Document [original]

https://doi.org/10.1007/s11040-021-09397-z

Constant Mean Curvature Surfaces Based

on Fundamental Quadrilaterals

Alexander I. Bobenko1

·Sebastian Heller2

·Nick Schmitt1

Received: 23 February 2021 / Accepted: 15 June 2021 /

©The Author(s) 2021

Abstract

We describe the construction of CMC surfaces with symmetries in S3and R3using

a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The

fundamental piece is constructed by the generalized Weierstrass representation using

a geometric flow on the space of potentials.

Keywords CMC surface ·Flat connections ·DPW method ·Tesselations

Mathematics Subject Classiﬁcation (2010) 53A10 ·53C42 ·53C43

1 Introduction

Surfaces with constant mean curvature (CMC) in euclidean 3-space and in the round

3-sphere can be investigated by methods of integrable systems. Their Gauss equation

is the elliptic sinh-Gordon equation

u +sinh u=0,(1.1)

which is one of the basic examples of integrable equations. Similar to minimal sur-

faces in euclidean 3-space, CMC surfaces possess 1-parameter (denoted usually by λ)

families of isometric associated surfaces obtained by rotating their Hopf differential.

Communicated by:Alexander P. Veselov

Sebastian Heller

seb[email protected]

Alexander I. Bobenko

[email protected]berlin.de

Nick Schmitt

[email protected]berlin.de

1Institut f¨ur Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

2Institut f¨ur Differentialgeometrie, Universit¨

at Hannover, 30167 Hannover, Germany

Published online: 6 November 2021

Math Phys Anal Geom (2021) 24: 37

This allows CMC surfaces to be described in terms of loop groups [4], so that

analytic methods of the theory of integrable systems can be applied. One of the pow-

erful methods of the construction of CMC surfaces is the generalized Weierstrass

representation (DPW) by Dorfmeister-Pedit-Wu [6]. It starts with an analytic dif-

ferential equation for the holomorphic frame z=ξ with a meromorphic DPW

potential ξ(z,λ) and the subsequent loop group factorization of , leading to immer-

sion formulas for the CMC surfaces. Control of the monodromy of the holomorphic

frame is of crucial importance for the construction of CMC surfaces with non-trivial

topology and symmetries.

A particularly important class of potentials is given by Fuchsian systems ξ(z),

those with only simple poles. In the simplest case of three singularities it reduces to

the hypergeometric equation (see, for example, [7]), whose monodromy group can

be described explicitly from the local residues. This leads to CMC surfaces based on

fundamental triangles [25]. From the geometric point of view, CMC surfaces con-

structed from fundamental quadrilaterals are more natural, since they come from the

curvature line parametrization. But for Fuchsian systems with more then three singu-

larities the monodromy cannot be computed explicitly in terms of the coefficients of

the system, introducing accessory parameters. Then the simplest holomorphic frame

equation is a Fuchsian system with four singularities on the Riemann sphere

z=



k=0

z−zk

. (1.2)

In Section 5of this paper we show how all periodic and compact surfaces based on

fundamental quadrilaterals can be constructed from the system (1.2). Our construc-

tions make explicit use of this Fuchsian DPW form. The relation of the monodromy

and the coefficients of the Fuchsian system is the famous Hilbert’s 21st problem,

which was intensely studied [1]. There exist many important partial results in the sim-

plest non-trivial case of four singularities. This case was investigated mostly within

the theory of isomonodromic deformations [7] and the Painlev´

e VI equation, where

the problem is to describe the coefficients Akas functions of the poles zjwhen the

monodromy group is preserved. The holomorphic frame (z, λ) of a CMC surface

lies in a loop group, and the main analytic problem is to construct solutions whose

monodromy group is unitary on the unit circle |λ|=1, giving global solutions of the

Gauss equation on the four-punctured sphere.

In general, it is a hard problem to control the intrinsic and extrinsic closing condi-

tions to obtain closed surfaces or surfaces with prescribed global properties. In recent

years, important progress has been made using a flow of DPW potentials [13,27,28]

or similar methods on spectral data [12]. By the very nature of these techniques, only

surfaces which are small perturbations of spheres or tori have been reached [13].

In [18] Lawson constructed the first compact minimal surfaces in the round 3-

sphere of genus g⩾2. A fundamental piece of a Lawson surface is obtained by

the Plateau solution of a specific geodesic polygon. The compact surface is then

built from the fundamental 4-gon by the finite group generated by rotations around

Math Phys Anal Geom (2021) 24: 37

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the geodesic edges of the polygon. Later, Karcher-Pinkall-Sterling [16] constructed

new minimal surfaces in the 3-sphere by starting with a tessellation of the 3-sphere

into tetrahedra. The minimal surfaces are obtained from fundamental minimal 4-gons

within such a tetrahedron which reflect across the geodesic boundaries. Constant

mean curvature (CMC) surfaces in R3have been constructed by adapting these

methods [8]; see also [10,22] for related computer experiments.

This paper constructs such fundamental patches of surfaces (Section 4) based

on the deformation of DPW potentials. In this paper the following new surfaces

are numerically constructed: triply periodic surfaces (Figs. 1band2a) and doubly

periodic surfaces (Figs. 16b, 17band18), new doubly periodic surfaces with Delau-

nay ends (Fig. 21), new surfaces with Delaunay ends of positive genus (Figs. 28,

29,30 and 42) as well as new KPS-type surfaces (Fig. 38). We also reconstruct by

these methods previously constructed surfaces based on doubly-periodic hexagonal,

square and triangular tilings of the plane, triply periodic cubic examples, cylinders

with ends [8,9], and the Lawson and KPS surfaces [16,18](Figs.3,4,5,6,7,8,9,

10,11,12 and 13).

The 3D-data of the surfaces constructed in this paper are available in the DGD Gallery [5].

2 Geometric Construction

2.1 The Construction

This paper reports on the experimental construction of CMC surfaces in R3with

non-trivial topology with and without Delaunay ends via the generalized Weierstrass

Fig. 1 Triply periodic CMC surfaces in R3

Math Phys Anal Geom (2021) 24: 37 Page 3 of 46 37

representation (DPW) [6]. The construction starts with a tetrahedron in R3

as shown which tessellates R3by the group generated by the reflections in

the four planes containing its faces. Each of the six edges of the tetrahedron

is marked with an integer n∈N⩾1∪{∞}specifying that the internal dihedral

angle between the two planes meeting at that edge is π/n. The tetrahedron can be

degenerate in the following ways:

•vertex at ∞

•parallel planes, with opposite outward normals: the edge of the tetrahedron

between the two planes is marked with ∞

•coincident planes, with the same outward normal: the edge of the tetrahedron

between the two planes is marked with 1.

In this tetrahedron construct a CMC quadrilateral as shown, such that

•each of the four edges of the quadrilateral lies in a plane of the

tetrahedron, and the surface reflects smoothly across this plane

•at each of the four vertices of the quadrilateral, application of the tessellation

group results in a surface with either an immersed point of the surface or a once-

wrapped Delaunay end at the vertex.

Then the surface constructed by application of the tessellation group is a CMC

immersion with optional Delaunay ends and the symmetries of that group. Its genus

is finite if the tessellation group is finite, and infinite if the group is infinite. These

surfaces are described in detail at the end of this section.

Fig. 2 Triply periodic CMC surfaces in R3

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The quadrilateral is constructed via a Fuchsian DPW potential on CP1with four

simple poles on S1(Section 4.1) and a reflection symmetry across S1. The unit disk is

the domain of a CMC quadrilateral which reflects in planes containing its boundaries.

The simple poles with constant or Delaunay residue eigenvalues insure that each

vertex of the quadrilateral after reflection is either immersed or a Delaunay end.

The four dihedral angles of the tetrahedron at the corners of the quadrilateral are

controlled by the four local monodromies of the potential, and the two remaining

dihedral angles by two global monodromies.

The potential has two accessory parameters which are computed by the unitary

flow (Section 4.3). Starting with an initial surface (Section 4.2) which satisfies the

intrinsic closing condition (unitary monodromy on S1

λ), the unitary flow, which pre-

serves this condition, is run through the space of potentials until the dihedral angles of

the planes reach the values prescribed by the tessellation. The dihedral angles are con-

trolled by certain monodromy traces at the evaluation point. The unitary flow is not

known in general to exist, but short time existence can be shown in some cases [12].

Hence we construct the surfaces numerically, giving evidence that the unitary flow

has long time existence.

The Lawson surfaces [18](Figs.33 and 34) and the surfaces of Karcher-Pinkall-

Sterling [16](Figs.35,36,37,38,39,40 and 41) have been constructed by solutions

of Plateau problems. The cubic lattice and some of the 2-dimensional lattices were

shown to exist by similar methods [8].

2.2 Tetrahedral Tessellations

The following theorem classifies the tetrahedral tessellation of S3(which are compact)

and of R3(which are compact, paracompact or degenerate). The tetrahedral tessellation

of H3, which can be determined by the same methods, are omitted for simplicity.

Theorem 2.1 (1) The tetrahedral tessellations of S3are as follows:

Math Phys Anal Geom (2021) 24: 37 Page 5 of 46 37

(2) The tetrahedral tessellations of R3are as follows:

Proof Necessary conditions that a compact tetrahedron tessellates one of the space-

forms S3,R3or H3are the following.

•Each edge of the tetrahedron is marked with an integer n∈N⩾2denoting that

the internal dihedral angle between the two faces meeting at that edge is π/n.

•At each vertex of the tetrahedron, the three integers marking the three edges

meeting at the vertex are (2,2,n),n∈N⩾2or (2,3,k),k∈{3,4,5}.

•The Gram matrix T∈Mn×n(R)defined by Tij =−cos π/nij has signature

(δ, 1,1,1),whereδ=1, 0 or −1forS3,R3and H3respectively.

The compact tetrahedra in S3,R3and H3are the following:

with identifications Aba =Aab,Bba =Bab,Dba =Dab and

A23 =B22 ,A

33 =D22 ,B

23 =C2,B

33 =D23 . (2.1)

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To see this, first consider those tetrahedra with at least one edge marked with 4 or 5.

Then at each of the vertices at the endpoints of that edge, the other two edges meeting

the vertex must be marked with 2 and 3. Hence all such tetrahedra with at least one 4

or 5 is one of the four types Aab,Bab,Caor Dab. The remaining tetrahedra have only

2 or 3 at each face. There are seven of these, namely A22,A23 =B22,B23 =C2,C3,

A33 =D22,B33 =D23 and D33.

Since its Gram matrix has positive determinant, the tetrahedron Aab is in S3.The

spaceforms for the other tetrahedra Bab,Caand Dab are determined by the signs of

the determinant of the Gram matrix as follows:

B2 345

2[+] + + +

3+++

4 0 −

5−

C2345

[+] + 0−

D2345

2[+] [+] + −

3 0 −−

4−−

5−

(2.2)

The [+] in the above tables denotes entries which are redundant due to the identifi-

cations (2.1)

Hence the tetrahedra which tessellate S3are Aab and the seven tetrahedra B23,

B24,B25,B33,B34,B35 and D24. These tessellate S3because they tessellate either a

sphere or a n-cell, which in turn tessellates S3.

The compact tetrahedra which tessellate R3are the three tetrahedra B44,C4and

D33. The first two tessellate a cube and the third tessellates a rhombic dodecahedron,

each of which in turn tessellates R3.

The paracompact tetrahedral tessellations of R3are classified similarly except that

the integer triple at each vertex is as in the compact case or one of (3,3,3),(2,4,4)

or (2,3,6).

The degenerate tetrahedral tessellations of R3are classified similarly except that

the integer triple at each vertex is as in the paracompact case or one of (2,2,∞)or

(1,n,n),n∈N⩾1∪{∞}.

2.3 The surfaces

This section describes the experimentally constructed minimal surfaces in S3and

CMC surfaces in R3. They are of the types:

In R3:

•triply periodic CMC surfaces R3without ends (Figs. 1,2,14 and 15)

•doubly periodic CMC surfaces R3without ends (e.g. Figs. 16,17,18,19 and 20)

•doubly periodic CMC surfaces R3with ends (Fig. 21)

•single periodic CMC surfaces in R3with Delaunay ends (cylinders, Figs. 22

and 23)

•CMC surfaces in R3with dihedral symmetry and Delaunay ends (tori, Figs. 24,

25,26 and 27)

•CMC surfaces in R3with Platonic symmetries and Delaunay ends (Figs. 28,29,

30 and 31)

•CMC spheres in R3with four Delaunay ends (fournoids, Figs. 30 and 32).

Math Phys Anal Geom (2021) 24: 37 Page 7 of 46 37

In S3:

•Lawson surfaces ξab in S3(Figs. 33 and 34)

•minimal surfaces in S3with Platonic symmetries (Figs. 35,36,37 and 38)

•minimal surfaces in S3with n-cell symmetries (Figs. 39,40,and41)

•minimal tori in S3with Delaunay ends (Fig. 42).

In all shown figures the lines on the surfaces are curvature lines. The surfaces in S3

are stereographically projected to R3.

Triply periodic surfaces in R3The simplest triply periodic surfaces can be thought of

as tubes along the edges of the standard cubic lattice in R3(Fig. 4). The genus of

those surfaces modulo translation is 3.

The triply periodic surfaces are constructed from the three compact tetrahedra

which tessellate R3. Since the CMC quadrilateral can be situated in each tetrahedron

in three ways, this gives nine different configurations, of which two are redundant

due to symmetry (Fig. 3).

Of these, a3and b2are not possible under our symmetry constraints (compare

with (4.2)), c1seems to devolve to a1, and the flow for c2degenerates (Figs. 2,14,

15 and 16).

Fig. 3 The tetrahedra for the seven possible triply periodic examples

Fig. 4 The simplest triply periodic surface and its tetrahedron

Fig. 5 Left: tetrahedron for doubly periodic surface, where (a,b,c) is a permutation of (3, 3, 3), (2, 3,

6) or (2, 4, 4). Middle: table of 2-dimensional lattices. The genus listed in the table is that of the surface

modulo translations. Right: fundamental piece for a 2-dimensional lattice

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Fig. 6 The six 2-dimensional lattices

Doubly periodic surfaces in R3The doubly periodic surfaces can be thought of

as tubes along the edges of a triangle tessellations of R2. The six 2-dimensional

lattices are constructed with the tetrahedron below where (a,b,c)are the indices of

a triangle tessellation of R2, that is, a permutation of (3,3,3),(2,4,4)or (2,3,6)

(Figs. 16,17,18,19 and 20).

Doubly periodic surfaces in R3with Delaunay ends The doubly periodic surfaces

with Delaunay ends are obtained from triangle tessellations of R2. Additional

freedom is given by the choice of vertices corresponding to Delaunay ends (Fig. 21).

Cylinders in R3with ends Cylinders with Delaunay ends can be constructed from

a degenerate tetrahedron with two parallel planes. Of course, the same construction

without Delaunay ends give the classical rotational symmetric periodic surfaces, i.e.,

Delaunay cylinders (Figs. 22 and 23).

Tori in R3with Delaunay ends The torus with nends is constructed via the diagram

below (Figs. 24,25,26 and 27). For large n, existence of those tori can be shown by

growing Delaunay ends in equidistance on one side of a cylinder.

Surfaces in R3with Platonic symmetry and Delaunay ends Given a triangle tessella-

tions of S2, the surface is the orbit of a tube along one edge of the triangle with a

Delaunay end at a vertex of the triangle. Equivalently, the surface is built from tubes

along the edges of one of the five Platonic solids, with ends emanating from the ver-

tices. The five tetrahedra are as in the diagram below, with (a,b,c)a permutation

of (2,3,k),k∈{3,4,5}(Figs. 28,29 and 30).

Fig. 7 Left: tetrahedron for a torus in R3with nDelaunay ends and order ncyclic symmetry and nends.

Right: fundamental piece of this torus

Math Phys Anal Geom (2021) 24: 37 Page 9 of 46 37

Fig. 8 Left: Tetrahedron for Platonic surfaces in R3, where (a,b,c) is a permutation of (2, 3, k). Right:

fundamental piece of the surfaces with Platonic symmetry and Delaunay ends

Lawson surfaces Classically, the Lawson surfaces [18] are constructed from Plateau

solutions of a geodesic polygon by reflection. The tetrahedron Aab and its inscribed

fundamental piece admit a rotational order 2 symmetry around a geodesic through

the vertices labeled by aand b. The geodesic arc is contained in the fundamental

piece. This observation relates the original construction with the construction carried

out in the present work, see also [16](Figs.33 and 34).

Surfaces in S3with Platonic symmetries Minimal surfaces in S3with Platonic sym-

metries have been constructed by Karcher-Pinkall-Sterling [16]. These surfaces can

be thought of as tubes along one edge of a triangle which tessellates S2. Note that [16]

does not list all possible surfaces, e.g. the alternate octahedron of genus 11 and the

alternate icosahedron of genus 29 are missing (Figs. 35,36,37 and 38).

Fig. 9 Tetrahedron for Lawson surface ζa−1,b−1

Fig. 10 Left: tetrahedron for surfaces in S3with Platonic symmetries. Right: table of these surfaces

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Fig. 11 Triangle tessellations of S2. Left to right: cyclic of order 5, dihedral of order 10, tetrahedral,

octahedral and icosahedra

Fig. 12 Left: tetrahedron for surfaces in S3with n-cell symmetries. Right: table of these surfaces

Fig. 13 The 5-, 8-, 16-, 24-, 120- and 600-cell tessellations of S3, stereographically projected to R3

Surfaces in S3with

-cell symmetries For each of the n-cell tessellations of S3there

is a surface which can be thought of as tubes along the edges of the cells. These

minimal surfaces have also been constructed by Karcher-Pinkall-Sterling [16]

(Figs. 39,40 and 41).

3 CMC Polygons via the DPW Method

3.1 The generalized Weierstrass representation (DPW)

Define the following loop groups (see [24] for details):

= smooth maps (loops) from S1

λto SL2C

u= the subgroup loops in which are in SU2on S1

+= the subgroup loops in which extend to the interior unit disk in CP1

−= the subgroup of loops in which extend to the exterior unit disk in CP1

Math Phys Anal Geom (2021) 24: 37 Page 11 of 46 37

−= the subgroup of g∈–1 such that g(0)is upper triangular with diagonal

in R+.

These can be generalized to loops on a circle of radius r∈(0,1);see[20].

ADPW potential ξon a Riemann surface is a sl(2,C)valued holomorphic

differential form on with ξ=∞

k=–1 ξkλk,detξ–1 =0. A meromorphic DPW

potential is defined analogously.

A CMC surface is constructed from a DPW potential as follows. Let be the

holomorphic frame solving d=ξ;generally has monodromy. Let =FB ∈

u˚

+be the Iwasawa factorization into the unitary frame Fand positive part B

(see [24]); for our case of SU2this factorization always exists. The CMC surface is

constructed via the formulas first obtained in [4]:

S3:f(λ

0,λ

1)=F(λ

0)F –1(λ1), λ

0,λ

1∈S1(3.1a)

H3:f(λ

0,λ

1)=F(λ

0)F –1(λ1), λ

1=λ–1

0∈C\S1(3.1b)

R3:f(λ

0)=–2

H˙

F(λ

0)F (λ0)–1 ,λ

0=1 or –1 (3.1c)

where in the case of R3the dot denotes the derivative with respect to θ,λ=eiθ.

The unitary frame Fyields a unitary potential μ=F−1dFwhich is well-defined on

the (Riemann) surface as opposed to Fwhich is well-defined only on the universal

covering. The unitary potential is also known as the associated family of flat

connections, see [14] and the literature therein.

A DPW potential ξ=∞

k=–1 ξkλkis adapted if ξ–1 is upper triangular (and hence

has zero diagonal because det ξ–1 =0). For adapted DPW potentials, the Hopf dif-

ferential is Q=ξ–1,ξ

0. For non-adapted potentials, the formula for Qis more

complicated.

Fig. 14 Triply periodic CMC surfaces in R3

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If ξis holomorphic at z0, the induced CMC surface is immersed at z0if and only

if ξ–1 does not vanish at z0.

This representation differs from the original representation [6] in that the potential

and loops are not twisted. It is slightly looser than the representation [25]inthatit

does not require the DPW potential to be adapted.

3.2 Delaunay ends

ADelaunay eigenvalue is

ν=1

21+λ–1(λ −λ0)(λ −λ1)w . (3.2)

where the evaluation points λ0,λ1and the end weight w∈R×chosen so that νis

real on S1. A DPW potential with a simple pole, unitary monodromy, and Delau-

nay eigenvalues of the residue induces a surface asymptotic to a half Delaunay

cylinder [17].

To construct surfaces, two types of closing conditions must be satisfied by a DPW

potential:

•The intrinsic closing condition is the condition that the monodromy group is

unitarizable on S1(or more generally, r-unitarizable on an circle of radius r∈

(0,1)). For the Fuchsian DPW potentials, this condition is not directly satisfiable

except in the case of 3 or 2 geometric poles; more than 3 requires the unitary

flow, which by definition preserves the intrinsic closing condition.

•The extrinsic closing conditions are conditions on the DPW potential on the mon-

odromy at the evaluation points, chosen to control the desired geometry of the

surface via (3.1). For surfaces constructed via tessellations these conditions are

given in Theorem 3.3.

Fig. 15 Triply periodic CMC surfaces in R3

Math Phys Anal Geom (2021) 24: 37 Page 13 of 46 37

3.3 Gauge

Consider a holomorphic DPW potential ξand a holomorphic map g:→.The

gauge action is

ξ→ g–1ξg +g–1dg. (3.3)

The point of the gauge action is that if d=ξ then d(g) =(g)(ξ.g). We allow

gauges to have monodromy ±1along paths; such multivalued gauges nevertheless

map single-values potentials to single-valued potentials.

ADPW gauge is one which maps DPW potentials to DPW potentials, that is,

g:→+is holomorphic in λ.Ifξis a DPW potential and gis DPW gauge, then

ξand ξ.ginduce the same surface in the sense that and g do. A DPW gauge gis

adapted if it preserves adapted DPW potentials, that is, g|λ=0is upper triangular.

Let ξbe a holomorphic DPW potential on . A meromorphic DPW gauge is

given by a meromorphic map g:→+. Then, ξ.gis a meromorphic DPW poten-

tial with so-called apparent singularities at the singular points of g. In general, the

singularities of a meromorphic DPW potential are not apparent.

3.4 Spin

For a DPW gauge gdefine the group homomorphism

spin :H1() →Z2={±1},(3.4a)

spin

g=+1ifghas monodromy +1 along γ

−1ifghas monodromy −1 along γ(3.4b)

Fig. 16 Doubly periodic CMC surfaces in R3

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that is, spinγg=+1 (resp. –1) if greturns to itself (resp. its negative) along γ.Then

spinγgh =spinγg·spinγh.

To define spin for DPW potentials consider the double cover

C2\{0}→{x∈sl(2,C)\{0}|det x=0},u

v→ u

v–vu

. (3.5)

For a DPW potential ξon a Riemann surface ,letξ–1 be its λ–1 coefficient. Define

the group homomorphism

spin :H1() →Z2={±1},(3.6a)

spin

ξ=+1 if the lift of ξ–1 along γis a closed cuvve

−1 otherwise (3.6b)

That is, spinγξis +1 (resp. –1) if the lift of ξ–1 returns to itself (resp. its negative)

along γ. Then spinγξ.g=spinγξ·spinγg.

The spin can similarly be defined for unitary potentials using the lift of the coef-

ficient of λ–1.If=FB is the holomorphic frame with potential ξ,andFis

the corresponding unitary frame with potential η, then spin η=spin ξbecause

spin B|λ=0=1sinceB|λ=0∈˚

+.

A geometric interpretation for the spin of a potential can be given in terms of a

coordinate frame, that is, a unitary frame Gsatisfying

N=Ge0G–1 ,f

x/v =Ge1G–1 ,f

y/v =Ge2G–1 (3.7)

where e0,e

1,e

2is a positively oriented orthonormal basis for su2,fis the CMC

immersion, vis the metric of f,andNis its normal. Then spin ξ=spin u,where

u=F–1Gis the gauge between the unitary frame Fand a coordinate frame G.

Fig. 17 Doubly periodic CMC surfaces in R3

Math Phys Anal Geom (2021) 24: 37 Page 15 of 46 37

Consider a meromorphic DPW potential ξon .Forz∈write spinzξto mean

spinγξalong a small circle γencircling z.Ifξis regular at zthen spinzξ=1. For a

DPW potential ξon CP1with finitely many singularities we have the total spin



z∈CP1

spin

ξ=1 . (3.8)

As an application of spin, when we construct CMC polygons whose boundaries

reflect in planes in Section 3.6, the spin is used to distinguish the internal and external

dihedral angles of the planes.

3.5 Symmetry

The following theorem and lemma detail how a symmetry of the potential descends

to a symmetry of the meromorphic frame, the unitary frame, and the CMC immersion

via (3.1).

Theorem 3.1 Let ξbe a DPW potential.

1. If for a holomorphic automorphism τof the domain, τ∗ξ=ξ.g,thenτ∗=

Rg for some R∈.IfRis unitary, then the CMC immersion has the

orientation preserving symmetry

S3and H3:τ∗f=Rf R–1(3.9a)

R3:τ∗f=Rf R–1−2

H˙

RR , (3.9b)

where in the case of R3the dot denotes the derivative with respect to θ,λ=eiθ.

Fig. 18 Doubly periodic CMC surfaces in R3

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2. If for an antiholomorphic automorphism τof the domain, τ∗ξ(λ) =ξ(λ).g,then

τ∗(λ) =R(λ)g for some R∈.IfRis unitary, then the CMC immersion

has the orientation reversing symmetry

S3and H3:τ∗f(λ

0,λ

1)=Rf(λ0, λ1)R–1,(3.10a)

R3:τ∗f(λ

0)=−Rf(λ0)R–1−2

H˙

RR . (3.10b)

In the orientation reversing case of the above theorem, the symmetry (3.10) relates

two associate CMC surfaces, which are the same surface if λ1=λ0(for S3), λ0∈R

and λ1∈R(for H3)andλ0∈{±1}(for R3).

Theorem 3.1 is of limited use without the knowledge that Rin that theorem is

unitary. One necessary condition that Ris unitary is given in the following lemma:

Lemma 3.2 If ξin Theorem 3.1 extends to S1

λand has irreducible unitary mon-

odromy, then Rin that theorem is unitary.

Proof Let f:U→be the universal cover, and τa lift of τto the universal cover,

so fτ =τ◦f.Letσbe a deck transformation, so fσ =f.Thenfτστ–1 =f

implying that τστ–1 is a deck transformation.

Let Mσthe monodromy of with respect to σ.andspin

σ1∈{±1}the

monodromy of gwith respect to σ. For the orientation reversing case,

σ∗τ∗=σ∗(Rg) =(spin

g)RMσg =(spin

g)RMσR–1τ∗(3.11)

τ–1∗σ∗τ∗=(spin

g)RMσR–1. (3.12)

Fig. 19 Doubly periodic CMC surfaces in R3

Math Phys Anal Geom (2021) 24: 37 Page 17 of 46 37

Since τστ–1 is a deck transformation, its monodromy is given by Nσ.

(spinσg)RMσR–1. Since by assumption the monodromy group is irreducible and

unitary, then Nσ∈ufor every deck transformation σ. Using that ξextends to S1

λ,

this implies R∈u.

The proof for the orientation preserving case is the same without the overline.

3.6 CMC Polygons

Let R∪{∞}be divided into nsegments s1,...,s

nat ndistinct consecutive points zij

dividing siand sj.Letξbe a meromorphic DPW potential on CP1with singularities

at these points zij . With ba basepoint in the upper halfplane, for i, j ∈{1,...,n}

let γij be a simple closed counterclockwise curve based at bwhich crosses the seg-

ments siand sj,andletMij ,i, j ∈{1,...,n},i<jbe the monodromy along γij .

The nlocal monodromies are those along paths which enclose one singularity; the

remaining monodromies are called global.

Theorem 3.3 Let ξbe a meromorphic DPW potential satisfying the conditions of

Lemma 3.2 with nsingularities on R∪{∞}as above. Assume ξadmits the reflection

symmetry τ∗ξ(λ) =ξ(λ) for τ(z) =z.Letθij ∈[0,π],i, j ∈{1,...,n},i<j.If

the monodromies Mij satisfy

2tr Mij |λ0=–(spin

γij

ξ)cos θij ,i,j∈{1,...,n},i<j (3.13)

then the CMC surface induced by ξwith the upper halfplane as domain is a n-

gon whose boundaries reflect in nplanes (respectively totally geodesic spheres)

P1,...,P

n, with internal dihedral angles θij between Piand Pj.

Fig. 20 Doubly periodic CMC surfaces in R3

Math Phys Anal Geom (2021) 24: 37

37 Page 18 of 46

Proof Let Fbe the unitary frame, Ga coordinate frame, with respect to a basis ˆe0,

ˆe1,ˆe2∈sl(2,C)and uthe unitary λ-independent gauge between them, so F=Gu.

By the proof of Theorem 3.1 (2), τ∗

kF=PkF,k∈{1,...,n}so

τ∗

kG=PkGQ–1

k,τ

∗u=Qku. (3.14)

Then with ρij and σij =spinγij u,

u=τ∗

kτ∗

ku=QkQku=⇒ QkQk=1(3.15a)

σij u=ρ∗u=τjτiu=QjQ–1

iu=⇒ Qj=σij Qi. (3.15b)

With pka fixed point of τk,defineuk=u(pk)so uk=Qkuk. Then for i, j ∈

{1,...,n},i= j,uj=Qjuj=σij Qiuj. Thus u–1

iuj=σij u–1

iuj.so

σij =1:u–1

iuj=u–1

iuj=⇒ u–1

iuje1=e1u–1

iuj(3.16a)

σij =–1 :u–1

iuj=−u–1

iuj=⇒ u–1

iuje1=−e1u–1

iuj. (3.16b)

Hence u–1

iuj=σij e1u–1

iujso uje1u–1

j=σij uie1u–1

Let e1=01

–1 0 .SinceGis a coordinate frame, we have (fy)k=Gkˆe2G–1

Define σk∈{±1}by Pke1=σk(fy)k.Then

Nk.

.=Pke1=FkF–1

ke1=Fke1F–1

k=Gkuke1u–1

kG–1

k=σkGkˆe2G–1

k(3.17)

uke1u–1

k=σkˆe2. (3.18)

Hence σiσj=σij .

Since d

dyis pointing into the upper half plane, (fy)kis an internal normal to the

plane. Thus σk=1ifNkis internal, and σk=–1 if Nkis external. Thus σij =

Fig. 21 Doubly periodic CMC surfaces in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 19 of 46 37

σiσj=1 if and only if Niand Njare both internal or both internal, and σij =

σiσj=–1 if and only if one of Niand Njis internal and one external.

This means

Ni,N

j=−σij cos θij (3.19)

where θij is the internal angle between planes iand j.SinceMij =PjP–1

i,then

2tr Mij |λ=λ0=Ni,N

j=−σij cos θij . (3.20)

Remark 3.4 Planes with dihedral angle θij =πare parallel. Constraining the planes

to coincide (for example, for the CMC torus with ends in R3) requires that the

extrinsic conditions of Theorem 3.3 be augmented with the additional condition

dλMij |λ=λ0=0 . (3.21)

It remains to control the vertices of the CMC polygon constructed in Theorem 3.3.

For this we use a DPW potential with simple poles:

Theorem 3.5 Let ξa DPW potential as in Theorem 3.3, zka simple pole of ξon R,

and νthe eigenvalue of resz=zkξ.

(1) If ν=1/(2n) or ν=1

2−1/(2n),n∈N⩾2, and ξ−1has a simple pole at zk,

then the CMC surface constructed from ξwith 2nreflections around the vertex

is immersed at the vertex.

(2) If ν=νDel/n,n∈N⩾2,whereνDel is a Delaunay eigenvalue (3.2),then

the CMC surface constructed from ξwith 2nreflections around the vertex is a

once-wrapped Delaunay end.

Fig. 22 CMC cylinders in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37

37 Page 20 of 46

Proof Assuming zk=0, write ξ=A–1dz/z+A0dz+.... The pullback with respect

to the local covering map f(w)=wnis

f∗ξ=nA–1dw/w +nA0wn−1dw+... . (3.22)

Proof of Eq. 3.5: Since by assumption A−1has a pole at λ=0, by a z-independent

local gauge of ξit may be assumed

A–1 =1/(2n) λ–1

0–1/(2n). (3.23)

Then the local gauge g=diag(w–1/2,w

1/2)removes the simple pole of f∗ξat

w=0, and

(f ∗ξ).g=0nλ−1

dw+... . (3.24)

Since (f ∗ξ).gis holomorphic at w=0 and its λ−1coefficient does not vanish at

w=0, then the CMC surface induced by (f ∗ξ).gis immersed at w=0. The proof

for ν=1

2−1/(2n) is analogous.

Proof of Eq. 3.5: Since the eigenvalue of A–1 is νDel/n, then the eigenvalue of

nA–1 is νDel, Unitary monodromy implies this is a once-wrapped Delaunay end.

4 Symmetric CMC Surfaces with Non-Trivial Topology

4.1 The potential

By applying a M¨

obius transformation we assume that the singular points of the CMC

polygon are on the unit circle. As the fundamental piece is a CMC quadrilateral, we

Fig. 23 CMC cylinders in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 21 of 46 37

restrict to the 4-punctured sphere in the following. We will see in Section 5that, at

least for surfaces without Delaunay ends, we can restrict without loss of generality to

a Fuchsian DPW potential of the 4-punctured sphere. The means it has four simple

poles and no pole at z=∞, and is of the form

ξ=



k=0

z−zk

dz(4.1)

as follows:

•The poles are z0∈S1in the open first quadrant, and (z1,z

2,z

3)is a permutation

of (1/z0,−z0,−1/z0).

•The residues are

A0=yλ

−1p

λ(ν2

0−y2)

p−y,A

2=−y(ν2

1−y2)

,(4.2a)

A1=σA0σ−1,A

3=σA2σ−1,σ=diag(i,−i). (4.2b)

•For surfaces without Delaunay ends, the eigenvalue ν0of A0and A1and ν1

of A2and A3are constants in (0,1/2). For surfaces with Delaunay ends, ν0

is of the form cνDel with c∈(0,1)and νDel is the eigenvalue of a Delaunay

unduloid

νDel =1

21+1

4λ–1(λ −1)2w, w∈(0,1]. (4.3)

Fig. 24 CMC tori in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37

37 Page 22 of 46

•The accessory parameters xand yare holomorphic functions of λon an open

disk Drof radius r>0 centered at the origin satisfying x(λ) =x(λ) and

y(λ) =y(λ). The function pis a monic polynomial in λsatisfying p(λ) =p(λ).

•The quotients λ(ν2

0−y2)/p and (ν2

1−y2)/x are holomorphic functions of λon

Dr.

The need for the last condition is as follows. The unitary flow, which preserves the

unitarizability of the monodromy of ξ, is implemented by evaluating the monodromy

of ξdirectly on the unit circle, and not by the numerically more problematic proce-

dure of computing the monodromy on an r<1 circle and then extending it to the

unit circle.

Let Mk(k∈{0,...,3}) be the local monodromy around zkbased at z=1.

The surfaces are constructed by running the unitary flow (see Section 4.3 below)

so that at the end of the flow for k=0,1

ν0|λ=λk=1

2n0,ν

1|λ=λk=1

2−1

2n1,(4.4a)

tr M0M1|λ=λk=−cos 2π

r,1

2tr M1M2|λ=λk=cos π

s. (4.4b)

Then, by Theorem 3.3 and 3.5, the unit z-disk maps to a CMC quadrilateral whose

edges reflect in planes (respectively geodesic 2-spheres) with internal

dihedral angles specified by the figure at right, and whose vertices after

these reflections are either immersed points or once-wrapped Delaunay

ends.

Fig. 25 CMC tori in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 23 of 46 37

In the special case ν0+ν1=1

2, the surfaces are constructed by running the unitary

flow so that at the end of the flow for k=0,1

ν0|λ=λk=1

2n0,ν

1|λ=λk=1

2−1

2n0,(4.5a)

tr M0M1|λ=λk=−cos 2π

r,1

2tr M1M2|λ=λk=cos 2π

s(4.5b)

Then the quarter disk in the first quadrant maps to a CMC quadrilateral

whose edges reflect in planes with internal dihedral angles specified by the

figure at right, and whose vertices after these reflections are immersed.

4.2 The initial condition

4.2.1 The initial condition

The initial condition for the unitary flow is a potential ξ0of the form in Section 4.1

with eigenvalues ν0=ν1=1

4with unitary monodromy on S1which induces a

Delaunay surface.

Lemma 4.1 (1) For CMC tori of spectral genus 0 the spectral curve π:CP1→

CP1can be chosen to be π(ξ) =ξ2. The involutions are

σ(ξ) =−ξ, ρ(ξ)=ξ−1,κ(ξ)=ξ. (4.6)

Fig. 26 CMC tori in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37

37 Page 24 of 46

The monodromy eigenvalues of the vacuum are exp(±ν1+2iπZ),exp(±ν2+

2iπZ)where

ν1(ξ) .

.=iπξ−ξ−1

ξ0−ξ−1

,ν

2(ξ) .

.=iπξ+ξ−1

ξ0+ξ−1

(4.7)

and π(±ξ0), π(±ξ−1

0)∈S1are the evaluation points.

(2) For some , m ∈Z+the monodromy eigenvalues of a Delaunay cylinder are

exp(±ν1+2iπZ),exp(±ν2+2iπZ)where

ν1(u) .

.=iπf1(u) −f2(u)

f1(u0)−f2(u0),(4.8a)

ν2(u) =1

2(f1(u) +f2(u)) , ν2(u0)=iπm , (4.8b)

where π(±u0),π(±u0+1

2ω1)∈S1are the evaluation points, and f1,f

2are

as in Eq. 4.9.

On the torus C/(Z+τZ),Let℘be the Weierstrass function and let ζ.

.=−℘.

Let {ω1,ω

2,ω

3}={1

2,1

2+τ

2,τ

2}.

Define on some torus with modulus τspec

h1(u) .

.=η1u−ω1ζ(u) , h2(u) .

.=f1(u −1

2ω1). (4.9)

The theta function

θ(x, τ) .

.=

k∈Z

exp2iπ(1

2n2τ+n(x −ω2)(4.10)

Fig. 27 CMC tori in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 25 of 46 37

is an entire function C→Cwith simple zeros at lattice points Z+τZandnoother

zeros, satisfying

θ(x +1)=θ(x) , θ(x +τ) =−exp(−2iπx)θ(x) , θ(τ −x) =θ(x) (4.11)

for all x∈C.Define

2g0(1

2x) =θ(x)

θ(x) −2θ(1

2x)

θ(1

2x) +iπ, (4.12a)

2gk(1

2x) =exp2iπxωk−ωk

τ−τθ(x +ωk)θ(0)

θ(x)θ(ωk),k∈{1,2,3}. (4.12b)

The initial condition is the potential in Section 4.1 with

x=λ(y +ν0)(y +ν1)1−u

1+u,y(b,a)=−

2iπ

τ−τ(b +a) +f0(b)

2g2(1

2b) (4.13a)

u(b) =−g1(1

2b)

g3(1

2b) ,v(b,a)=

2iπ

τ−τ(b +a) +f0(b)

g2(1

2b) (4.13b)

a=2iπ

τ−τ

2h1,b=2iπ

τ−τ

2h2(4.13c)

ν0=ν1=1

4,[z0,z

1,z

2,z

3]=u(ω2)2,p=1 . (4.13d)

The initial condition is computed numerically from Eq. 4.13a as Laurent series on S1

by computing its Fourier coefficients. The initial data can be computed from Lemma

(4.1) using results from [11]

Fig. 28 CMC surfaces in R3with Platonic symmetry and Delaunay ends

Math Phys Anal Geom (2021) 24: 37

37 Page 26 of 46

4.2.2 Conﬁgurations of the initial condition

Permuting the lattice generators in the initial condition creates different arrangements

of residues of the DPW potential on the Delaunay cylinder. For the configurations

used in this report, the two circle arcs (z0,z

1)and (z2,z

3)are mapped to semicircles

(resp. profile curves) on the Delaunay surface, while the other two circle arcs (z1,z

and (z3,z

0)are mapped to profile curves (resp. semicircles) on the Delaunay surface.

The first of these configurations is used to compute the 2-dimensional lattices and the

cubic lattices; the second is used to compute the tori and Platonic surfaces with ends.

4.2.3 Neck and bulge

For the initial potential ξ0above, the poles of the Fuchsian DPW potential are at necks

of the Delaunay surface. The initial potential with poles at bulges is constructed as a

gauge of ξ0by the gauge diag((λ −λ0)−1/2,(λ−λ0)1/2), where the λ0is a common

zero of xand y2−1/16. This gauge is not a DPW gauge, but a so-called dressing

transformation.

4.3 The unitary ﬂow

4.3.1 The unitary ﬂow

The unitary flow is a flow through the space of potentials of Section 4.1 preserv-

ing the intrinsic closing condition. It starts at a potential in the space with unitary

monodromy, and flows until the monodromy at the evaluation points reach some

desired extrinsic closing conditions.

Fig. 29 CMC surfaces in R3with Platonic symmetry and Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 27 of 46 37

Given a smooth function F=F(t, x) :R1+n→Rnencoding nconditions on

a flow parameter t∈[0,1]and nvariables x,ifdetdF

dx= 0, then x(t) satisfying

F(t, x(t)) =0 can be computed by the implicit ODE dF

dt+dF

dt=0. The solution

for the infinite case can be computed numerically by truncating to F:R1+n→Rm,

m⩾n, and solving the resulting finite dimensional ODE by least squares methods.

The variables xparametrizing the potential consists of:

•the conformal type [z0,z

1,z

2,z

•the local eigenvalues ν0|λ=1and ν1|λ=1

•the end weight w0

•the polynomial p

•the accessory parameters xand y.

The accessory parameters, which are holomorphic functions of λ, are approx-

imated by truncating their power series at λ=0. We always assume that these

functions extend to a disc Drwith r>1 so that the first bullet point below can be

checked directly on the unit circle.

The constraints F=0 are of two types:

•the intrinsic closing conditions: the halftraces tij =1

2tr MiMj,i, j ∈{0,...,3},

i<jare real on S1. This is a necessary condition by Theorem 4.4.

•geometric constraints which choose a path through the space of geometric

parameters to reach the desired extrinsic closing conditions at the end of the flow.

By Theorem 4.4 the monodromy is unitarizable if all halftraces along the unit circle

are real and of absolute value less or equal to 1. As the components of irreducible

Fig. 30 CMC surfaces in R3with Platonic symmetry and Delaunay ends

Math Phys Anal Geom (2021) 24: 37

37 Page 28 of 46

SL(2,C)-representations of the 4-punctured sphere with real traces consists entirely

of either SL(2,R)or SU(2)representations, and since ξ0is unitarizable, we can

ignore the condition that all traces are of absolute value less or equal to 1 during the

unitary flow.

The intrinsic closing conditions on S1

λare approximated by evaluation at finitely

many equally spaced sample points on S1

λ. In the following we describe the other

constraints in more detail:

4.3.2 Geometric constraints

The simplest configuration of the geometric constraints are as follows. The two local

and two global eigenvalues depend linearly on the flow parameter tto reach the

desired values at t=1. If the surface has no ends, the end weight w0is set to 0;

otherwise it depends linearly on tstarting at 0 and reaching a heuristically chosen

value at t=1. In this configuration the conformal type is fixed during the flow.

It is possible that the flow with this simple configuration breaks down, in which

case the path must be modified in some heuristically determined way, for example

by making the conformal type depend on the flow parameter.

In practice each geometric parameter is of one of three types:

•fixed during the flow

•depending linearly on the flow parameter t

•free (unconstrained).

Then the fixed variables, and the variables depending on t, being computable from t,

can be omitted from x.

Fig. 31 CMC spheres in R3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 29 of 46 37

4.4 Irreducibility and unitarizability

For a subgroup G⊂SL2Cgenerated by three elements, this section proves

•a necessary and sufficient condition for the irreducibility of G,and

•a necessary condition for the SU2unitarizability of G, assuming Gis irreducible.

Here, a group Gis reducible if all elements have a common eigenline, and is SU2

unitarizable if there exists C∈SL2Csuch that CGC–1 ⊂SU2. The methods used in

the proofs can be generalized to any finitely generated group.

The proof depends on the following Lemma 4.2, which determines to what extend

three elements of C3are determined by their standard C3inner products.

With −,− the standard inner product on C3,letL={v∈C3|v, v=0}.

Let X=(x0,x

1,x

2)∈M3×3C, with columns x0,x

1,x

2∈C3.LetW=XX∈

SymnC,soWij =xi,x

j.

Lemma 4.2 With Xand Was above,

(1) ker X∩L={0}if and only if rank W⩾2.

(2) Assuming (a), if for some Y∈M3×3C,XX=YYand det X=det Y,then

there exists a unique S∈SO3Csuch that Y=SX.

Proof By the rank-nullity theorem applied to X|image X,

rank X=dim(ker X∩image X) +rank W(4.14)

from which it follows that rank X⩾rank W, and rank W=3ifrankX=3.

Fig. 32 CMC spheres in R3with Delaunay ends

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37 Page 30 of 46

Moreover, if rank X⩾2, then

ker X∩image X=ker X∩L. (4.15)

To prove (4.2), assume rank W⩾2, so rank X=2. By Eq. 4.14,dim(ker X∩

image X) =0. By Eq. 4.15,kerX∩L={0}.

Conversely, assume ker X∩L={0}.ThenrankX⩾2 because every 2-

dimensional subspace of C3intersects L.ByEq.4.15,kerX∩L={0}.ByEq.4.14,

rank W=rank X⩾2.

To prove (4.2) in the case rank W=3, since rank X=rank Y=3, define S.

YX−1.ThenS∈SO3Cby XX=YYand det X=det Y.

To prove (4.2) in the case rank W=2, let xa,x

bbe two independent columns of

Xand let ˆ

X=(xa,x

b,x

a×xb)and ˆ

Y=(ya,y

b,y

a×yb).Sincexa×xb∈ker X,

then by the assumption and Lemma 4.2 (4.2), xa×xb∈ L.Thendetˆ

X=xa×

xb,x

a×xb = 0sorank ˆ

X=3. Moreover, since xa×xb,x

a×xb=ya×yb,y

a×

yb,then ˆ

Xˆ

X=ˆ

Yˆ

Y.ThenS.

.=ˆ

Yˆ

X−1is in SO3C,andY=SX.

Identify C4with gl(2,C)by identifying the standard basis E0,E

1,E

2,E

3with

1,e

0,e

1,e

2,where

e0.

.=i0

0–

i,e

.=01

−10

,e

.=0i

–i0. (4.16)

Under this identification, the standard inner product on C4is

x, y=1

2tr xadj(y) , adjab

cd.

.=d−b

−ca(4.17)

Fig. 33 Minimal Lawson surfaces in S3

Math Phys Anal Geom (2021) 24: 37 Page 31 of 46 37

and SU2⊂SL2Cis identified with R4⊂C4. In particular, for X, Y ∈sl(2,C)it

holds x, y=−1

2tr xy.

In order to treat irreducibility, the following lemma translates the notion of

eigenline to a more convenient form. With a

b⊥.

.=[−ba]consider the double cover

ˆx∈C2\{0}→{x∈sl(2,C)|det x=0},ˆx→ x=ˆxˆx⊥. (4.18)

Lemma 4.3 ∈C2\{0}is an eigenvector of the invertible matrix x∈SL2Cif and

only if x, ⊥=0.

Proof For any p, q ∈C2\{0},

tr qp⊥=p⊥q=det(p, q) . (4.19)

So with p=,q=x,andy=⊥

2x, y=2x, ⊥=det(x, ) (4.20)

so x, y=0 if and only if x and are dependent, that is, if and only if is an

eigenline of x.

Let Pbe the group generated by P0=1,P

1,P

2,P

3∈SL2C. Under the above

identification C4∼

=gl(2,C)let P=(P0,P

1,P

2,P

3)∈M4×4Cbeamatrixwith

columns Pk∈C4.LetT=PP∈Sym4C,soTij =Pi,P

j.

Theorem 4.4 With Pand Tas above,

(1) Pis irreducible if and only if rank T⩾3.

Fig. 34 Minimal Lawson surface in S3

Math Phys Anal Geom (2021) 24: 37

37 Page 32 of 46

(2) Assuming (4.4),Pis SU2unitarizable if and only if Tis real positive semidefi-

nite.

Proof We have the factorization

T=10

V110

0X10

0X1V

01,P=1V

0X. (4.21)

To prove (4.4), let X=(x0,x

1,x

2)∈M3×3Cbe the lower right 3 ×3 submatrix

of P, that is the matrix with columns given by the tracefree parts of P1,P

2,P

3,and

let Y.

.=XX. By Lemma 4.3, Pis irreducible if and only if ker X∩L={0}.By

Lemma 4.2 this is if and only if rank Y⩾2. Since rank T=1+rank Y,thisisifand

only if rank T⩾3.

To prove (4.4), if Pis SU2unitarizable, it may be assumed without loss of gen-

erality that P⊂SU2.ThenP∈M4×4R,soT=TT∈Sym4Ris real positive

semidefinite.

Conversely, if Tis real positive semidefinite, then W=XX=YYfor some

Y∈M3×3R. Replacing Y→ –Yif necessary, then det X=det Y,sobyLemma4.2

(4.2), there exists S∈SO3Csuch that X=SY.LetC∈SL2Cbe a lift of Svia the

double cover SL2C→ SO3Cdefined with respect to 1,e

1,e

2,e

3. Note that this

double cover is given by conjugation on sl(2,C)∼

=C3.ThenCunitarizes P.

4.5 Constructing the surface

Once the potential for a surface is obtained via the unitary flow, the surface is

constructed as follows:

Fig. 35 Minimal surface in S3with Platonic symmetry

Math Phys Anal Geom (2021) 24: 37 Page 33 of 46 37

•Compute the unitarizer of the monodromy.

•Compute curvature lines.

•Compute the fundamental piece of the surface via the DPW construction

•Build the surface from the fundamental piece by reflections.

4.5.1 The unitarizer

Due to the symmetry (4.2) of the potential and of the monodromies with basepoint

z=1 the unitarizer is diagonal and can be computed as follows. With the notation

a∗(λ) =a(1/λ) write

M0=ab

∗. (4.22)

By the unitarizability of M0by a diagonal loop, p.

.=−c∗/b =−c/b∗takes values

in R+along S1away from its zeros and poles, which are even. Let f=(λ −

α)/ (λ −β)so that f∗fhas the same zeros and poles as p.Thenq=p/(f ∗f)

takes values in R+along S1without zeros or poles. Let y∗y=qbe the scalar GL1C

Birkhoff factorization, so yis holomorphic in the unit disk. Then with x=fy the

loop diag(x1/2,x

−1/2)is the required unitarizer, holomorphic on the open unit disk.

4.5.2 Curvature lines

Let Q=q(z)dz2the Hopf differential of the CMC surface. The curvature line coor-

dinate vsatisfies dv2=Q(z)dz2. Curvature line coordinates can be computed by

computing √Q(z) dzover the domain.

Fig. 36 Minimal surfaces in S3with Platonic symmetry

Math Phys Anal Geom (2021) 24: 37

37 Page 34 of 46

The surface is computed numerically by dividing the domain into polygons (trian-

gles or quadrilaterals) and mapping via the CMC immersion these triangles to R3.In

the computation of curvature lines described above, the polygon edges are unrelated

to the curvature lines.

Quadrilaterals whose edges are along curvature lines can be computed as follows.

Divide the domain into quadrilaterals whose edges are curvature lines and such that

the umbilics are at corners of the quadrilaterals. For each quadrilateral, pull back the

potential to curvature line coordinates.

This computation is complicated by the fact that the maps from curvature lines

rectangles to the domain are singular at the umbilics, and the potential is singular

at the umbilics. The potential can be desingularized locally at an umbilic z0by a

coordinate change of the form z=z0+wnand a gauge.

4.5.3 Building the surface

In general the position of the surface in space is not controlled, so to build the surface

it must first be put into a standard position, where a group of standard reflections

can be applied. To do so, compute the four generating reflections Rkin the isometry

group Iso R3of R3. Conjugate them to standard reflections Skvia CRkC−1=Sk.

Then the surface after being moved via x→ Cx has the standard reflections Skas

symmetries.

4.5.4 The bulge count for families of CMC surfaces

The surfaces constructed in this paper allow for non-trivial 1-parameter deformations

within the space of CMC surfaces with the same combinatorics. A natural question,

Fig. 37 Minimal surfaces in S3with Platonic symmetry

Math Phys Anal Geom (2021) 24: 37 Page 35 of 46 37

also considered by [8], is whether different surfaces with the same combinatorics, but

which swap neck and bulge, belong to the same family of CMC surfaces, for example

Figs. 1a, b and 16a, b. It turns out that these examples belong to different families.

We denote by a leg of the surface a cylindrical piece obtained from the trajectories

of the Hopf differential, that is from the curvature line parametrisation. Although the

images are labeled according to whether there are bulges or necks where the legs

meet, in this section we rather count the number of bulges on each leg.

We show that this number is an invariant in the case of surfaces without ends. Note

that in this case, there is a covering →CP1by a compact Riemann surface 

on which the pullback of the DPW potential has only apparent singularities. Phrased

differently, is the surface on which the first and second fundamental forms are

well-defined and smooth, that is for compact CMC surfaces in the 3-sphere, is just

the underlying Riemann surface, and in the case of periodic CMC surfaces in R3,

is the Riemann surface quotient of the CMC surface by the translational symmetries.

We construct surfaces starting from Delaunay cylinders by deforming the eigen-

values νi. In the case of cylinders without umbilics, all four eigenvalues are νi=1

At the starting point, is a torus and the relevant moduli space of flat connections

∇on has only reducible points. The underlying holomorphic bundles (equipped

with the (0,1)-parts ¯

∂∇of the connections) are semistable, i.e. if they admit holomor-

phic line subbundles of degree 0. A holomorphic structure (on a rank 2 bundle over

a compact Riemann surface of degree 0) is called unstable if there exist a holomor-

phic line subbundle of positive degree and they are called stable if every holomorphic

line subbundle has negative degree. This notion is relevant to us since an unstable

holomorphic structure does not admit a flat unitary connection. Spectral parameters

λat which the holomorphic structure is unstable are isolated in the spectral plane.

Fig. 38 Minimal surfaces in S3with Platonic symmetry

Math Phys Anal Geom (2021) 24: 37

37 Page 36 of 46

Moreover, for CMC surfaces based on quadrilaterals, the number of those values of

spectral parameters within a bounded region is always finite and can only change

during a deformation by values crossing the boundary of that region. Values of the

spectral parameter at which the holomorphic structure is unstable cannot cross the

unit circle, as the connections on the unit circle are unitary. For the initial torus

the bundle is semistable for all spectral values. The number of values at which the

holomorphic bundle becomes unstable within infinitesimal deformation of the eigen-

values νican be identified with the number of bulges on the leg of the initial Delaunay

cylinder; for more details see [12]. Actually this number coincides with the num-

ber of zeros of the holomorphic function xin Eq. 4.2 inside the unit circle; see also

[11,15]

5 Fuchsian DPW potentials

The aim of this section is to prove the existence of Fuchsian DPW potentials of

the form Eq. 4.1 for CMC quadrilaterals without Delaunay ends. This generalizes

previous work by the second author [14] for the Lawson genus 2 surface. Similar

results have been obtained by Manca [19]. Our arguments are more geometric and

prove the existence of a Fuchsian potential on a 4-punctured sphere for all surfaces

obtained by CMC quadrilaterals.

5.1 Setup

Let f:→M(where M∈{S3,R3}) be a complete CMC surface without Delau-

nay ends. Assume that fis build from a fundamental piece Pby the group G

generated by the reflections across totally geodesic subspaces along geodesic arcs

contained in P. Assume that Phas the topology of a (closed) disc.

The surface fis equivariant with respect to the (discrete) group Gacting on 

by conformal transformations and on the ambient space Mby a representation ρinto

the space of isometries. Let Go⊂Gbe the subgroup of orientation preserving (i.e.

holomorphic) symmetries on .

5.2 Local theory

The first step in our derivation of a Fuchsian DPW potential is the converse of Theo-

rem 3.5. This means that at fixed points of a rotational symmetry there always exists

DPW potentials with Fuchsian singularity on the quotient.

Let p∈be a fixed point of some rotation given by an element in Go. Then there

exists k∈Nand g∈Goof order ksuch that g(p) =pand such that for any h∈Go

with h(p) =pthere exists l∈Nwith gl=h.

Lemma 5.1 There exists D∈SU2of order 2kand a local DPW potential ηfor f

on an open g-invariant neighbourhood of psuch that

g∗η=DηD−1. (5.1)

Math Phys Anal Geom (2021) 24: 37 Page 37 of 46 37

Proof Consider Dorfmeister’s normalized potential (see for example [29]) which

takes the form

ηnor =0λ−1f(z,0)

f(z,0)0dz (5.2)

where zis a local holomorphic coordinate centered in psuch that g∗z=e2πi

kz,

Q=q(dz)2is the Hopf differential and f(z,w)is a holomorphic function such that

f(z,¯z)dzd ¯zis the induced metric of the surface. As g∗dz =e2πi

kdz and g∗d¯z=

e2πi

kd¯zthe result follows.

Proposition 5.2 There exists a local meromorphic DPW potential of fon /Go

with a Fuchsian singularity at pmod Go. The eigenvalues of the residue are ±1

2k,

independently of λ,wherekis the order of the stabilizer group of p.

Likewise, there exists a local meromorphic DPW potential of fon /Gowith

Fuchsian singularity at pmod Gosuch that the eigenvalues of the residue are

±k−1

2k.

Proof Consider w=zkwhich is a holomorphic coordinate centered at pmod Go∈

/Go. Consider the positive gauge e=diag √z, 1

√zof spin −1. Then

(d +ηnor ).e=d+e−1de +e−1ηnor e(5.3)

is a well-defined meromorphic DPW potential with apparent Fuchsian singularity at

p. As this potential is clearly invariant under pull-back by gwe have proven the first

part of the proposition.

Fig. 39 Minimal surfaces in S3with n-cell symmetry

Math Phys Anal Geom (2021) 24: 37

37 Page 38 of 46

For the second part and k=2l+1 consider the gauge ˜e=diag (z−l,z

l)while for

k=2lconsider the gauge ˜e=diag (z−l+1

2,z

l−1

2), and proceed as in the first part of

the proof.

5.3 Global theory

Our aim is to construct a DPW potential on the /Go. Recall that by assumption the

fundamental piece Pof the Riemann surface is of the topological type of a disc.

Lemma 5.3 The Riemann surface /Gois the projective line.

Proof By the Riemann mapping theorem there exists a holomorphic map from

Pto the unit disc. Schwarzian reflection yields a holomorphic map from to

CP1, branched at the fixed points of Go. By its construction, this map is invariant

under Go.

For simplicity of the arguments, we will assume that nis even in the following.

Lemma 5.4 Let nbe even. There exists a unitary potential μon the n-punctured

Riemann sphere such that

•μis singular exactly at the branch values of →/Go=CP1;

•the pull-back of μgenerates fon the covering .

Proof Let {z1,...,z

n}⊂CP1be the branch values of →/Go=CP1and

S⊂its preimage. Denote the reflection planes of the fundamental piece by

Fig. 40 Minimal surfaces in S3with n-cell symmetry

Math Phys Anal Geom (2021) 24: 37 Page 39 of 46 37

P0,...,P

n−1,with outward oriented unit normals N0,...,N

n−1,respectively, such

that

zm∈Pm−1∩Pm∀m∈{1,...,n}. (5.4)

Denote by gmthe compositions of the reflection across Pm−1and Pm.ThenG0is

generated by {gm|m∈{1,...,n}}.

Let Mbe euclidean 3-space or the 3-sphere, and let d=3andd=4 accordingly,

so that Iso(M) =SO4Ror Iso(M) =SO3RR

Consider the group

H⊂Spinn×Iso(M) (5.5)

generated by the elements

ˆgm:= (Nm·Nm−1,g

m), m =1,...,n (5.6)

where ·denotes Clifford multiplication. This gives a group extension

{id}→Z2→H→G0→{id}. (5.7)

Note that ˆgmhas order 2kif gmhas order k. Similarly, since nis even, the product

ˆgn... ˆg1is trivial. Consequently, we have a representation

h:π1(CP1\{z1,...,z

n},∗)→H. (5.8)

\S→CP1\{z1,...,z

n}(5.9)

is a (unbranched) covering, the fundamental group of \S(with appropriate base

point) is a subgroup of the first fundamental group of CP1\{z1,...,z

n}with corre-

sponding base point. By construction, the induced representation of π1( \S,∗)→

G0is trivial, and the induced representation of htakesvaluesin

Z2=Z2×{id}⊂Spind×Iso(M) (5.10)

Fig. 41 Minimal surfaces in S3with n-cell symmetry

Math Phys Anal Geom (2021) 24: 37

37 Page 40 of 46

such that a simple closed curve around any one of the points in Sis mapped to the

non-trivial element in Z2.

By Riemann surface covering theory, we obtain a 2-fold covering

→, (5.11)

branched over the points in S, with an action of Hby holomorphic automorphisms

on ˆ

such that

→ˆ

/H =CP1(5.12)

is branched over {z1,...,z

n}. Denote its preimage of Sby ˆ

S⊂ˆ

. Note that Hacts

faithfully on ˆ

\ˆ

Consider the pull-back ωon ˆ

of the unitary potential η=F−1dFof f.Note

that, for minimal f:→S3the unitary potential is given by

η=λ−1+−∗−λ∗(5.13)

where

=1

2(f −1df )1,0and ∗=1

2(f −1df )0,1(5.14)

and similarly for CMC surfaces f:→R3.Letπ:H→SU2be the projection

to the rotational part of the symmetry. From the construction (5.13) of the unitary

potential,

h∗ηλ=ηλ.π(h) (5.15)

for a holomorphic automorphism h∈H(where, on the right hand side, the gauge

action of the constant matrix π(h) is given by conjugation).

Consider the free action of Hon ˆ

\ˆ

S×C2given by

(p, v).h=(p.h, π(h−1)(v)) . (5.16)

Fig. 42 Minimal torus in S3with Delaunay ends

Math Phys Anal Geom (2021) 24: 37 Page 41 of 46 37

The quotient

V=(ˆ

\ˆ

S×C2)/H (5.17)

is a trivial smooth vector bundle of rank 2 over CP1\{z1,...,z

n}. We claim that the

unitary potential ωyields a well-defined potential μon this vector bundle: in fact,

the connection 1-form acts on [p, v]∈(ˆ

\ˆ

S×C2)/H as

[p, ωp(v)](5.18)

which is well-defined since

(p, ωp(v)).h=(ph, h−1ωphh−1(v)) =(ph, ωph(h−1v)) . (5.19)

Proposition 5.5 Let nbe even. There exist a meromorphic DPW potential ξon CP1

with simple poles at z1,...,z

nand possible apparent singularity at z=∞.

Proof From Lemma 5.4 we obtain a unitary potential μon the n-punctured sphere.

Let l∈{1,...,n}. By Proposition 5.2 there exist a DPW gauge locally well-defined

on a punctured disc around zlwhich gauges μinto a meromorphic potential with a

Fuchsian singularity at zl. Of course, the holomorphic structures (i.e. the (0,1)-part)

of a meromorphic potential extends to the singular points. Note that these gauges are

well-defined (i.e. have spin 1) as we have chosen the representation s:π1( \S,Z2)

to have local monodromy −1 around every point in S.

Using these gauges as cocycles, we obtain a holomorphic C∗-family of flat SL2C-

connections d+ˆμwith the following properties:

•the induced family of holomorphic structures extends to λ=0togivea

holomorphic rank 2 bundle E0→CP1with trivial holomorphic determinant;

•the connections d+ˆμhave Fuchsian singularities with λ-independent eigenval-

ues ±1

2k;

•the complex linear part of the family of connections has a first order pole at

λ=0, i.e., λ→ λ( ˆμ)1,0(λ) extends to λ=0.

Note that all bundles have trivial determinant. Hence, by the Birkhoff-Grothendieck

theorem, the bundle type of E0is O(d) ⊕O(−d) for some d∈N.

First consider the case d=0. Then, the bundle type Eλis locally constant on an

open disc near λ=0. In particular, there exists a smooth positive family of gauge

transformations gλ(holomorphic in λ) such that

((d +ˆμ).gλ)0,1=d0,1(5.20)

is the trivial holomorphic structure on the rank 2 vector bundle C2→CP1. Thus

d+ξ:= (d +ˆμ).gλ(5.21)

is the meromorphic DPW potential which has only Fuchsian singularities and no

apparent singularity at ∞.

Let d>0. Let z:CP1\{∞}→Cbe an affine holomorphic coordinate and

assume without loss of generality that zl=∞ ∀l∈{1,...,n}. There exists an

integer 0 ⩽s⩽dsuch that on a punctured disc D\{0}around λ=0 all bundles

Math Phys Anal Geom (2021) 24: 37

37 Page 42 of 46

are of the holomorphic type O(s) ⊕O(−s). By the family version of Birkhoff-

Grothendieck, there exists a holomorphic function r:D→Cwith r(0)=0

such that the holomorphic bundle ((d +ˆμ).gλ)0,1has the cocycle (for the covering

U+:= C,U−:= CP1\{∞}of CP1)

z−dzsr(λ)

0zd=r(λ)zs0

zd1

r(λ)z−s1

r(λ)z−d−s1

−10

,(5.22)

where the equality obviously holds only for r(λ) = 0. Again, there exists a DPW

gauge gλwhich gauges Eλinto the above form Eq. 5.22. This means that there

exists a pair (g+

λ,g

−

λ)of DPW gauges on U+respectively U−whichdifferbythe

gauge (5.22) and gauge (d +ˆμ)0,1on U±to the trivial holomorphic structure on

C2→U±. Then,

(d +ˆμ).g+

λ=: d+ξ(5.23)

yields the meromorphic potential ξwith Fuchsian singularities at zkand an apparent

singularity at ∞.

5.3.1 CMC quadrilaterals

Finally, we consider the case of CMC quadrilaterals, i.e., n=4. We show that

these are always determined by a Fuchsian DPW potential (4.1), which, assuming an

additional symmetry, is of the form (4.2).

Lemma 5.6 For n=4the bundle type of E0is either trivial or O(1)⊕O(−1)→

CP1.

Proof Assume the bundle type at λ=0isO(d) ⊕O(−d) →CP1for some d>1.

The Higgs field := resλ=0ξis a meromorphic section of the bundle

K⊗End0(E0)→CP1(5.24)

where K=O(−2)is the canonical bundle of CP1and End0(E0)denotes the trace-

free endomorphisms of E0. Moreover, is nilpotent as the immersion is conformal,

has at most simple poles at z1,...,z

4by construction and does not vanish on CP1as

fis an immersion. Using the decomposition E0=O(d) ⊕O(−d) the Higgs field is

of the form

=ab

c−a(5.25)

where a,b,c are meromorphic sections in O(−2),O(−2+2d),O(−2−2d), respec-

tively, with at most simple poles at z1,...,z

4. Hence c=0. As is nilpotent a=0

as well. For d>1, −2+2d>0andbwould have a zero contradicting the fact that

is nowhere vanishing.

Theorem 5.7 Let fbe a complete CMC surface without Delaunay ends in S3or R3.

If fis built from a CMC quadrilateral in a fundamental tetrahedron of a tessellation

of the ambient space then it is obtained from a Fuchsian DPW potential (4.1)on the

4-punctured sphere.

Math Phys Anal Geom (2021) 24: 37 Page 43 of 46 37

Proof We give a proof by contradiction. Assume that the bundle type at λ=0is

E0=O(−1)⊕O(1). (5.26)

By the proof of Proposition 5.2 the nilpotent λ−1-part =ξ−1of the meromorphic

potential has no zeros, and poles of order 1 at the 4 branch points z1,...,z

4. Thus,

with respect to Eq. 5.26,itmustbeoftheform

=0s−D

(5.27)

where s−D∈M(CP1,O(−4)) is the unique meromorphic section (up to scaling)

with simple poles at D=z1+···+z4. Moreover, the positive eigenvalues νiof the

residues of the connections are contained in the respective kernels of the residues of

ξ−1. This equips E0with a parabolic structure (see for example [3,11,21,23,26]for

definitions and further references) which is unstable. We denote the parabolic bundle

also by E0. The pair (E0,)is a stable strongly parabolic Higgs pair. Note that



νi<1 . (5.28)

It is easy to see (compare with [11]) that (E0,)is the only stable strongly parabolic

Higgs pair with nilpotent Higgs field on the 4-punctured sphere with unstable under-

lying parabolic bundle. Consider the compact Riemann surface X→CP1on which

the rotational symmetry is trivial. Its Fuchsian monodromy (given by uniformization)

corresponds by the Hitchin-Kobayashi correspondence to a stable nilpotent Higgs

pair

S∗⊕S,

01

 ,(5.29)

where S2=KX. Its underlying holomorphic structure is unstable. As the rotational

symmetries act on Xwe obtain, in the same manner as for f, an strongly parabolic

nilpotent Higgs pair with underlying parabolic structure. As the holomorphic struc-

ture is unstable, the parabolic structure must be unstable as well; see [2], and hence

it must be (E0,). Thus, the holomorphic Higgs pair of f, i.e., (∂∇,) would be

gauge equivalent to (S∗⊕S,01

). This is only possible if the Hopf differential of

the minimal (respectively CMC) surface vanishes (compare with [14, sections 2 and

3]), which gives a contradiction.

Finally, we show under which conditions the Fuchsian DPW potential ξcan be

gauged into the form (4.2).

Note that a Fuchsian potential for a CMC quadrilateral defining a compact embed-

ded CMC surface cannot be adapted if all the 4 positive eigenvalues of the residues

are contained in (0,1

4).

Corollary 5.8 Assume that the potential of Corollary 5.7 has equal pairs of eigen-

values. Then, there exist a coordinate change and a gauge such that the potential is

of the form (4.2).

Math Phys Anal Geom (2021) 24: 37

37 Page 44 of 46

Proof We only sketch the proof. Assume that the eigenvalues at z0and z1, respec-

tively z2and z3are equal. First, apply a so-called flip gauge which flips the

eigenvalues at z2and z3by adding ∓1

2. This can be achieved by conjugating the

potential by a DPW gauge which is constant in zsuch that the residues at z2and z3

are lower respectively upper triangular, and then gauge with diag z−z2

z−z3,z−z3

z−z2.

Denote the residues of the potential ˜

ξobtained in this way by Rk,and find Tsuch

that R3=TD

−1T−1R2TDT−1. Then, T−1˜

ξT turns out to be of the form (4.2).

Acknowledgements The first author is partially supported by the DFG Collaborative Research Center

TRR 109 Discretization in Geometry and Dynamics. The second author is supported by the DFG grant

HE 6829/3-1 of the DFG priority program SPP 2026 Geometry at Infinity. The third author is supported

by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics.

Funding Open Access funding enabled and organized by Projekt DEAL.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,

which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as

you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons

licence, and indicate if changes were made. The images or other third party material in this article are

included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the

material. If material is not included in the article’s Creative Commons licence and your intended use is not

permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly

from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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