with p1,i > 0 for i = 1, 2, . . . , n1 and p2,i < 0 for i =1, 2, . . . , n2 then the Jacobian matrix ∂Fk,i /∂pj,h is a (n1 + n2 ) × (n1 + n2 )matrix, and it is easy to see that the submatrix obtained by removing thelast row and column is strongly diagonally dominant. Hence, the Jacobianmatrix has rank n1 +n2 −1, and the configuration pk,i is non-degenerate. Otherwise, the non-degeneracy of a given balanced configuration can bechecked on a case by case basis.
4. Configurations of type (2, n)Consider the case when N = 2, n1 = 2, and n2 = n ≥ 2. After rescaling
and translating, we can assume that p1,2 = 1/p1,1 . ThenP1 (z) = (z − p1,1 )(z − p1,2 ) = z 2 − αz + 1,nn
YXP2 (z) =(z − p2,i ) =ai z i ,i=1

andi=0Q(z) =4(z 3 − αz 2 + z)P200 (z) + 4 (1 − 2n)z 2 + (αn − α)z + 1 P20 (z)+ n2 (4z − α)P2 (z).

Finding balanced (2, n) configurations corresponds to finding a α ∈ R
and polynomial P2 (z) such that Q(z) ≡ 0 and the roots of P1 (z)P2 (z) aredistinct.
In this case, Q(z) is a polynomial of degree at most n + 1. If we re-express
Q(z) =n+1
Xbi z ii=0thenbk = 4ak−1 (k − n − 1)2 − αak (2k − n)2 + 4ak+1 (k + 1)2
with ak = 0 for k > n.
We want Q(z) ≡ 0, which is the same as bk = 0 for 0 ≤ k ≤ n + 1, and124

PETER CONNOR

bn+1 = 0 ⇔ 4an (n + 1 − n − 1)2 = 0andαak (2k − n)2 − 4ak+1 (k + 1)2bk = 0 ⇔ ak−1 =4(k − n − 1)2αn2for k = 1, 2, . . . , n. Starting with an = 1 and an−1 =, we can recursively
4
define an−k for k = 1, 2, . . . , n. Each ak is a polynomial with respect to α ofdegree at most n − k, call them ak = Ak (α).
Thus, {pk,i } provides a balanced (2, n) configuration if!
nXP1 P2 (z) = z 2 − αz + 1
Ai (α)z ii=0

has distinct roots. This can be checked on a case by case basis. Numericalevidence suggests that for each n ∈ N there is one balanced configurationof type (2, 2n + 1) with p1,1 , p1,2 > 0 and p2,i < 0 for i = 1, 2, . . . , n. By
proposition 3.2, this configuration is non-degenerate. The non-degeneracyof other examples can be checked on a case by case basis.
Note that the ODE
Q(z)=034(z − αz 2 + z)
has 4 regular singularities at z = 0, p1,1 , 1/p1,1 , and ∞. Therefore, this ODE
is the Heun equation.
Example 3.2 in [2] shows that the only non-degenerate (2, 2) balanced
configuration is equivalent to the non-degenerate (1, 1) balanced configuration. The first examples of (2, n) balanced configurations are of type (2, 3),
which were discussed in example 3.3 in [2]. The remainder of this section
examines balanced configurations of type (2, n) for n = 4, 5, 6, 7.
Note that the Weierstrass data can be recovered from a balanced configuration using the techniques discussed in chapter 4 of [1] and section 3 of[6]. These methods were used to produce the images in figures 1.1, 2.1, 4.3,4.6, and 5.2.
4.1. (2,4) Balanced Configurations. If n = 4 then Q(z) ≡ 0 whenb4 = 0 ⇔ a3 = 4αb3 = 0 ⇔ a2 = α2 − 4
b2 = 0 ⇔ a1 = −4α1b1 = 0 ⇔ a0 = − (α2 − 2)2

BALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

1251α(α2 − 4) = 0,8and 18 α(α2 − 4) = 0 has roots 0 and ±2. However, ±2 don’t work because
then P1 (z) has repeated root z = 1 or z = −1. If α = 0 thenb0 = 0 ⇔P1 (z) = z 2 + 1, P2 (z) = z 4 − 4z 2 + 1.
pp√√Then P1 (z) has roots ±i and P2 (z) has roots ± 2 − 3, ± 2 + 3. Hence,
the nodes are located at√ 1√ 1√√
1
π 1± i, log(2 + 3), log(2 − 3), log(2 + 3) + πi, log(2 − 3) + πi.
2 2222√However, √the (1, 2) configuration has p1,1 = 1, p2,1 = −2 + 3, andp2,2 = −2 − 3, with the location of the nodes√√0, log(2 + 3) + πi, log(2 − 3) + πi.
Thus, if we rescale the (2, 4) configuration by 2 and translate by πi, we getthe (1, 2) configuration. See figure 4.1.

(a) (2, 4) balanced configuration(b) (1, 2) balanced configuration

Figure 4.1. The circles and squares represent the nodes atlevels one and two, respectively.
4.2. (2,5) Balanced Configurations. If n = 5 then Q(z) ≡ 0 when25αb5 = 0 ⇔ a4 =425(9α2 − 16)b4 = 0 ⇔ a3 =6425α(9α2 − 1040)
b3 = 0 ⇔ a2 =2304
425(9α − 12704α2 + 20736)
b2 = 0 ⇔ a1 =147456
α(81α4 − 123552α2 + 1251584)
b1 = 0 ⇔ a0 =589824

126PETER CONNOR

b0 = 0 ⇔ 9α3 − 324α2 − 1040α + 576 = 0 or
9α3 + 324α2 − 1040α − 576 = 0.

Because of the symmetries of the solutions, there are three balanced configurations corresponding to the positive solutions to the b0 = 0 equation:α ≈ 0.48233788, α ≈ 3.40867116, or α ≈ 38.92633327. See figures 4.2 and
4.3.

(a) α ≈ 0.48233788(b) α ≈ 3.40867116(c) α ≈ 38.92633327

Figure 4.2. (2, 5) balanced configurations

Figure 4.3. Surface corresponding to the (2, 5) balanced
configuration in figure 4.2c
4.3. (2,6) Balanced Configurations. If n = 6 then Q(z) ≡ 0 whenb6 = 0 ⇔ a5 = 9αb5 = 0 ⇔ a4 = 9(α2 − 1)b4 = 0 ⇔ a3 = α(α2 − 26)

b3 = 0 ⇔ a2 = −9(α2 − 1)
9b2 = 0 ⇔ a1 = − (2α3 − 27α)25BALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

b1 = 0 ⇔ a0 =1271(−2α4 + 52α2 − 25)251α(α2 − 26)(α2 − 1) = 0.
b0 = 0 ⇔50√So, α can be 0, ±1, or ± 26. There are only two new configurations. Seefigure 4.4. The α = 0 configuration
is equivalent to the (1, 3) configuration,√by√ a factor of 2, the α = − 26 configuration is equivalent to the α =26 configuration, by a translation of πi, and the α = −1 configuration isequivalent to the α = 1 configuration, by a translation of πi.

(a) α = 1(b) α =√26

Figure 4.4. (2, 6) balanced configurations
4.4. (2,7) Balanced Configurations. If n = 7 then Q(z) ≡ 0 when49α
449α(25α2 − 16)a5 =6449α(25α2 − 272)
a4 =25649(25α4 − 10272α2 + 6400)
a3 =
16384
449α(25α − 112672α2 + 1120512)
a2 =1638400
49(25α6 − 122672α4 + 5229312α2 − 2560000)
a1 =26214400
6α(625α − 3073200α4 + 159576832α2 − 350851072)
a0 =104857600
43225α − 1600α − 10272α + 17408α + 6400 = 0 or
25α4 + 1600α3 − 10272α2 − 17408α + 6400 = 0.

b7 = 0 ⇔ a6 =b6 = 0 ⇔b5 = 0 ⇔b4 = 0 ⇔b3 = 0 ⇔b2 = 0 ⇔b1 = 0 ⇔b0 = 0 ⇔Because of the symmetries of the solutions, there are four balanced configurations corresponding to the positive solutions to the b0 = 0 equation:

128

PETER CONNOR

α ≈ 0.312754, α ≈ 1.65533, α ≈ 7.08968, or α ≈ 69.7471. See figures 4.5
and 4.6.

(a) α ≈ 0.312754(b) α ≈ 1.65533(c) α ≈ 7.08968(d) α ≈ 69.7471

Figure 4.5. (2, 7) balanced configurations

Figure 4.6. Surface corresponding to the (2, 7) balanced
configuration in figure 4.5d

BALANCE EQUATIONS FOR DOUBLY PERIODIC MINIMAL SURFACES

129

5. (3,4) Configurations
The (3, 4) balance equations are the smallest for which there is a balanced configuration such that the location of the nodes has no nontrivial
symmetries. Here,p1 (z) = z 3 + (a1 − 1)z 2 − (a1 + a0 )z + a0 , p2 (z) = z 4 + b3 z 3 + b2 z 2 + b1 z + b0 ,where we assume that p1 (1) = 0. Q(z) is a polynomial of degree six withcoefficients ck such thatc0 = 0 ⇔ −16a0 b0 − 16a1 b0 + 9a0 b1 = 0
c1 = 0 ⇔ −64b0 + 64a1 b0 − a0 b1 − a1 b1 + 36a0 b2 = 0
c2 = 0 ⇔ 144b0 − 25b1 + 25a1 b1 − 4a0 b2 − 4a1 b2 + 81a0 b3 = 0
c3 = 0 ⇔ 144a0 + 81b1 − 4b2 + 4a1 b2 − 25a0 b3 − 25a1 b3 = 0
c4 = 0 ⇔ −64a0 − 64a1 + 36b2 − b3 + a1 b3 = 0
c5 = 0 ⇔ −16 + 16a1 + 9b3 = 0.

As with the (2, n) balance equations, we can solve iteratively, starting with
c5 = 0 down to c0 = 0:16(1 − a1 )c5 =0 ⇔ b3 =9

4 1 + 36a0 + 34a1 + a21
c4 =0 ⇔ b2 =81

16 1 − 468a0 + 258a1 − 261a0 a1 − 258a21 − a31
c3 =0 ⇔ b1 =6561
25 − 70668a0 + 2916a20 + 6506a1 + 69894a0 a1 − 10146a21
c2 =0 ⇔ b0 =59049
236606a0 a1 + 6506a1 + 25a41.
+59049This reduces the coefficients c1 and c0 of Q(z) to fifth degree polynomials in
a0 and a1 . Solving c0 = c1 = 0 numerically, there are four distinct balanced
configurations. See figures 5.1 and 5.2. The smallest balanced configuration
with no non-trivial symmetries is shown in figure 5.1a.

(a)

(b)

(c)

Figure 5.1. (3, 4) balanced configurations

(d)

130

PETER CONNOR

Figure 5.2. Surface corresponding to the (3, 4) balanced
configuration in figure 5.1dAcknowledgements
The author is grateful to the referee for the constructive input and helpfulcomments.
References[1] P. Connor. Adding handles to Wei’s genus two doubly periodic minimal surface. PhD
thesis, Indiana University, 2009.
[2] P. Connor and M. Weber. The construction of doubly periodic minimal surfaces via
balance equations. Amer. J. Math., 134:1275–1301, 2012.
[3] H. Karcher. Embedded minimal surfaces derived from Scherk’s examples. Manuscripta
Math., 62:83–114, 1988.
[4] W. H. Meeks III and H. Rosenberg. The global theory of doubly periodic minimalsurfaces. Inventiones Math., 97:351–379, 1989.
[5] M. Traizet. An embedded minimal surface with no symmetries. J. Differential Geometry, 60:103–153, 2002.
[6] M. Traizet. Exploring the space of embedded minimal surfaces of finite total curvature.
Exp. Math., 17:2:205–221, 2008.
[7] F. Wei. Some existence and uniqueness theorems for doubly periodic minimal surfaces.
Invent. Math., 109:113–136, 1992.
Peter Connor
Department of Mathematical Sciences
Indiana University South Bend
South Bend, IN 46634 USA
e-mail address: pconnor@iusb.edu
URL: http://pages.iu.edu/∼pconnor/Home.html(Received April 25, 2016 )(Accepted May 11, 2016 )