Constant mean curvature surfaces based on fundamental quadrilaterals

We describe the construction of CMC surfaces with symmetries in $\mathbb S^3$ and $\mathbb R^3$ using 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.


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 which is one of the basic examples of integrable equations. Similar to minimal surfaces in euclidean 3-space, CMC surfaces possess 1-parameter (denoted usually by λ) families of isometric associated surfaces obtained by rotating their Hopf differential. 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 powerful methods of the construction of CMC surfaces is the generalized Weierstrass representation (DPW) by Dorfmeister-Pedit-Wu [6]. It starts with an analytic differential equation for the holomorphic frame Φ z = Φξ with a meromorphic DPW potential ξ(z, λ) and the subsequent loop group factorization of Φ, leading to immersion 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 constructed from fundamental quadrilaterals are more natural, since they come from the curvature line parametrization. But for Fuchsian systems with more then three singularities 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 In section 4 of this paper we show how all periodic and compact surfaces based on fundamental quadrilaterals can be constructed from the system (0.2). Our constructions 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 simplest nontrivial case of four singularities. This case was investigated mostly within the theory of isomonodromic deformations [7] and the Painlevé VI equation, where the problem is to describe the coefficients A k as functions of the poles z j when the monodromy group is preserved. The holomorphic frame Φ(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 conditions to obtain closed surfaces or surfaces with prescribed global properties. In recent Date: February 8, 2021. years, important progress has been made using a flow of DPW potentials [27,28,13] 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 3sphere 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 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 R 3 have been constructed by adapting these methods [8]; see also [22,10]  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 [18,16].
The 3D-data of the surfaces constructed in this paper are available in the DGD Gallery [5]. 1.1. The construction. This report reports on the experimental construction of CMC surfaces in R 3 with genus with and without Delaunay ends via the generalized Weierstrass representation (DPW) [6]. The construction starts with a tetrahedron in R 3 as shown which tessellates R 3 by 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:

Geometric construction
• 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.
The polygon is constructed via a Fuchsian DPW potential on CP 1 with four simple poles on S 1 (section 3.1) and a reflection symmetry across S 1 . The unit disk is the domain of a CMC quadrilateral which reflects in planes containing its boundaries. The 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 A ab , B ab , C a or D ab . The remaining tetrahedra have only 2 or 3 at each face. There are seven of these, namely Since its Gram matrix has positive determinant, the tetrahedron A ab is in S 3 . The spaceforms for the other tetrahedra B ab , C a and D ab are determined by the signs of the determinate of the Gram matrix as follows: The [+] in the above tables denotes entries which are redundant due to the identifications (1.1) Hence the tetrahedra which tessellate S 3 are A ab and the seven tetrahedra B 23 , B 24 , B 25 , B 33 , B 34 , B 35 and D 24 . These tessellate S 3 because they tessellate either a sphere or a n-cell, which in turn tessellates S 3 .
The compact tetrahedra which tessellate R 3 are the three tetrahedra B 44 , C 4 and D 33 . The first two tessellate a cube and the third tessellates a rhombic dodecahedron, each of which in turn tessellates R 3 .
The degenerate tetrahedral tessellations of R 3 are 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 ∪ {∞}.
1.3. The surfaces. This section describes the experimentally constructed minimal surfaces in S 3 and CMC surfaces in R 3 . They are of the types: In R 3 : • triply periodic CMC surfaces R 3 without ends (figures 1 and 2) • doubly periodic CMC surfaces R 3 without ends (e.g. figure 3) • doubly periodic CMC surfaces R 3 with ends (figure 11) • single periodic CMC surfaces in R 3 with Delaunay ends (cylinders, figure 16)  • minimal tori in S 3 with Delaunay ends (figure 30). In all shown figures the lines on the surfaces are curvature lines. The surfaces in S 3 are stereographically projected to R 3 .
Triply periodic surfaces in R 3 . The simplest triply periodic surfaces can be thought of as tubes along the edges of the standard cubic lattice in R 3 (figure 7). The genus of those surfaces modulo translation is 3.
The triply periodic surfaces are constructed from the three compact tetrahedra which tessellate R 3 . 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 (figure 6). Of these, a 3 and b 2 are not possible under our symmetry constraints (compare with (3.2)), c 1 seems to devolve to a 1 , and the flow for c 2 degenerates (figures 1 and 2). Doubly periodic surfaces in R 3 . The doubly periodic surfaces can be thought of as tubes along the edges of a triangle tessellations of R 2 . The six 2-dimensional lattices are constructed with the tetrahedron below where (a, b, c) are the indices of a triangle tessellation of R 2 , that is, a permutation of (3, 3, 3), (2,4,4)    Cylinders in R 3 with 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 (figure 16).
Tori in R 3 with Delaunay ends. The torus with n ends is constructed via the diagram below (figures 19 and 20). For large n, existence of those tori can be shown by growing Delaunay ends in equidistance on one side of a cylinder. Surfaces in R 3 with Platonic symmetry and Delaunay ends. Given a triangle tessellations of S 2 , 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 vertices. The five tetrahedra are as in the diagram below, with (a, b, c) a permutation of (2, 3, k), k ∈ {3, 4, 5} (figures 22 and 23). Lawson surfaces. Classically, the Lawson surfaces [18] are constructed from Plateau solutions of a geodesic polygon by reflection. The tetrahedron A ab and its inscribed fundamental piece admit a rotational order 2 symmetry around a geodesic through the vertices labeled by a and 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] (figure 25). Surfaces in S 3 with Platonic symmetries. Minimal surfaces in S 3 with Platonic symmetries have been constructed by Karcher-Pinkall-Sterling [16]. These surfaces can be thought of as tubes along one edge of a triangle which tessellates S 2 . 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 (figures 26 and 27).

CMC polygons via the DPW method
2.1. The generalized Weierstrass representation (DPW). Define the following loop groups (see [24] for details): Λ = smooth maps (loops) from S 1 λ to SL 2 C Λ u = the subgroup loops in Λ which are in SU 2 on S 1 Λ + = the subgroup loops in Λ which extend to the interior unit disk in CP 1 λ Λ − = the subgroup of loops in Λ which extend to the exterior unit disk in CP 1 λ Λ − = the subgroup of g ∈ Λ -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]. A DPW potential ξ on a Riemann surface Σ is a Λsl(2, C) valued holomorphic differential form on Σ with ξ = ∞ k=-1 ξ k λ k , det ξ -1 = 0. A meromorphic DPW potential is defined analogously.
A CMC surface is constructed from a DPW potential as follows. Let Φ be the holomorphic frame solving dΦ = Φξ; Φ generally has monodromy. Let Φ = F B ∈ Λ uΛ+ be the Iwasawa factorization into the unitary frame F and positive part B (see [24]); for our case of SU 2 this factorization always exists. The CMC surface is constructed via the formulas first obtained in [4]: where in the case of R 3 the dot denotes the derivative with respect to θ, λ = e iθ . The unitary frame F yields a unitary potential µ = F −1 dF which is well-defined on the (Riemann) surface as opposed to F which 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.
If ξ is holomorphic at z 0 , the induced CMC surface is immersed at z 0 if and only if ξ -1 does not vanish at z 0 .
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] in that it does not require the DPW potential to be adapted.

Delaunay ends.
A Delaunay eigenvalue is where the evaluation points λ 0 , λ 1 and the end weight w ∈ R × chosen so that ν is real on S 1 . A DPW potential with a simple pole, unitary monodromy, and Delaunay 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 S 1 (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 monodromy at the evaluation points, chosen to control the desired geometry of the surface via (2.1). For surfaces constructed via tessellations these conditions are given in theorem 2.3.

2.3.
Gauge. Consider a holomorphic DPW potential ξ and a holomorphic map g : Σ → Λ. The gauge action is The point of the gauge action is that if dΦ = Φξ then d(Φg) = (Φg)(ξ.g). We allow gauges to have monodromy ± 1 along paths; such multivalued gauges nevertheless map single-values potentials to single-valued potentials.
A DPW gauge is one which maps DPW potentials to DPW potentials, that is, g : Σ → Λ + is holomorphic in λ. If ξ is a DPW potential and g is DPW gauge, then ξ and ξ.g induce the same surface in the sense that Φ and Φg do. A DPW gauge g is adapted if it preserves adapted DPW potentials, that is, g| λ=0 is upper triangular.
Let ξ be a holomorphic DPW potential on Σ. A meromorphic DPW gauge is given by a meromorphic map g : Σ → Λ + . Then, ξ.g is a meromorphic DPW potential with so-called apparent singularities at the singular points of g. In general, the singularities of a meromorphic DPW potential are not apparent.
To define spin for DPW potentials consider the double cover For a DPW potential ξ on a Riemann surface Σ, let ξ -1 be its λ -1 coefficient. Define the group homomorphism That is, spin γ ξ is +1 (resp. -1) if the lift of ξ -1 returns to itself (resp. its negative) along γ. Then spin γ ξ.g = spin γ ξ · spin γ g.
The spin can similarly be defined for unitary potentials using the lift of the coefficient of λ -1 . If Φ = F B is the holomorphic frame with potential ξ, and F is the corresponding unitary frame with potential η, then spin η = spin ξ because spin B| λ=0 = 1 since B| λ=0 ∈Λ + .
A geometric interpretation for the spin of a potential can be given in terms of a coordinate frame, that is, a unitary frame G satisfying where e 0 , e 1 , e 2 is a positively oriented orthonormal basis for su 2 , f is the CMC immersion, v is the metric of f , and N is its normal. Then spin ξ = spin u, where u = F -1 G is the gauge between the unitary frame F and a coordinate frame G. Consider a meromorphic DPW potential ξ on Σ. For z ∈ Σ write spin z ξ to mean spin γ ξ along a small circle γ encircling z. If ξ is regular at z then spin z ξ = 1. For a DPW potential ξ on CP 1 with finitely many singularities we have the total spin As an application of spin, when we construct CMC polygons whose boundaries reflect in planes in section 2.6, the spin is used to distinguish the internal and external dihedral angles of the planes.
2.5. 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 (2.1).
Theorem 2.1. Let ξ be a DPW potential.
(1) If for a holomorphic automorphism τ of the domain, τ * ξ = ξ.g, then τ * Φ = RΦg for some R ∈ Λ. If R is unitary, then the CMC immersion has the orientation preserving symmetry where in the case of R 3 the dot denotes the derivative with respect to θ, λ = e iθ .
Theorem 2.1 is of limited use without the knowledge that R in that theorem is unitary. One necessary condition that R is unitary is given in the following lemma:  Proof. Let f : U → Σ be the universal cover, and τ a lift of τ to the universal cover, Let M σ the monodromy of Φ with respect to σ. and spin σ 1 ∈ {± 1} the monodromy of g with respect to σ. For the orientation reversing case, Since τ στ -1 is a deck transformation, its monodromy is given by N σ . . = (spin σ g)RM σ R -1 .
Since by assumption the monodromy group is irreducible and unitary, then N σ ∈ Λ u for every deck transformation σ. Using that ξ extends to S 1 λ , this implies R ∈ Λ u . The proof for the orientation preserving case is the same without the overline.
2.6. CMC polygons. Let R∪{∞} be divided into n segments s 1 , . . . , s n at n distinct consecutive points z ij dividing s i and s j . Let ξ be a meromorphic DPW potential on CP 1 with singularities at these points z ij . With b a basepoint in the upper halfplane, for i, j ∈ {1, . . . , n} let γ ij be a simple closed counterclockwise curve based at b which crosses the segments s i and s j , and let M ij , i, j ∈ {1, . . . , n}, i < j be the monodromy along γ ij . The n local monodromies are those along paths which enclose one singularity; the remaining monodromies are called global.
then the CMC surface induced by ξ with the upper halfplane as domain is a n-gon whose boundaries reflect in n planes (respectively totally geodesic spheres) P 1 , . . . , P n , with internal dihedral angles θ ij between P i and P j .
Proof. Let F be the unitary frame, G a coordinate frame, with respect to a basisê 0 , e 1 ,ê 2 ∈ sl(2, C) and u the unitary λ-independent gauge between them, so F = Gu. By the proof of theorem 2.1 (2) Then with ρ ij and σ ij = spin γij u, Since d dy is pointing into the upper half plane, (f y ) k is an internal normal to the plane. Thus σ k = 1 if N k is internal, and σ k = -1 if N k is external. Thus σ ij = σ i σ j = 1 if and only if N i and N j are both internal or both internal, and σ ij = σ i σ j = -1 if and only if one of N i and N j is internal and one external. This means where θ ij is the internal angle between planes i and j. Since M ij = P j P -1 i , then (2.20) Remark 2.4. Planes with dihedral angle θ ij = π are parallel. Constraining the planes to coincide (for example, for the CMC torus with ends in R 3 ) requires that the extrinsic conditions of theorem 2.3 be augmented with the additional condition It remains to control the vertices of the CMC polygon constructed in theorem 2.3. For this we use a DPW potential with simple poles: Theorem 2.5. Let ξ a DPW potential as in theorem 2.3, z k a simple pole of ξ on R, and ν the eigenvalue of res z=z k ξ.
(2) If ν = ν Del /n, n ∈ N ≥2 , where ν Del is a Delaunay eigenvalue (2.2), then the CMC surface constructed from ξ with 2n reflections around the vertex is a once-wrapped Delaunay end.
Since (f * ξ).g is holomorphic at w = 0 and its λ −1 coefficient does not vanish at w = 0, then the CMC surface induced by (f * ξ).g is immersed at w = 0. The proof for ν = 1 2 − 1/(2n) is analogous. Proof of (2): Since the eigenvalue of A -1 is ν Del /n, then the eigenvalue of nA -1 is ν Del , Unitary monodromy implies this is a once-wrapped Delaunay end.

Symmetric CMC surfaces with genus
3.1. The potential. By applying a Möbius 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 restrict to the 4-punctured sphere in the following. We will see in section 4 that, 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 as follows: • The poles are z 0 ∈ S 1 in the open first quadrant, and (z 1 , z 2 , z 3 ) is a permutation of (1/z 0 , −z 0 , −1/z 0 ). • The residues are • For surfaces without Delaunay ends, the eigenvalue ν 0 of A 0 and A 1 and ν 1 of A 2 and A 3 are constants in (0, 1/2). For surfaces with Delaunay ends, ν 0 is of the form cν Del with c ∈ (0, 1) and ν Del is the eigenvalue of a Delaunay unduloid • The accessory parameters x and y are holomorphic functions of λ on an open disk D r of radius r > 0 centered at the origin satisfying x(λ) = x(λ) and y(λ) = y(λ). The function p is a monic polynomial in λ satisfying p(λ) = p(λ). • The quotients λ(ν 2 0 − y 2 )/p and (ν 2 1 − y 2 )/x are holomorphic functions of λ on D r . 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 ξ directly on the unit circle, and not by the numerically more problematic procedure of computing the monodromy on an r < 1 circle and then extending it to the unit circle.
Let M k (k ∈ {0, . . . , 3}) be the local monodromy around z k based at z = 1. The surfaces are constructed by running the unitary flow (see section 3.3 below) so that at the end of the flow for k = 0, 1 Then, by theorems 2.3 and 2.5, 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.
In the special case ν 0 + ν 1 = 1 2 , the surfaces are constructed by running the unitary flow so that at the end of the flow for k = 0, 1 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.
(2) For some , m ∈ Z + the monodromy eigenvalues of a Delaunay cylinder are exp(±ν 1 + 2iπZ), exp(±ν 2 + 2iπZ) where where π(±u 0 ), π(±u 0 + 1 2 ω 1 ) ∈ S 1 are the evaluation points, and f 1 , f 2 are as in (3.9). On the torus C/(Z + τ Z), Let ℘ be the Weierstrass function and let ζ . . = − ℘. Let {ω 1 , ω 2 , ω 3 } = { 1 2 , 1 2 + τ 2 , τ 2 }. Define on some torus with modulus τ spec (3.9) h 1 (u) . . = η 1 u − ω 1 ζ(u) , h 2 (u) . . = f 1 (u − 1 2 ω 1 ) . The theta function (3.10) θ(x, τ ) . . = k∈Z exp 2iπ( 1 2 n 2 τ + n(x − ω 2 ) is an entire function C → C with simple zeros at lattice points Z + τ Z and no other zeros, satisfying for all x ∈ C. Define The initial condition is the potential in section 3.1 with The initial condition is computed numerically from (3.13a) as Laurent series on S 1 by computing its Fourier coefficients. The initial data can be computed from Lemma (3.1) using results from [11] 3.2.2. 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 (z 0 , z 1 ) and (z 2 , z 3 ) are mapped to semicircles (resp. profile curves) on the Delaunay surface, while the other two circle arcs (z 1 , z 2 ) and (z 3 , 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.

Neck and bulge.
For the initial potential ξ 0 above, 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 ξ 0 by the gauge diag((λ − λ 0 ) −1/2 , (λ − λ 0 ) 1/2 ), where the λ 0 is a common zero of x and y 2 − 1/16. This gauge is not a DPW gauge, but a so-called dressing transformation. • the polynomial p • the accessory parameters x and y.
The accessory parameters, which are holomorphic functions of λ, are approximated by truncating their power series at λ = 0. We always assume that these functions extend to a disc D r with 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 t ij = 1 2 tr M i M j , i, j ∈ {0, . . . , 3}, i < j are real on S 1 . This is a necessary condition by theorem 3.4.
• 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 3.4 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 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 ξ 0 is 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 S 1 λ are approximated by evaluation at finitely many equally spaced sample points on S 1 λ . In the following we describe the other constraints in more detail: 3.3.2. 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 t to reach the desired values at t = 1. If the surface has no ends, the end weight w 0 is set to 0; otherwise it depends linearly on t starting 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.

Irreducibility and unitarizability.
For a subgroup G ⊂ SL 2 C generated by three elements, this section proves • a necessary and sufficient condition for the irreducibility of G, and • a necessary condition for the SU 2 unitarizability of G, assuming G is irreducible.
Here, a group G is reducible if all elements have a common eigenline, and is SU 2 unitarizable if there exists C ∈ SL 2 C such that CGC -1 ⊂ SU 2 . The methods used in the proofs can be generalized to any finitely generated group.
The proof depends on the following lemma 3.2, which determines to what extend three elements of C 3 are determined by their standard C 3 inner products.
With −, − the standard inner product on (2) Assuming (a), if for some Y ∈ M 3×3 C, X X = Y Y and det X = det Y , then there exists a unique S ∈ SO 3 C such that Y = SX.
Proof. By the rank-nullity theorem applied to X | image X , (3.14) rank X = dim(ker X ∩ image X) + rank W from which it follows that rank X ≥ rank W , and rank W = 3 if rank X = 3. Moreover, if rank X ≥ 2, then (3.15) ker X ∩ image X = ker X ∩ L .
To prove (2) in the case rank W = 3, since rank X = rank Y = 3, define S . . = Y X −1 . Then S ∈ SO 3 C by X X = Y Y and det X = det Y .
To prove (2) in the case rank W = 2, let x a , x b be two independent columns of X and letX = (x a , x b , x a ×x b ) andŶ = (y a , y b , y a ×y b ). Since x a ×x b ∈ ker X , then by the assumption and lemma 3.2(1), x a ×x b ∈ L. Then detX = x a ×x b , x a ×x b = 0 so rankX = 3. Moreover, since x a ×x b , x a ×x b = y a ×y b , y a ×y b , thenX X =Ŷ Ŷ . Then S . . =ŶX −1 is in SO 3 C, and Y = SX.
Identify and SU 2 ⊂ SL 2 C is identified with R 4 ⊂ C 4 . In particular, for X, Y ∈ sl(2, C) it holds x, y = − 1 2 tr xy. In order to treat irreducibility, the following lemma translates the notion of eigenline to a more convenient form. With a b ⊥ . . = [ −b a ] consider the double cover Proof. For any p, q ∈ C 2 \ {0}, (3.19) tr qp ⊥ = p ⊥ q = det(p, q) .
So with p = , q = x , and y = ⊥ Let P be the group generated by P 0 = 1, P 1 , P 2 , P 3 ∈ SL 2 C. Under the above identification C 4 ∼ = gl(2, C) let P = (P 0 , P 1 , P 2 , P 3 ) ∈ M 4×4 C be a matrix with columns P k ∈ C 4 . Let T = P P ∈ Sym 4 C, so T ij = P i , P j .
Theorem 3.4. With P and T as above, (1) P is irreducible if and only if rank T ≥ 3.
(2) Assuming (1), P is SU 2 unitarizable if and only if T is real positive semidefinite.
Proof. We have the factorization To prove (1), let X = (x 0 , x 1 , x 2 ) ∈ M 3×3 C be the lower right 3 × 3 submatrix of P , that is the matrix with columns given by the tracefree parts of P 1 , P 2 , P 3 , and let Y . . = X X. By lemma 3.3, P is irreducible if and only if ker X ∩ L = {0}. By lemma 3.2 this is if and only if rank Y ≥ 2. Since rank T = 1 + rank Y , this is if and only if rank T ≥ 3.
To prove (2), if P is SU 2 unitarizable, it may be assumed without loss of generality that P ⊂ SU 2 . Then P ∈ M 4×4 R, so T = T T ∈ Sym 4 R is real positive semidefinite.
Conversely, if T is real positive semidefinite, then W = X X = Y Y for some Y ∈ M 3×3 R. Replacing Y → -Y if necessary, then det X = det Y , so by lemma 3.2(2), there exists S ∈ SO 3 C such that X = SY . Let C ∈ SL 2 C be a lift of S via the double cover SL 2 C → SO 3 C defined with respect to 1, e 1 , e 2 , e 3 . Note that this double cover is given by conjugation on sl(2, C) ∼ = C 3 . Then C unitarizes P.

3.5.
Constructing the surface. Once the potential for a surface is obtained via the unitary flow, the surface is constructed as follows: • Compute the unitarizer of the monodromy.
• Compute the fundamental piece of the surface via the DPW construction • Build the surface from the fundamental piece by reflections.
3.5.1. The unitarizer. Due to the symmetry (3.2) 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 By the unitarizability of M 0 by a diagonal loop, p . . = −c * /b = −c/b * takes values in R + along S 1 away from its zeros and poles, which are even. Let f = (λ − α)/ (λ − β) so that f * f has the same zeros and poles as p. Then q = p/(f * f ) takes values in R + along S 1 without zeros or poles. Let y * y = q be the scalar GL 1 C Birkhoff factorization, so y is holomorphic in the unit disk. Then with x = f y the loop diag(x 1/2 , x −1/2 ) is the required unitarizer, holomorphic on the open unit disk.
3.5.2. Curvature lines. Let Q = q(z)dz 2 the Hopf differential of the CMC surface. The curvature line coordinate v satisfies dv 2 = Q(z)dz 2 . Curvature line coordinates can be computed by computing Q(z) dz over the domain. The surface is computed numerically by dividing the domain into polygons (triangles or quadrilaterals) and mapping via the CMC immersion these triangles to R 3 . 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 z 0 by a coordinate change of the form z = z 0 + w n and a gauge.
3.5.3. 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 R k in the isometry group Iso R 3 of R 3 . Conjugate them to standard reflections S k via CR k C −1 = S k . Then the surface after being moved via x → Cx has the standard reflections S k as symmetries.
3.5.4. 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, 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 figure 1a-b, and figure 3a-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 Σ → CP 1 by a compact Riemann surface Σ on which the pullback of the DPW potential has only apparent singularities. Phrased differently, Σ is the surface on which the first and second fundamental forms are welldefined and smooth, that is for compact CMC surfaces in the 3-sphere, Σ is just the underlying Riemann surface, and in the case of periodic CMC surfaces in R 3 , Σ is the Riemann surface quotient of the CMC surface by the translational symmetries. We construct surfaces starting from Delaunay cylinders by deforming the eigenvalues ν i . In the case of cylinders without umbilics, all four eigenvalues are ν i = 1 4 . At the starting point, Σ is a torus and the relevant moduli space of flat connections ∇ on Σ has only reducible points. The underlying holomorphic bundles (equipped with the (0, 1)-parts∂ ∇ of the connections) are semistable, i.e. if they admit holomorphic 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 holomorphic 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 λ at which the holomorphic structure is unstable are isolated in the spectral plane. 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 eigenvalues ν i can 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 number of zeros of the holomorphic function x in (3.2) inside the unit circle; see also [11,15] 4. Fuchsian DPW potentials The aim of this section is to prove the existence of Fuchsian DPW potentials of the form (3.1) 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.

4.1.
Setup. Let f : Σ → M (where M ∈ {S 3 , R 3 }) be a complete CMC surface without Delaunay ends. Assume that f is build from a fundamental piece P by the group G generated by the reflections across totally geodesic subspaces along geodesic arcs contained in P . Assume that P has the topology of a (closed) disc.
The surface f is equivariant with respect to the (discrete) group G acting on Σ by conformal transformations and on the ambient space M by a representation ρ into the space of isometries. Let G o ⊂ G be the subgroup of orientation preserving (i.e. holomorphic) symmetries on Σ.

4.2.
Local theory. The first step in our derivation of a Fuchsian DPW potential is the converse of theorem 2.5. This means that at fixed points of a rotational symmetry there always exists DPW potentials with Fuchsian singularity on the quotient.
Let p ∈ Σ be a fixed point of some rotation given by an element in G o . Then there exists k ∈ N and g ∈ G o of order k such that g(p) = p and such that for any h ∈ G o with h(p) = p there exists l ∈ N with g l = h.  Proof. Consider Dorfmeister's normalized potential (see for example [29]) which takes the form where z is a local holomorphic coordinate centered in p such that g * z = e 2πi k z, Q = q(dz) 2 is the Hopf differential and f (z, w) is a holomorphic function such that f (z,z)dzdz is the induced metric of the surface. As g * dz = e   Likewise, there exists a local meromorphic DPW potential of f on Σ/G o with Fuchsian singularity at p mod G o such that the eigenvalues of the residue are ± k−1 2k . Proof. Consider w = z k which is a holomorphic coordinate centered at p mod G o ∈ Σ/G o . Consider the positive gauge e = diag ( √ z, 1 √ z ) of spin −1. Then (4.3) (d + η nor ).e = d + e −1 de + e −1 η nor e is a well-defined meromorphic DPW potential with apparent Fuchsian singularity at p. As this potential is clearly invariant under pull-back by g we have proven the first part of the proposition. For the second part and k = 2l + 1 consider the gaugeẽ = diag (z −l , z l ) while for k = 2l consider the gaugeẽ = diag (z −l+ 1 2 , z l− 1 2 ), and proceed as in the first part of the proof.

Global theory.
Our aim is to construct a DPW potential on the Σ/G o . Recall that by assumption the fundamental piece P of the Riemann surface Σ is of the topological type of a disc. Proof. By the Riemann mapping theorem there exists a holomorphic map from P to the unit disc. Schwarzian reflection yields a holomorphic map from Σ to CP 1 , branched at the fixed points of G o . By its construction, this map is invariant under G o .
For simplicity of the arguments, we will assume that n is even in the following. where · denotes Clifford multiplication. This gives a group extension Note thatĝ m has order 2k if g m has order k. Similarly, since n is even, the product g n . . .ĝ 1 is trivial. Consequently, we have a representation such that a simple closed curve around any one of the points in S is mapped to the non-trivial element in Z 2 . By Riemann surface covering theory, we obtain a 2-fold covering (4.11)Σ → Σ , branched over the points in S, with an action of H by holomorphic automorphisms on Σ such that (4.12)Σ →Σ/H = CP 1 is branched over {z 1 , . . . , z n }. Denote its preimage of S byŜ ⊂Σ. Note that H acts faithfully onΣ \Ŝ.
Consider the pull-back ω onΣ of the unitary potential η = F −1 dF of f . Note that, for minimal f : Σ → S 3 the unitary potential is given by where (4.14) Φ = 1 2 (f −1 df ) 1,0 and Φ * = 1 2 (f −1 df ) 0,1 and similarly for CMC surfaces f : Σ → R 3 . Let π : H → SU 2 be the projection to the rotational part of the symmetry. From the construction (4.13) of the unitary potential, 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 H onΣ \Ŝ × C 2 given by The quotient which is well-defined since (4.19) (p, ω p (v)).h = (ph, h −1 ω p hh −1 (v)) = (ph, ω ph (h −1 v)) .
Proposition 4.5. Let n be even. There exist a meromorphic DPW potential ξ on CP 1 with simple poles at z 1 , . . . , z n and possible apparent singularity at z = ∞.
Proof. From lemma 4.4 we obtain a unitary potential µ on the n-punctured sphere. Let l ∈ {1, . . . , n}. By proposition 4.2 there exist a DPW gauge locally well-defined on a punctured disc around z l which gauges µ into a meromorphic potential with a Fuchsian singularity at z l . 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 : π 1 (Σ \ S, Z 2 ) to have local monodromy −1 around every point in S. Using these gauges as cocycles, we obtain a holomorphic C * -family of flat SL 2 Cconnections d +μ with the following properties: • the induced family of holomorphic structures extends to λ = 0 to give a holomorphic rank 2 bundle E 0 → CP 1 with trivial holomorphic determinant; • the connections d +μ have Fuchsian singularities with λ-independent eigenvalues ± 1 2k ; • the complex linear part of the family of connections has a first order pole at λ = 0, i.e., λ → λ(μ) 1,0 (λ) extends to λ = 0.
Note that all bundles have trivial determinant. Hence, by the Birkhoff-Grothendieck theorem, the bundle type of First consider the case d = 0. Then, the bundle type E λ is locally constant on an open disc near λ = 0. In particular, there exists a smooth positive family of gauge transformations g λ (holomorphic in λ) such that (4.20) ((d +μ).g λ ) 0,1 = d 0,1 is the trivial holomorphic structure on the rank 2 vector bundle C 2 → CP 1 . Thus (4.21) d + ξ := (d +μ).g λ is the meromorphic DPW potential which has only Fuchsian singularities and no apparent singularity at ∞. Let d > 0. Let z : CP 1 \ {∞} → C be an affine holomorphic coordinate and assume without loss of generality that z l = ∞ ∀ l ∈ {1, . . . , n}. There exists an integer 0 ≤ s ≤ d such that on a punctured disc D \ {0} around λ = 0 all bundles are of the holomorphic type O(s) ⊕ O(−s). By the family version of Birkhoff-Grothendieck, there exists a holomorphic function r : D → C with r(0) = 0 such that the holomorphic bundle ((d +μ).g λ ) 0,1 has the cocycle (for the covering U + := C, where the equality obviously holds only for r(λ) = 0. Again, there exists a DPW gauge g λ which gauges E λ into the above form (4.22). This means that there exists a pair (g + λ , g − λ ) of DPW gauges on U + respectively U − which differ by the gauge (4.22) and gauge (d +μ) 0,1 on U ± to the trivial holomorphic structure on C 2 → U ± . Then, (4.23) (d +μ).g + λ =: d + ξ yields the meromorphic potential ξ with Fuchsian singularities at z k and an apparent singularity at ∞. 4.3.1. 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 (3.1), which, assuming an additional symmetry, is of the form (3.2). Theorem 4.7. Let f be a complete CMC surface without Delaunay ends in S 3 or R 3 . If f is 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 (3.1) on the 4-punctured sphere.
Proof. We give a proof by contradiction. Assume that the bundle type at λ = 0 is By the proof of proposition 4.2 the nilpotent λ −1 -part Φ = ξ −1 of the meromorphic potential has no zeros, and poles of order 1 at the 4 branch points z 1 , . . . , z 4 . Thus, with respect to (4.26), it must be of the form  4)) is the unique meromorphic section (up to scaling) with simple poles at D = z 1 + · · · + z 4 . Moreover, the positive eigenvalues ν i of the residues of the connections are contained in the respective kernels of the residues of ξ −1 . This equips E 0 with a parabolic structure (see for example [21,26,23,11,3] for definitions and further references) which is unstable. We denote the parabolic bundle also by E 0 . The pair (E 0 , Φ) is a stable strongly parabolic Higgs pair. Note that (4.28) It is easy to see (compare with [11]) that (E 0 , Φ) is the only stable strongly parabolic Higgs pair with nilpotent Higgs field on the 4-punctured sphere with unstable underlying parabolic bundle. Consider the compact Riemann surface X → CP 1 on which the rotational symmetry is trivial. Its Fuchsian monodromy (given by uniformization) corresponds by the Hitchin-Kobayashi correspondence to a stable nilpotent Higgs pair (4.29) S * ⊕ S , 0 1 0 0 , where S 2 = K X . Its underlying holomorphic structure is unstable. As the rotational symmetries act on X we obtain, in the same manner as for f , an strongly parabolic nilpotent Higgs pair with underlying parabolic structure. As the holomorphic structure is unstable, the parabolic structure must be unstable as well; see [2], and hence it must be (E 0 , Φ). Thus, the holomorphic Higgs pair of f , i.e., (∂ ∇ , Φ) would be gauge equivalent to (S * ⊕S, 0 1 0 0 ). 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 ξ can be gauged into the form (3.2).
Note that a Fuchsian potential for a CMC quadrilateral defining a compact embedded CMC surface cannot be adapted if all the 4 positive eigenvalues of the residues are contained in (0, 1 4 ). Corollary 4.8. Assume that the potential of theorem 4.7 has equal pairs of eigenvalues. Then, there exist a coordinate change and a gauge such that the potential is of the form (3.2).
Proof. We only sketch the proof. Assume that the eigenvalues at z 0 and z 1 , respectively z 2 and z 3 are equal. First, apply a so-called flip gauge which flips the eigenvalues at z 2 and z 3 by adding ∓ 1 2 . This can be achieved by conjugating the potential by a DPW gauge which is constant in z such that the residues at z 2 and z 3 are lower respectively upper triangular, and then gauge with diag( z−z2 z−z3 , z−z3 z−z2 ). Denote the residues of the potentialξ obtained in this way by R k , and find T such that R 3 = T D −1 T −1 R 2 T DT −1 . Then, T −1ξ T turns out to be of the form (3.2).