Constant mean curvature surfaces based on fundamental quadrilaterals
(( a ) cube (bulge) ( b ) cube (neck) ( c ) octahedron Figure 1 . Triply periodic CMC surfacesin R CONSTANT MEAN CURVATURE SURFACES BASED ONFUNDAMENTAL QUADRILATERALS
ALEXANDER I. BOBENKO, SEBASTIAN HELLER, AND NICK SCHMITT
Abstract.
We describe the construction of CMC surfaces with symmetries in S and R using a CMC quadrilateral in a fundamental tetrahedron of a tessellationof the space. The fundamental piece is constructed by the generalized Weierstrassrepresentation using a geometric flow on the space of potentials. Introduction
Surfaces with constant mean curvature (CMC) in euclidean 3-space and in the round3-sphere can be investigated by methods of integrable systems. Their Gauss equationis the elliptic sinh-Gordon equation(0.1) ∆ u + sinh u = 0 , which is one of the basic examples of integrable equations. Similar to minimal surfacesin euclidean 3-space, CMC surfaces possess 1-parameter (denoted usually by λ ) familiesof isometric associated surfaces obtained by rotating their Hopf differential. This allowsCMC surfaces to be described in terms of loop groups [4], so that analytic methodsof the theory of integrable systems can be applied. One of the powerful methods ofthe construction of CMC surfaces is the generalized Weierstrass representation (DPW)by Dorfmeister-Pedit-Wu [6]. It starts with an analytic differential equation for theholomorphic frame Φ z = Φ ξ with a meromorphic DPW potential ξ ( z, λ ) and thesubsequent loop group factorization of Φ, leading to immersion formulas for the CMCsurfaces. Control of the monodromy of the holomorphic frame is of crucial importancefor the construction of CMC surfaces with non-trivial topology and symmetries.A particularly important class of potentials is given by Fuchsian systems ξ ( z ), thosewith only simple poles. In the simplest case of three singularities it reduces to thehypergeometric equation (see, for example, [7]), whose monodromy group can be de-scribed explicitly from the local residues. This leads to CMC surfaces based on fun-damental triangles [25]. From the geometric point of view, CMC surfaces constructedfrom fundamental quadrilaterals are more natural, since they come from the curvatureline parametrization. But for Fuchsian systems with more then three singularities themonodromy cannot be computed explicitly in terms of the coefficients of the system,introducing accessory parameters. Then the simplest holomorphic frame equation is aFuchsian system with four singularities on the Riemann sphere(0.2) Φ z = Φ (cid:88) k =0 A k z − z k . In section 4 of this paper we show how all periodic and compact surfaces based on fun-damental quadrilaterals can be constructed from the system (0.2). Our constructionsmake explicit use of this Fuchsian DPW form. The relation of the monodromy andthe coefficients of the Fuchsian system is the famous Hilbert’s 21st problem, which wasintensely studied [1]. There exist many important partial results in the simplest non-trivial case of four singularities. This case was investigated mostly within the theoryof isomonodromic deformations [7] and the Painlev´e VI equation, where the problem isto describe the coefficients A k as functions of the poles z j when the monodromy groupis 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 isunitary on the unit circle | λ | = 1, giving global solutions of the Gauss equation on thefour-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 Date : February 8, 2021. a r X i v : . [ m a t h . DG ] F e b a ) alternate octahedron ( b ) alternate octahedron ( c ) alternate cube Figure 2 . Triply periodic CMC surfacesin R CMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 2 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, onlysurfaces 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 ≥ . A fundamental piece of a Lawson surface is obtained by thePlateau solution of a specific geodesic polygon. The compact surface is then builtfrom the fundamental 4-gon by the finite group generated by rotations around thegeodesic edges of the polygon. Later, Karcher-Pinkall-Sterling [16] constructed newminimal surfaces in the 3-sphere by starting with a tessellation of the 3-sphere intotetrahedra. The minimal surfaces are obtained from fundamental minimal 4-gonswithin such a tetrahedron which reflect across the geodesic boundaries. Constant meancurvature (CMC) surfaces in R have been constructed by adapting these methods [8];see also [22, 10] for related computer experiments.This paper constructs such fundamental patches of surfaces (section 3) based onthe deformation of DPW potentials. In this paper the following new surfaces arenumerically constructed: triply periodic surfaces (figure 1 b-c , figure 2) and doublyperiodic surfaces (figure 3 b-c , figure 4 b-c , figure 5 b-c ), new doubly periodic surfaceswith Delaunay ends (figure 11), new surfaces with Delaunay ends of positive genus(figures 22, 23 and 30) as well as new KPS-type surfaces (figure 27 a,d ). We alsoreconstruct by these methods previously constructed surfaces based on doubly-periodichexagonal, 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 DGDGallery [5]. Acknowledgements
The first author is partially supported by the DFG Collaborative Research CenterTRR 109
Discretization in Geometry and Dynamics . The second author is supportedby the DFG grant HE 6829/3-1 of the DFG priority program SPP 2026
Geometry atInfinity . The third author is supported by the DFG Collaborative Research CenterTRR 109
Discretization in Geometry and Dynamics . Geometric construction a b cd e f
The construction.
This report reports on the experimental construc-tion of CMC surfaces in R with genus with and without Delaunay endsvia the generalized Weierstrass representation (DPW) [6]. The constructionstarts with a tetrahedron in R as shown which tessellates R by the groupgenerated by the reflections in the four planes containing its faces. Each ofthe six edges of the tetrahedron is marked with an integer n ∈ N ≥ ∪ {∞} specifying that the internal dihedral angle between the two planes meeting at thatedge is π/n . The tetrahedron can be degenerate in the following ways: • vertex at ∞• parallel planes, with opposite outward normals: the edge of the tetrahedronbetween the two planes is marked with ∞• coincident planes, with the same outward normal: the edge of the tetrahedronbetween the two planes is marked with 1. a b cd e f In this tetrahedron construct a CMC quadrilateral as shown, such that • each of the four edges of the quadrilateral lies in a plane of thetetrahedron, and the surface reflects smoothly across this plane • at each of the four vertices of the quadrilateral, application of thetessellation group results in a surface with either an immersed pointof the surface or a once-wrapped Delaunay end at the vertex.Then the surface constructed by application of the tessellation group is a CMC immer-sion with optional Delaunay ends and the symmetries of that group. Its genus is finiteif the tessellation group is finite, and infinite if the group is infinite. These surfacesare described in detail at the end of this section.The polygon is constructed via a Fuchsian DPW potential on C P with four simplepoles on S (section 3.1) and a reflection symmetry across S . The unit disk is thedomain of a CMC quadrilateral which reflects in planes containing its boundaries. The a ) hexagon (bulge) ( b ) hexagon (neck) ( c ) alternate hexagon Figure 3 . Doubly periodic CMC sur-faces in R CMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 3 simple poles with constant or Delaunay residue eigenvalues insure that each vertex ofthe quadrilateral after reflection is either immersed or a Delaunay end. The fourdihedral angles of the tetrahedron at the corners of the quadrilateral are controlled bythe four local monodromies of the potential, and the two remaining dihedral angles bytwo global monodromies.The potential has two accessory parameters which are computed by the unitary flow(section 3.3). Starting with an initial surface (section 3.2) which satisfies the intrinsicclosing condition (unitary monodromy on S λ ), the unitary flow, which preserves thiscondition, is run through the space of potentials until the dihedral angles of the planesreach the values prescribed by the tessellation. The dihedral angles are controlled bycertain monodromy traces at the evaluation point. The unitary flow is not known ingeneral to exist, but short time existence can be shown in some cases [12]. Hence weconstruct the surfaces numerically, giving evidence that the unitary flow has long timeexistence.The Lawson surfaces [18] (figure 25) and the surfaces of Karcher-Pinkall-Sterling [16](figures 26 and 27 and figures 28 and 29) have been constructed by solutions of Plateauproblems. The cubic lattice and some of the 2-dimensional lattices were shown to existby similar methods [8].1.2. Tetrahedral tessellations.
The following theorem classifies the tetrahedral tes-sellation of S (which are compact) and of R (which are compact, paracompact ordegenerate). The tetrahedral tessellation of H , which can be determined by the samemethods, are omitted for simplicity. Theorem 1.1. (1)
The tetrahedral tessellations of S are as follows: a b a, b ∈ N ≥ Minimal Lawson surfaces ξ ab k k ∈ { , , } surfaces with respective tetrahedral, oc-tahedral, and icosahedral symmetries k k ∈ { , , } surfaces with respective -cell, / -celland / -cell symmetries surface with -cell symmetry subgroup of -cell (2) The tetrahedral tessellations of R are as follows: triply periodic surfaces a c b ( a, b, c ) one of (3 , , , (2 , , or (2 , , doubly periodic surfaces n ∞ n ∈ N ≥ ∪ {∞} singly periodic surfaces b a cb c ( a, b, c ) a permutationof (2 , , , (2 , , or (2 , , surfaces with Platonic symmetries andDelaunay ends b a cb c ( a, b, c ) a permutationof (3 , , , (2 , , or (2 , , n
22 1 2 2 1 2 n n n ∈ N ≥ ∪ {∞} tori with Delaunay ends n nn n n n n n ∈ N ≥ ∪ {∞} Proof.
Necessary conditions that a compact tetrahedron tessellates one of the space-forms S , R or H are the following. a ) square (bulge) ( b ) square (neck) ( c ) alternate square Figure 4 . Doubly periodic CMC sur-faces in R CMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 4 • Each edge of the tetrahedron is marked with an integer n ∈ N ≥ denoting thatthe 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 edgesmeeting at the vertex are (2 , , n ), n ∈ N ≥ or (2 , , k ), k ∈ { , , } . • The Gram matrix T ∈ M n × n ( R ) defined by T ij = − cos π/n ij has signature( δ, , , δ = 1, 0 or − S , R and H respectively.The compact tetrahedra in S , R and H are the following: b a a, b ∈ N ≥ A ab b a a, b ∈ { , , , } B ab a a ∈ { , , , } C a b a a, b ∈ { , , , } D ab with identifications A ba = A ab , B ba = B ab , D ba = D ab and(1.1) A = B , A = D , B = C , B = D . 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 meetingthe vertex must be marked with 2 and 3. Hence all such tetrahedra with at least one4 or 5 is one of the four types A ab , B ab , C a or D ab . The remaining tetrahedra haveonly 2 or 3 at each face. There are seven of these, namely A , A = B , B = C , C , A = D , B = D and D .Since its Gram matrix has positive determinant, the tetrahedron A ab is in S . Thespaceforms for the other tetrahedra B ab , C a and D ab are determined by the signs ofthe determinate of the Gram matrix as follows: (1.2) B − − C − D − − − − − − The [+] in the above tables denotes entries which are redundant due to the identifica-tions (1.1)Hence the tetrahedra which tessellate S are A ab and the seven tetrahedra B , B , B , B , B , B and D . These tessellate S because they tessellate either a sphereor a n -cell, which in turn tessellates S .The compact tetrahedra which tessellate R are the three tetrahedra B , C and D . The first two tessellate a cube and the third tessellates a rhombic dodecahedron,each of which in turn tessellates R .The paracompact tetrahedral tessellations of R are classified similarly except thatthe integer triple at each vertex is as in the compact case or one of (3 , , , , , , R are classified similarly except thatthe integer triple at each vertex is as in the paracompact case or one of (2 , , ∞ ) or(1 , n, n ), n ∈ N ≥ ∪ {∞} . (cid:3) The surfaces.
This section describes the experimentally constructed minimalsurfaces in S and CMC surfaces in R . They are of the types:In R : • triply periodic CMC surfaces R without ends (figures 1 and 2) • doubly periodic CMC surfaces R without ends (e.g. figure 3) • doubly periodic CMC surfaces R with ends (figure 11) • single periodic CMC surfaces in R with Delaunay ends (cylinders, figure 16) • CMC surfaces in R with dihedral symmetry and Delaunay ends (tori, fig-ures 19 and 20) • CMC surfaces in R with Platonic symmetries and Delaunay ends (figures 22and 23) • CMC spheres in R with four Delaunay ends (fournoids, figure 24).In S : • Lawson surfaces ξ ab in S (figure 25) • minimal surfaces in S with Platonic symmetries (figures 26 and 27) • minimal surfaces in S with n -cell symmetries (figures 28 and 29) a ) triangle (bulge) ( b ) triangle (neck) ( c ) rhombus Figure 5 . Doubly periodic CMC sur-faces in R CMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 5 • minimal tori in S with Delaunay ends (figure 30).In all shown figures the lines on the surfaces are curvature lines. The surfaces in S are stereographically projected to R . Triply periodic surfaces in R . The simplest triply periodic surfaces can be thought of as tubes along the edgesof the standard cubic lattice in R (figure 7). The genus of those surfaces modulotranslation is 3.The triply periodic surfaces are constructed from the three compact tetrahedrawhich tessellate R . Since the CMC quadrilateral can be situated in each tetrahedronin three ways, this gives nine different configurations, of which two are redundant dueto symmetry (figure 6). cube a oct a a alt cube b alt oct b c c Figure 6 . The tetrahedra for the seven possible triply periodic examples.
Of these, a and b are not possible under our symmetry constraints (compare with(3.2)), c seems to devolve to a , and the flow for c degenerates (figures 1 and 2). Figure 7 . The simplest triply periodic surface and its tetrahedron.
Doubly periodic surfaces in R . The doubly periodic surfaces can be thought of as tubes along the edges of a triangletessellations of R . The six 2-dimensional lattices are constructed with the tetrahe-dron below where ( a, b, c ) are the indices of a triangle tessellation of R , that is, apermutation of (3 , , , ,
4) or (2 , ,
6) (figures 3 to 5). a c b lattice (a, b, c) genushexagon (2, 3, 6) 2square (2, 3, 6) 2triangle (2, 3, 6) 3alt hexagon (3, 3, 3) 2alt square (4, 4, 2) 2rhombus (3, 6, 2) 2
Figure 8 . Left: tetrahedron for doubly periodic surface,, where ( a, b, c ) is a permu-tation of (3 , , , ,
6) or (2 , , Figure 9 . The six 2-dimensional lattices.
Doubly periodic surfaces in R with Delaunay ends. The doubly periodicsurfaces with Delaunay ends are obtained from triangle tessellations of R . Additionalfreedom is given by the choice of vertices corresponding to Delaunay ends (figure 11). a ) hexagon lattice with ends (bulge) ( b ) hexagon lattice with ends (neck) Figure 11 . Doubly periodic CMC sur-faces in R with Delaunay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 6 Cylinders in R with ends. Cylinders with Delaunay ends can be constructedfrom a degenerate tetrahedron with two parallel planes. Of course, the same construc-tion without Delaunay ends give the classical rotational symmetric periodic surfaces,i.e., Delaunay cylinders (figure 16).
Tori in R with Delaunay ends. The torus with n ends is constructed via thediagram below (figures 19 and 20). For large n , existence of those tori can be shownby growing Delaunay ends in equidistance on one side of a cylinder. n
22 1 2
Figure 10 . Left: tetrahedron for a torus in R with n Delaunay ends and order n cyclic symmetry and n ends. Right: fundamental piece of this torus. Surfaces in R with Platonic symmetry and Delaunay ends. Given a triangle tessellations of S , the surface is the orbit of a tube along oneedge of the triangle with a Delaunay end at a vertex of the triangle. Equivalently, thesurface is built from tubes along the edges of one of the five Platonic solids, with endsemanating from the vertices. The five tetrahedra are as in the diagram below, with( a, b, c ) a permutation of (2 , , k ), k ∈ { , , } (figures 22 and 23). a c ba b surface (a, b, c) genustetrahedron (2, 3, 6) 3octahedron (2, 3, 6) 7icosahedron (2, 3, 6) 19cube (2, 3, 6) 5dodecahedron (2, 3, 6) 11 Figure 12 . Left: Tetrahedron for Platonic surfaces in R , where ( a, b, c ) is a permu-tation of (2 , , k ). Right: fundamental piece of the surfaces with Platonic symmetryand Delaunay ends. Lawson surfaces.
Classically, the Lawson surfaces [18] are constructed fromPlateau solutions of a geodesic polygon by reflection. The tetrahedron A ab and itsinscribed fundamental piece admit a rotational order 2 symmetry around a geodesicthrough the vertices labeled by a and b . The geodesic arc is contained in the funda-mental piece. This observation relates the original construction with the constructioncarried out in the present work, see also [16] (figure 25). a b Figure 13 . Tetrahedron for Lawson surface ξ a − ,b − . Surfaces in S with Platonic symmetries. Minimal surfaces in S with Platonic symmetries have been constructed by Karcher-Pinkall-Sterling [16]. These surfaces can be thought of as tubes along one edge of atriangle which tessellates S . Note that [16] does not list all possible surfaces, e.g. thealternate octahedron of genus 11 and the alternate icosahedron of genus 29 are missing(figures 26 and 27). a ) cylinder 2 ∞ ( b ) cylinder 3 ∞ ( c ) cylinder 4 ∞ Figure 16 . CMC cylinders in R withDelaunay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 7 k symmetry genustetrahedron 3cube 5octahedron 7alt octahedron 11dodecahedron 11icosahedron 19alt icosahedron 29 Figure 14 . Left: tetrahedron for surfaces in S with Platonic symmetries. Right:table of these surfaces. Figure 15 . Triangle tessellations of S . Left to right: cyclic of order 5, dihedral oforder 10, tetrahedral, octahedral and icosahedra. Surfaces in S with n -cell symmetries. For each of the n -cell tessellations of S there is a surface which can be thought of astubes along the edges of the cells. These minimal surfaces have also been constructedby Karcher-Pinkall-Sterling [16] (figures 28 and 29). k symmetry genus5-cell 616-cell 1724-cell 73600-cell 601 Figure 17 . Left: tetrahedron for surfaces in S with n -cell symmetries. Right: tableof these surfaces. Figure 18 . The 5-, 8-, 16-, 24-, 120- and 600-cell tessellations of S , stereographicallyprojected to R . CMC polygons via the DPW method
The generalized Weierstrass representation (DPW).
Define the followingloop groups (see [24] for details):Λ = smooth maps (loops) from S λ to SL C Λ u = the subgroup loops in Λ which are in SU on S Λ + = the subgroup loops in Λ which extend to the interior unit disk in C P λ Λ − = the subgroup of loops in Λ which extend to the exterior unit disk in C P λ ˚Λ − = the subgroup of g ∈ Λ –1 such that g (0) is upper triangular with diagonalin R + .These can be generalized to loops on a circle of radius r ∈ (0 , DPW potential ξ on a Riemann surface Σ is a Λ sl (2 , C ) valued holomorphicdifferential form on Σ with ξ = (cid:80) ∞ k =–1 ξ k λ k , det ξ –1 = 0. A meromorphic DPWpotential is defined analogously.A CMC surface is constructed from a DPW potential as follows. Let Φ be the holomorphic frame solving dΦ = Φ ξ ; Φ generally has monodromy. Let Φ = F B ∈ a ) torus 3 (bulge) ( b ) torus 3 (neck) Figure 19 . CMC tori in R with Delau-nay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 8 Λ u ˚Λ + be the Iwasawa factorization into the unitary frame F and positive part B (see [24]); for our case of SU this factorization always exists. The CMC surface isconstructed via the formulas first obtained in [4]: S : f ( λ , λ ) = F ( λ ) F –1 ( λ ) , λ , λ ∈ S (2.1a) H : f ( λ , λ ) = F ( λ ) F –1 ( λ ) , λ = λ –10 ∈ C \ S (2.1b) R : f ( λ ) = – H ˙ F ( λ ) F ( λ ) –1 , λ = 1 or –1(2.1c)where in the case of R the dot denotes the derivative with respect to θ , λ = e i θ . Theunitary frame F yields a unitary potential µ = F − d F 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.A DPW potential ξ = (cid:80) ∞ k =–1 ξ k λ k is adapted if ξ –1 is upper triangular (and hencehas zero diagonal because det ξ –1 = 0). For adapted DPW potentials, the Hopf dif-ferential is Q = (cid:104) ξ –1 , ξ (cid:105) . For non-adapted potentials, the formula for Q is morecomplicated.If ξ is holomorphic at z , the induced CMC surface is immersed at z if and only if ξ –1 does not vanish at z .This representation differs from the original representation [6] in that the potentialand loops are not twisted. It is slightly looser than the representation [25] in that itdoes not require the DPW potential to be adapted.2.2. Delaunay ends. A Delaunay eigenvalue is(2.2) ν = (cid:112) λ –1 ( λ − λ )( λ − λ ) w . where the evaluation points λ , λ and the end weight w ∈ R × chosen so that ν isreal on S . A DPW potential with a simple pole, unitary monodromy, and Delaunayeigenvalues 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 DPWpotential: • The intrinsic closing condition is the condition that the monodromy groupis unitarizable on S (or more generally, r -unitarizable on an circle of radius r ∈ (0 , unitary flow , which by definition preserves the intrinsic closing condition. • The extrinsic closing conditions are conditions on the DPW potential on themonodromy at the evaluation points, chosen to control the desired geometry ofthe surface via (2.1). For surfaces constructed via tessellations these conditionsare 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(2.3) ξ (cid:55)→ g –1 ξg + g –1 d g . The point of the gauge action is that if dΦ = Φ ξ then d(Φ g ) = (Φ g )( ξ.g ). We allowgauges to have monodromy ± along paths; such multivalued gauges nevertheless mapsingle-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 Φ g do. A DPW gauge g isadapted 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 givenby a meromorphic map g : Σ → Λ + . Then, ξ.g is a meromorphic DPW potential withso-called apparent singularities at the singular points of g . In general, the singularitiesof a meromorphic DPW potential are not apparent.2.4. Spin.
For a DPW gauge g define the group homomorphismspin : H (Σ) → Z = {± } , (2.4a) spin γ g = (cid:40) +1 if g has monodromy +1 along γ − g has monodromy − γ (2.4b) a ) torus 4 (bulge) ( b ) torus 6 (bulge) ( c ) torus 8 (bulge) Figure 20 . CMC tori in R with Delau-nay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 9 that is, spin γ g = +1 (resp. –1) if g returns to itself (resp. its negative) along γ . Thenspin γ gh = spin γ g · spin γ h .To define spin for DPW potentials consider the double cover(2.5) C \ { } → { x ∈ sl (2 , C ) \ { } | det x = 0 } , (cid:2) uv (cid:3) (cid:55)→ (cid:2) uv (cid:3)(cid:2) – v u (cid:3) . For a DPW potential ξ on a Riemann surface Σ, let ξ –1 be its λ –1 coefficient. Definethe group homomorphism spin : H (Σ) → Z = {± } , (2.6a) spin γ ξ = (cid:40) +1 if the lift of ξ –1 along γ is a closed cuvve − 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 coefficientof λ –1 . If Φ = F B is the holomorphic frame with potential ξ , and F is the correspond-ing 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 acoordinate frame, that is, a unitary frame G satisfying(2.7) N = Ge G –1 , f x /v = Ge G –1 , f y /v = Ge G –1 where e , e , e is a positively oriented orthonormal basis for su , f is the CMCimmersion, 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 meanspin γ ξ along a small circle γ encircling z . If ξ is regular at z then spin z ξ = 1 . For aDPW potential ξ on C P with finitely many singularities we have the total spin(2.8) (cid:89) z ∈ C P spin z ξ = 1 . As an application of spin, when we construct CMC polygons whose boundariesreflect in planes in section 2.6, the spin is used to distinguish the internal and externaldihedral angles of the planes.2.5.
Symmetry.
The following theorem and lemma detail how a symmetry of thepotential descends to a symmetry of the meromorphic frame, the unitary frame, andthe 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 = R Φ g for some R ∈ Λ . If R is unitary, then the CMC immersion has theorientation preserving symmetry S and H : τ ∗ f = Rf R – (2.9a) R : τ ∗ f = Rf R – − H ˙ RR , (2.9b) where in the case of R the dot denotes the derivative with respect to θ , λ = e i θ . (2) If for an antiholomorphic automorphism τ of the domain, τ ∗ ξ ( λ ) = ξ ( λ ) .g ,then τ ∗ Φ( λ ) = R Φ( λ ) g for some R ∈ Λ . If R is unitary, then the CMCimmersion has the orientation reversing symmetry S and H : τ ∗ f ( λ , λ ) = Rf ( λ , λ ) R – , (2.10a) R : τ ∗ f ( λ ) = − Rf ( λ ) R – − H ˙ RR . (2.10b)In the orientation reversing case of the above theorem, the symmetry (2.10) relatestwo associate CMC surfaces, which are the same surface if λ = λ (for S ), λ ∈ R and λ ∈ R (for H ) and λ ∈ {± } (for R ).Theorem 2.1 is of limited use without the knowledge that R in that theorem isunitary. One necessary condition that R is unitary is given in the following lemma: Lemma 2.2. If ξ in theorem 2.1 extends to S λ and has irreducible unitary monodromy,then R in that theorem is unitary. a ) torus 4 (neck) ( b ) torus 6 (neck) ( c ) torus 8 (neck) Figure 21 . CMC tori in R with Delau-nay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 10 Proof.
Let f : U → Σ be the universal cover, and τ a lift of τ to the universal cover,so f τ = τ ◦ f . Let σ be a deck transformation, so f σ = f . Then f τ στ –1 = f implyingthat τ στ –1 is a deck transformation.Let M σ the monodromy of Φ with respect to σ . and spin σ ∈ {± } the monodromyof g with respect to σ . For the orientation reversing case,(2.11) σ ∗ τ ∗ Φ = σ ∗ ( R Φ g ) = (spin σ g ) RM σ Φ g = (spin σ g ) RM σ R –1 τ ∗ Φso(2.12) τ –1 ∗ σ ∗ τ ∗ Φ = (spin σ g ) RM σ R –1 Φ . 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 λ , this implies R ∈ Λ u .The proof for the orientation preserving case is the same without the overline. (cid:3) CMC polygons.
Let R ∪{∞} be divided into n segments s , . . . , s n at n distinctconsecutive points z ij dividing s i and s j . Let ξ be a meromorphic DPW potential on C P with singularities at these points z ij . With b a basepoint in the upper halfplane,for i, j ∈ { , . . . , 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 ∈ { , . . . , n } , i < j be themonodromy along γ ij . The n local monodromies are those along paths which encloseone singularity; the remaining monodromies are called global . Theorem 2.3.
Let ξ be a meromorphic DPW potential satisfying the conditions oflemma 2.2 with n singularities on R ∪ {∞} as above. Assume ξ admits the reflectionsymmetry τ ∗ ξ ( λ ) = ξ ( λ ) for τ ( z ) = z . Let θ ij ∈ [0 , π ] , i, j ∈ { , . . . , n } , i < j . If themonodromies M ij satisfy (2.13) tr M ij | λ = – (spin γ ij ξ ) cos θ ij , i, j ∈ { , . . . , n } , i < j then the CMC surface induced by ξ with the upper halfplane as domain is a n -gonwhose boundaries reflect in n planes (respectively totally geodesic spheres) P , . . . , 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 basis ˆ e ,ˆ e , ˆ e ∈ sl (2 , C ) and u the unitary λ -independent gauge between them, so F = Gu .By the proof of theorem 2.1(2), τ ∗ k F = P k F , k ∈ { , . . . , n } so(2.14) τ ∗ k G = P k GQ –1 k , τ ∗ u = Q k u . Then with ρ ij and σ ij = spin γ ij u , u = τ ∗ k τ ∗ k u = Q k Q k u = ⇒ Q k Q k = (2.15a) σ ij u = ρ ∗ u = τ j τ i u = Q j Q –1 i u = ⇒ Q j = σ ij Q i . (2.15b)With p k a fixed point of τ k , define u k = u ( p k ) so u k = Q k u k . Then for i, j ∈ { , . . . , n } , i (cid:54) = j , u j = Q j u j = σ ij Q i u j . Thus u –1 i u j = σ ij u –1 i u j . so σ ij = 1 : u –1 i u j = u –1 i u j = ⇒ u –1 i u j e = e u –1 i u j (2.16a) σ ij = –1 : u –1 i u j = − u –1 i u j = ⇒ u –1 i u j e = − e u –1 i u j . (2.16b)Hence u –1 i u j = σ ij e u –1 i u j so u j e u –1 j = σ ij u i e u –1 i .Let e = (cid:2) (cid:3) . Since G is a coordinate frame, we have ( f y ) k = G k ˆ e G –1 k . Define σ k ∈ {± } by P k e = σ k ( f y ) k . Then(2.17) N k .. = P k e = F k F –1 k e = F k e F –1 k = G k u k e u –1 k G –1 k = σ k G k ˆ e G –1 k so(2.18) u k e u –1 k = σ k ˆ e . Hence σ i σ j = σ ij .Since dd y is pointing into the upper half plane, ( f y ) k is an internal normal to theplane. 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 = 1if and only if N i and N j are both internal or both internal, and σ ij = σ i σ j = –1 if andonly if one of N i and N j is internal and one external. a ) tetrahedron (bulge) ( b ) tetrahedron (neck) ( c ) octahedron Figure 22 . CMC surfaces in R withPlatonic symmetry and Delaunay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 11 This means(2.19) (cid:104) N i , N j (cid:105) = − σ ij cos θ ij where θ ij is the internal angle between planes i and j . Since M ij = P j P –1 i , then (cid:3) (2.20) tr M ij | λ = λ = (cid:104) N i , N j (cid:105) = − σ ij cos θ ij . Remark 2.4.
Planes with dihedral angle θ ij = π are parallel. Constraining the planesto coincide (for example, for the CMC torus with ends in R ) requires that the extrinsicconditions of theorem 2.3 be augmented with the additional condition(2.21) dd λ M ij | λ = λ = 0 . 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 . (1) If ν = 1 / (2 n ) or ν = − / (2 n ) , n ∈ N ≥ , and ξ − has a simple pole at z k ,then the CMC surface constructed from ξ with n reflections around the vertexis immersed at the vertex. (2) If ν = ν Del /n , n ∈ N ≥ , where ν Del is a Delaunay eigenvalue (2.2) , then theCMC surface constructed from ξ with n reflections around the vertex is aonce-wrapped Delaunay end.Proof. Assuming z k = 0, write ξ = A –1 d z/z + A d z + . . . . The pullback with respectto the local covering map f ( w ) = w n is(2.22) f ∗ ξ = nA –1 d w/w + nA w n − d w + . . . . Proof of (1): Since by assumption A − has a pole at λ = 0, by a z -independentlocal gauge of ξ it may be assumed(2.23) A –1 = (cid:20) / (2 n ) λ –1 / (2 n ) (cid:21) . Then the local gauge g = diag( w –1 / , w / ) removes the simple pole of f ∗ ξ at w = 0,and(2.24) ( f ∗ ξ ) .g = (cid:20) nλ − (cid:21) d w + . . . . Since ( f ∗ ξ ) .g is holomorphic at w = 0 and its λ − 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 = − / (2 n ) 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. (cid:3) Symmetric CMC surfaces with genus
The potential.
By applying a M¨obius transformation we assume that the sin-gular points of the CMC polygon are on the unit circle. As the fundamental piece is aCMC quadrilateral, we restrict to the 4-punctured sphere in the following. We will seein section 4 that, at least for surfaces without Delaunay ends, we can restrict withoutloss of generality to a Fuchsian DPW potential of the 4-punctured sphere. The meansit has four simple poles and no pole at z = ∞ , and is of the form(3.1) ξ = (cid:88) k =0 A k z − z k d z as follows: • The poles are z ∈ S in the open first quadrant, and ( z , z , z ) is a permu-tation of (1 /z , − z , − /z ). • The residues are A = (cid:34) y λ − p λ ( ν − y ) p − y (cid:35) , A = (cid:34) − y ( ν − y ) x x y (cid:35) , (3.2a) A = σA σ − , A = σA σ − , σ = diag( i , − i ) . (3.2b) a ) cube ( b ) icosahedron ( c ) dodecahedron Figure 23 . CMC surfaces in R withPlatonic symmetry and Delaunay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 12 • For surfaces without Delaunay ends, the eigenvalue ν of A and A and ν of A and A are constants in (0 , / ν is of the form cν Del with c ∈ (0 ,
1) and ν Del is the eigenvalue of a Delaunayunduloid(3.3) ν Del = (cid:113) λ –1 ( λ − w , w ∈ (0 , . • The accessory parameters x and y are holomorphic functions of λ on an opendisk D r of radius r > 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 − y ) /p and ( ν − y ) /x are holomorphic functions of λ on D r .The need for the last condition is as follows. The unitary flow, which preserves theunitarizability of the monodromy of ξ , is implemented by evaluating the monodromyof ξ directly on the unit circle, and not by the numerically more problematic procedureof computing the monodromy on an r < M k ( k ∈ { , . . . , } ) 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) sothat at the end of the flow for k = 0 , ν | λ = λ k = n , ν | λ = λ k = − n , (3.4a) tr M M | λ = λ k = − cos πr , tr M M | λ = λ k = cos πs . (3.4b) n s n r Then, by theorems 2.3 and 2.5, the unit z -disk maps to a CMC quadrilateralwhose edges reflect in planes (respectively geodesic 2-spheres) with internaldihedral angles specified by the figure at right, and whose vertices after thesereflections are either immersed points or once-wrapped Delaunay ends.In the special case ν + ν = , the surfaces are constructed by runningthe unitary flow so that at the end of the flow for k = 0 , ν | λ = λ k = n , ν | λ = λ k = − n , (3.5a) tr M M | λ = λ k = − cos πr , tr M M | λ = λ k = cos πs (3.5b) s n r Then the quarter disk in the first quadrant maps to a CMC quadrilateralwhose edges reflect in planes with internal dihedral angles specified by thefigure at right, and whose vertices after these reflections are immersed.3.2.
The initial condition.
The initial condition.
The initial condition for the unitary flow is apotential ξ of the form in section 3.1 with eigenvalues ν = ν = withunitary monodromy on S which induces a Delaunay surface. Lemma 3.1. (1)
For CMC tori of spectral genus the spectral curve π : C P → C P can bechosen to be π ( ξ ) = ξ . The involutions are (3.6) σ ( ξ ) = − ξ , ρ ( ξ ) = ξ − , κ ( ξ ) = ξ . The monodromy eigenvalues of the vacuum are exp( ± ν + 2 i π Z ) , exp( ± ν +2 i π Z ) where (3.7) ν ( ξ ) .. = i π ξ − ξ − ξ − ξ − , ν ( ξ ) .. = i π ξ + ξ − ξ + ξ − and π ( ± ξ ) , π ( ± ξ − ) ∈ S are the evaluation points. (2) For some (cid:96), m ∈ Z + the monodromy eigenvalues of a Delaunay cylinder are exp( ± ν + 2 i π Z ) , exp( ± ν + 2 i π Z ) where ν ( u ) .. = i π f ( u ) − f ( u ) f ( u ) − f ( u ) , (3.8a) ν ( u ) = (cid:96) ( f ( u ) + f ( u )) , ν ( u ) = i πm , (3.8b) where π ( ± u ) , π ( ± u + ω ) ∈ S are the evaluation points, and f , f are asin (3.9) . a ) fournoid ( b ) fournoid ( c ) fournoid ( d ) fournoid Figure 24 . CMC sphere in R with De-launay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 13 On the torus C / ( Z + τ Z ), Let ℘ be the Weierstrass function and let ζ .. = − (cid:82) ℘ . Let { ω , ω , ω } = { , + τ , τ } .Define on some torus with modulus τ spec (3.9) h ( u ) .. = η u − ω ζ ( u ) , h ( u ) .. = f ( u − ω ) . The theta function(3.10) θ ( x, τ ) .. = (cid:88) k ∈ Z exp (cid:0) i π ( n τ + n ( x − ω ) (cid:1) is an entire function C → C with simple zeros at lattice points Z + τ Z and no otherzeros, satisfying(3.11) θ ( x + 1) = θ ( x ) , θ ( x + τ ) = − exp( − i πx ) θ ( x ) , θ ( τ − x ) = θ ( x )for all x ∈ C . Define g ( x ) = θ (cid:48) ( x ) θ ( x ) − θ (cid:48) ( x ) θ ( x ) + i π , (3.12a) g k ( x ) = exp (cid:0) i πx ω k − ω k τ − τ (cid:1) θ ( x + ω k ) θ (cid:48) (0) θ ( x ) θ ( ω k ) , k ∈ { , , } . (3.12b)The initial condition is the potential in section 3.1 with x = λ ( y + ν )( y + ν ) 1 − u u , y ( b, a ) = − i πτ − τ ( b + a ) + f ( b )2 g ( b )(3.13a) u ( b ) = − g ( b ) g ( b ) , v ( b, a ) = i πτ − τ ( b + a ) + f ( b ) g ( b )(3.13b) a = i πτ − τ h , b = i πτ − τ h (3.13c) ν = ν = , [ z , z , z , z ] = u ( ω ) , p = 1 . (3.13d)The initial condition is computed numerically from (3.13a) as Laurent series on S 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 theinitial condition creates different arrangements of residues of the DPW potential onthe Delaunay cylinder. For the configurations used in this report, the two circle arcs( z , z ) and ( z , z ) are mapped to semicircles (resp. profile curves) on the Delaunaysurface, while the other two circle arcs ( z , z ) and ( z , z ) are mapped to profilecurves (resp. semicircles) on the Delaunay surface. The first of these configurations isused to compute the 2-dimensional lattices and the cubic lattices; the second is usedto compute the tori and Platonic surfaces with ends.3.2.3. Neck and bulge.
For the initial potential ξ above, the poles of the FuchsianDPW potential are at necks of the Delaunay surface. The initial potential with polesat bulges is constructed as a gauge of ξ by the gauge diag(( λ − λ ) − / , ( λ − λ ) / ),where the λ is a common zero of x and y − /
16. This gauge is not a DPW gauge,but a so-called dressing transformation.3.3.
The unitary flow.
The unitary flow.
The unitary flow is a flow through the space of potentialsof section 3.1 preserving the intrinsic closing condition. It starts at a potential inthe space with unitary monodromy, and flows until the monodromy at the evaluationpoints reach some desired extrinsic closing conditions.Given a smooth function F = F ( t, (cid:126)x ) : R n → R n encoding n conditions on a flowparameter t ∈ [0 ,
1] and n variables (cid:126)x , if det d F d x (cid:54) = 0, then x ( t ) satisfying F ( t, x ( t )) = 0can be computed by the implicit ODE d F d t + d F d x d x d t = 0 . The solution for the infinitecase can be computed numerically by truncating to F : R n → R m , m ≥ n , andsolving the resulting finite dimensional ODE by least squares methods.The variables (cid:126)x parametrizing the potential consists of: • the conformal type [ z , z , z , z ] • the local eigenvalues ν | λ =1 and ν | λ =1 • the end weight w a ) Lawson ξ ( b ) Lawson ξ ( c ) Lawson ξ Figure 25 . Minimal Lawson surfaces in S CMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 14 • the polynomial p • the accessory parameters x and y .The accessory parameters, which are holomorphic functions of λ , are approximatedby truncating their power series at λ = 0. We always assume that these functionsextend to a disc D r with r > F = 0 are of two types: • the intrinsic closing conditions: the halftraces t ij = tr M i M j , i, j ∈ { , . . . , } , i < j are real on S . This is a necessary condition by theorem 3.4. • geometric constraints which choose a path through the space of geometricparameters to reach the desired extrinsic closing conditions at the end of theflow.By theorem 3.4 the monodromy is unitarizable if all halftraces along the unit circleare real and of absolute value less or equal to 1. As the components of irreducibleSL(2 , C )-representations of the 4-punctured sphere with real traces consists entirely ofeither SL(2 , R ) or SU(2) representations, and since ξ is unitarizable, we can ignorethe condition that all traces are of absolute value less or equal to 1 during the unitaryflow.The intrinsic closing conditions on S λ are approximated by evaluation at finitelymany equally spaced sample points on S λ . In the following we describe the otherconstraints in more detail:3.3.2. Geometric constraints.
The simplest configuration of the geometric constraintsare as follows. The two local and two global eigenvalues depend linearly on the flowparameter t to reach the desired values at t = 1. If the surface has no ends, the endweight w is set to 0; otherwise it depends linearly on t starting at 0 and reaching aheuristically chosen value at t = 1. In this configuration the conformal type is fixedduring the flow.It is possible that the flow with this simple configuration breaks down, in whichcase the path must be modified in some heuristically determined way, for example bymaking 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 (cid:126)x .3.4. Irreducibility and unitarizability.
For a subgroup
G ⊂ SL C generated bythree elements, this section proves • a necessary and sufficient condition for the irreducibility of G , and • a necessary condition for the SU unitarizability of G , assuming G is irreducible.Here, a group G is reducible if all elements have a common eigenline, and is SU unitarizable if there exists C ∈ SL C such that C G C –1 ⊂ SU . The methods used inthe proofs can be generalized to any finitely generated group.The proof depends on the following lemma 3.2, which determines to what extendthree elements of C are determined by their standard C inner products.With (cid:104)− , −(cid:105) the standard inner product on C , let L = { v ∈ C |(cid:104) v, v (cid:105) = 0 } . Let X = ( x , x , x ) ∈ M × C , with columns x , x , x ∈ C . Let W = X (cid:62) X ∈ Sym n C ,so W ij = (cid:104) x i , x j (cid:105) . Lemma 3.2.
With X and W as above, (1) ker X (cid:62) ∩ L = { } if and only if rank W ≥ . (2) Assuming (a), if for some Y ∈ M × C , X (cid:62) X = Y (cid:62) Y and det X = det Y ,then there exists a unique S ∈ SO C such that Y = SX .Proof. By the rank-nullity theorem applied to X (cid:62) | image X ,(3.14) rank X = dim(ker X (cid:62) ∩ image X ) + rank W from which it follows that rank X ≥ rank W , and rank W = 3 if rank X = 3. a ) tetrahedron ( b ) octahedron ( c ) cube Figure 26 . Minimal surfaces in S withPlatonic symmetryCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 15 Moreover, if rank X ≥
2, then(3.15) ker X (cid:62) ∩ image X = ker X (cid:62) ∩ L . To prove (1), assume rank W ≥
2, so rank X = 2. By (3.14), dim(ker X (cid:62) ∩ image X ) = 0. By (3.15), ker X (cid:62) ∩ L = { } .Conversely, assume ker X (cid:62) ∩L = { } . Then rank X ≥ C intersects L . By (3.15), ker X (cid:62) ∩ L = { } . By (3.14), rank W =rank X ≥ W = 3, since rank X = rank Y = 3, define S .. = Y X − .Then S ∈ SO C by X (cid:62) X = Y (cid:62) 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 let ˆ X = ( x a , x b , x a × x b ) and ˆ Y = ( y a , y b , y a × y b ). Since x a × x b ∈ ker X (cid:62) , then bythe assumption and lemma 3.2(1), x a × x b (cid:54)∈ L . Then det ˆ X = (cid:104) x a × x b , x a × x b (cid:105) (cid:54) = 0 sorank ˆ X = 3. Moreover, since (cid:104) x a × x b , x a × x b (cid:105) = (cid:104) y a × y b , y a × y b (cid:105) , then ˆ X (cid:62) ˆ X = ˆ Y (cid:62) ˆ Y .Then S .. = ˆ Y ˆ X − is in SO C , and Y = SX . (cid:3) Identify C with gl (2 , C ) by identifying the standard basis E , E , E , E with , e , e , e , where(3.16) e = (cid:2) i
00 – i (cid:3) , e = (cid:2) − (cid:3) , e = (cid:2) i – i (cid:3) . Under this identification, the standard inner product on C is(3.17) (cid:104) x, y (cid:105) = tr x adj( y ) , adj (cid:2) a bc d (cid:3) .. = (cid:2) d − b − c a (cid:3) and SU ⊂ SL C is identified with R ⊂ C . In particular, for X, Y ∈ sl (2 , C ) it holds (cid:104) x, y (cid:105) = − tr xy .In order to treat irreducibility, the following lemma translates the notion of eigenlineto a more convenient form. With (cid:2) ab (cid:3) ⊥ .. = [ − b a ] consider the double cover(3.18) ˆ x ∈ C \ { } → { x ∈ sl (2 , C ) | det x = 0 } , ˆ x (cid:55)→ x = ˆ x ˆ x ⊥ . Lemma 3.3. (cid:96) ∈ C \ { } is an eigenvector of the invertible matrix x ∈ SL C if andonly if (cid:104) x, (cid:96)(cid:96) ⊥ (cid:105) = 0 .Proof. For any p, q ∈ C \ { } ,(3.19) tr qp ⊥ = p ⊥ q = det( p, q ) . So with p = (cid:96) , q = x(cid:96) , and y = (cid:96)(cid:96) ⊥ (3.20) 2 (cid:104) x, y (cid:105) = 2 (cid:104) x, (cid:96)(cid:96) ⊥ (cid:105) = det( x(cid:96), (cid:96) )so (cid:104) x, y (cid:105) = 0 if and only if x(cid:96) and (cid:96) are dependent, that is, if and only if (cid:96) is aneigenline of x . (cid:3) Let P be the group generated by P = , P , P , P ∈ SL C . Under the aboveidentification C ∼ = gl (2 , C ) let P = ( P , P , P , P ) ∈ M × C be a matrix withcolumns P k ∈ C . Let T = P (cid:62) P ∈ Sym C , so T ij = (cid:104) P i , P j (cid:105) . Theorem 3.4.
With P and T as above, (1) P is irreducible if and only if rank T ≥ . (2) Assuming (1) , P is SU unitarizable if and only if T is real positive semidefi-nite.Proof. We have the factorization(3.21) T = (cid:20) V (cid:62) (cid:21) (cid:20) X (cid:62) (cid:21) (cid:20) X (cid:21) (cid:20) V (cid:21) , P = (cid:20) V X (cid:21) . To prove (1), let X = ( x , x , x ) ∈ M × C be the lower right 3 × P , that is the matrix with columns given by the tracefree parts of P , P , P , andlet Y .. = X (cid:62) X . By lemma 3.3, P is irreducible if and only if ker X (cid:62) ∩ L = { } . Bylemma 3.2 this is if and only if rank Y ≥
2. Since rank T = 1 + rank Y , this is if andonly if rank T ≥ P is SU unitarizable, it may be assumed without loss of generalitythat P ⊂ SU . Then P ∈ M × R , so T = T (cid:62) T ∈ Sym R is real positive semidefinite.Conversely, if T is real positive semidefinite, then W = X (cid:62) X = Y (cid:62) Y for some Y ∈ M × R . Replacing Y (cid:55)→ – Y if necessary, then det X = det Y , so by lemma 3.2(2),there exists S ∈ SO C such that X = SY . Let C ∈ SL C be a lift of S via the double a ) alternate octahedron ( b ) icosahedron ( c ) dodecahedron ( d ) alternate icosahedron Figure 27 . Minimal surfaces in S withPlatonic symmetryCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 16 cover SL C (cid:55)→ SO C defined with respect to , e , e , e . Note that this double coveris given by conjugation on sl (2 , C ) ∼ = C . Then C unitarizes P . (cid:3) Constructing the surface.
Once the potential for a surface is obtained via theunitary flow, the surface is constructed as follows: • 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.3.5.1.
The unitarizer.
Due to the symmetry (3.2) of the potential and of the mon-odromies with basepoint z = 1 the unitarizer is diagonal and can be computed asfollows. With the notation a ∗ ( λ ) = a (1 /λ ) write(3.22) M = (cid:20) a bc a ∗ (cid:21) . By the unitarizability of M by a diagonal loop, p .. = − c ∗ /b = − c/b ∗ takes values in R + along S away from its zeros and poles, which are even. Let f = (cid:81) ( λ − α ) / (cid:81) ( 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 without zeros or poles. Let y ∗ y = q be the scalar GL C Birkhoff factorization,so y is holomorphic in the unit disk. Then with x = f y the loop diag( x / , x − / ) isthe required unitarizer, holomorphic on the open unit disk.3.5.2. Curvature lines.
Let Q = q ( z )d z the Hopf differential of the CMC surface.The curvature line coordinate v satisfies d v = Q ( z )d z . Curvature line coordinatescan be computed by computing (cid:82)(cid:112) Q ( z ) d z over the domain.The surface is computed numerically by dividing the domain into polygons (trianglesor quadrilaterals) and mapping via the CMC immersion these triangles to R . In thecomputation of curvature lines described above, the polygon edges are unrelated tothe 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 thatthe umbilics are at corners of the quadrilaterals. For each quadrilateral, pull back thepotential to curvature line coordinates.This computation is complicated by the fact that the maps from curvature linesrectangles to the domain are singular at the umbilics, and the potential is singularat the umbilics. The potential can be desingularized locally at an umbilic z by acoordinate change of the form z = z + w n and a gauge.3.5.3. Building the surface.
In general the position of the surface in space is not con-trolled, so to build the surface it must first be put into a standard position, where agroup of standard reflections can be applied. To do so, compute the four generatingreflections R k in the isometry group Iso R of R . Conjugate them to standard reflec-tions S k via CR k C − = S k . Then the surface after being moved via x (cid:55)→ Cx has thestandard reflections S k as symmetries.3.5.4. The bulge count for families of CMC surfaces.
The surfaces constructed in thispaper allow for non-trivial 1-parameter deformations within the space of CMC surfaceswith the same combinatorics. A natural question, also considered by [8], is whetherdifferent surfaces with the same combinatorics, but which swap neck and bulge, belongto the same family of CMC surfaces, for example figure 1 a-b , and figure 3 a-b . It turnsout that these examples belong to different families. We denote by a leg of the surfacea cylindrical piece obtained from the trajectories of the Hopf differential, that is fromthe curvature line parametrisation. Although the images are labeled according towhether there are bulges or necks where the legs meet, in this section we rather countthe number of bulges on each leg.We show that this number is an invariant in the case of surfaces without ends. Notethat in this case, there is a covering Σ → C P by a compact Riemann surface Σ onwhich the pullback of the DPW potential has only apparent singularities. Phraseddifferently, Σ 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, Σ is just theunderlying Riemann surface, and in the case of periodic CMC surfaces in R , Σ is theRiemann surface quotient of the CMC surface by the translational symmetries. a ) 5-cell ( b ) 16- or 8-cell ( c ) 600- or 120-cell Figure 28 . Minimal surfaces in S with n -cell symmetryCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 17 We construct surfaces starting from Delaunay cylinders by deforming the eigenval-ues ν i . In the case of cylinders without umbilics, all four eigenvalues are ν i = . Atthe 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 withthe (0 , ∂ ∇ of the connections) are semistable, i.e. if they admit holomorphicline subbundles of degree 0. A holomorphic structure (on a rank 2 bundle over acompact Riemann surface of degree 0) is called unstable if there exist a holomorphicline subbundle of positive degree and they are called stable if every holomorphic linesubbundle has negative degree. This notion is relevant to us since an unstable holo-morphic structure does not admit a flat unitary connection. Spectral parameters λ atwhich the holomorphic structure is unstable are isolated in the spectral plane. More-over, for CMC surfaces based on quadrilaterals, the number of those values of spectralparameters within a bounded region is always finite and can only change during adeformation by values crossing the boundary of that region. Values of the spectralparameter 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 issemistable for all spectral values. The number of values at which the holomorphicbundle becomes unstable within infinitesimal deformation of the eigenvalues ν i can beidentified with the number of bulges on the leg of the initial Delaunay cylinder; formore details see [12]. Actually this number coincides with the number of zeros of theholomorphic function x in (3.2) inside the unit circle; see also [11, 15] Fuchsian DPW potentials
The aim of this section is to prove the existence of Fuchsian DPW potentials of theform (3.1) for CMC quadrilaterals without Delaunay ends. This generalizes previouswork by the second author [14] for the Lawson genus 2 surface. Similar results havebeen obtained by Manca [19]. Our arguments are more geometric and prove theexistence of a Fuchsian potential on a 4-punctured sphere for all surfaces obtained byCMC quadrilaterals.4.1.
Setup.
Let f : Σ → M (where M ∈ { S , R } ) be a complete CMC surface with-out 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 arcscontained 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 Σby conformal transformations and on the ambient space M by a representation ρ intothe 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 isthe converse of theorem 2.5. This means that at fixed points of a rotational symmetrythere 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 thereexists 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 . Lemma 4.1.
There exists D ∈ SU of order k and a local DPW potential η for f onan open g -invariant neighbourhood of p such that (4.1) g ∗ η = DηD − . Proof.
Consider Dorfmeister’s normalized potential (see for example [29]) which takesthe form(4.2) η nor = (cid:20) λ − f ( z, qf ( z, (cid:21) dz where z is a local holomorphic coordinate centered in p such that g ∗ z = e πik z , Q = q ( dz ) is the Hopf differential and f ( z, w ) is a holomorphic function such that f ( z, ¯ z ) dzd ¯ z is the induced metric of the surface. As g ∗ dz = e π i k dz and g ∗ d ¯ z = e πik d ¯ z the result follows. (cid:3) a ) 24-cell ( b ) 16- or 8-cell ( c ) 600- or 120-cell Figure 29 . Minimal surfaces in S with n -cell symmetryCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 18 Proposition 4.2.
There exists a local meromorphic DPW potential of f on Σ /G o with a Fuchsian singularity at p mod G o . The eigenvalues of the residue are ± k , independently of λ , where k is the order of the stabilizer group of p .Likewise, there exists a local meromorphic DPW potential of f on Σ /G o with Fuch-sian singularity at p mod G o such that the eigenvalues of the residue are ± k − k .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, √ z ) of spin −
1. Then(4.3) ( d + η nor ) .e = d + e − de + e − 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 firstpart of the proposition.For the second part and k = 2 l + 1 consider the gauge ˜ e = diag ( z − l , z l ) while for k = 2 l consider the gauge ˜ e = diag ( z − l + 12 , z l − ), and proceed as in the first part ofthe proof. (cid:3) Global theory.
Our aim is to construct a DPW potential on the Σ /G o . Recallthat by assumption the fundamental piece P of the Riemann surface Σ is of thetopological type of a disc. Lemma 4.3.
The Riemann surface Σ /G o is the projective line.Proof. By the Riemann mapping theorem there exists a holomorphic map from P tothe unit disc. Schwarzian reflection yields a holomorphic map from Σ to C P , branchedat the fixed points of G o . By its construction, this map is invariant under G o . (cid:3) For simplicity of the arguments, we will assume that n is even in the following. Lemma 4.4.
Let n be even. There exists a unitary potential µ on the n -puncturedRiemann sphere such that • µ is singular exactly at the branch values of Σ → Σ /G o = C P ; • the pull-back of µ generates f on the covering Σ .Proof. Let { z , . . . , z n } ⊂ C P be the branch values of Σ → Σ /G o = C P and S ⊂ Σits preimage. Denote the reflection planes of the fundamental piece by P , . . . , P n − , with outward oriented unit normals N , . . . , N n − , respectively, such that(4.4) z m ∈ P m − ∩ P m ∀ m ∈ { , . . . , n } . Denote by g m the compositions of the reflection across P m − and P m . Then G isgenerated by { g m | m ∈ { , . . . , n }} .Let M be euclidean 3-space or the 3-sphere, and let d = 3 and d = 4 accordingly,so that Iso( M ) = SO R or Iso( M ) = SO R (cid:110) R .Consider the group(4.5) H ⊂ Spin n × Iso( M )generated by the elements(4.6) ˆ g m := ( N m · N m − , g m ) , m = 1 , . . . , n where · denotes Clifford multiplication. This gives a group extension(4.7) { id } → Z → H → G → { id } . Note that ˆ g m has order 2 k if g m has order k . Similarly, since n is even, the productˆ g n . . . ˆ g is trivial. Consequently, we have a representation(4.8) h : π ( C P \ { z , . . . , z n } , ∗ ) → H .
As(4.9) Σ \ S → C P \ { z , . . . , z n } is a (unbranched) covering, the fundamental group of Σ \ S (with appropriate basepoint) is a subgroup of the first fundamental group of C P \ { z , . . . , z n } with corre-sponding base point. By construction, the induced representation of π (Σ \ S, ∗ ) → G is trivial, and the induced representation of h takes values in(4.10) Z = Z × { id } ⊂ Spin d × Iso( M ) a ) torus 8 Figure 30 . Minimal torus in S withDelaunay endsCMC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 19 such that a simple closed curve around any one of the points in S is mapped to thenon-trivial element in Z .By Riemann surface covering theory, we obtain a 2-fold covering(4.11) ˆΣ → Σ , branched over the points in S , with an action of H by holomorphic automorphisms onˆΣ such that(4.12) ˆΣ → ˆΣ /H = C P is branched over { z , . . . , z n } . Denote its preimage of S by ˆ S ⊂ ˆΣ. Note that H actsfaithfully on ˆΣ \ ˆ S .Consider the pull-back ω on ˆΣ of the unitary potential η = F − d F of f . Note that,for minimal f : Σ → S the unitary potential is given by(4.13) η = λ − Φ + Φ − Φ ∗ − λ Φ ∗ where(4.14) Φ = ( f − df ) , and Φ ∗ = ( f − df ) , and similarly for CMC surfaces f : Σ → R . Let π : H → SU be the projection to therotational part of the symmetry. From the construction (4.13) of the unitary potential,(4.15) h ∗ η λ = η λ .π ( h )for a holomorphic automorphism h ∈ H (where, on the right hand side, the gaugeaction of the constant matrix π ( h ) is given by conjugation).Consider the free action of H on ˆΣ \ ˆ S × C given by(4.16) ( p, v ) .h = ( p.h, π ( h − )( v )) . The quotient(4.17) V = ( ˆΣ \ ˆ S × C ) /H is a trivial smooth vector bundle of rank 2 over C P \ { z , . . . , z n } . We claim that theunitary potential ω yields a well-defined potential µ on this vector bundle: in fact, theconnection 1-form acts on [ p, v ] ∈ ( ˆΣ \ ˆ S × C ) /H as(4.18) [ p, ω p ( v )]which is well-defined since(4.19) ( p, ω p ( v )) .h = ( ph, h − ω p hh − ( v )) = ( ph, ω ph ( h − v )) . (cid:3) Proposition 4.5.
Let n be even. There exist a meromorphic DPW potential ξ on C P with simple poles at z , . . . , 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 ∈ { , . . . , n } . By proposition 4.2 there exist a DPW gauge locally well-definedon a punctured disc around z l which gauges µ into a meromorphic potential with aFuchsian singularity at z l . Of course, the holomorphic structures (i.e. the (0 , s : π (Σ \ S, Z ) tohave local monodromy − S .Using these gauges as cocycles, we obtain a holomorphic C ∗ -family of flat SL C -connections d + ˆ µ with the following properties: • the induced family of holomorphic structures extends to λ = 0 to give a holo-morphic rank 2 bundle E → C P with trivial holomorphic determinant; • the connections d + ˆ µ have Fuchsian singularities with λ -independent eigen-values ± k ; • the complex linear part of the family of connections has a first order pole at λ = 0, i.e., λ (cid:55)→ λ (ˆ µ ) , ( λ ) extends to λ = 0. MC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 20
Note that all bundles have trivial determinant. Hence, by the Birkhoff-Grothendiecktheorem, the bundle type of E is O ( d ) ⊕ O ( − d ) for some d ∈ N .First consider the case d = 0. Then, the bundle type E λ is locally constant on anopen disc near λ = 0. In particular, there exists a smooth positive family of gaugetransformations g λ (holomorphic in λ ) such that(4.20) (( d + ˆ µ ) .g λ ) , = d , is the trivial holomorphic structure on the rank 2 vector bundle C → C P . Thus(4.21) d + ξ := ( d + ˆ µ ) .g λ is the meromorphic DPW potential which has only Fuchsian singularities and no ap-parent singularity at ∞ .Let d >
0. Let z : C P \ {∞} → C be an affine holomorphic coordinate and assumewithout loss of generality that z l (cid:54) = ∞ ∀ l ∈ { , . . . , n } . There exists an integer0 ≤ s ≤ d such that on a punctured disc D \ { } around λ = 0 all bundles are ofthe 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 holomorphicbundle (( d + ˆ µ ) .g λ ) , has the cocycle (for the covering U + := C , U − := C P \ {∞} of C P )(4.22) (cid:20) z − d z s r ( λ )0 z d (cid:21) = (cid:20) r ( λ ) z s z d r ( λ ) z − s (cid:21) (cid:20) r ( λ ) z − d − s − (cid:21) , where the equality obviously holds only for r ( λ ) (cid:54) = 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) andgauge ( d + ˆ µ ) , on U ± to the trivial holomorphic structure on C → U ± . Then,(4.23) ( d + ˆ µ ) .g + λ =: d + ξ yields the meromorphic potential ξ with Fuchsian singularities at z k and an apparentsingularity at ∞ . (cid:3) 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). Lemma 4.6.
For n = 4 the bundle type of E is either trivial or O (1) ⊕O ( − → C P .Proof. Assume the bundle type at λ = 0 is O ( d ) ⊕ O ( − d ) → C P for some d >
1. TheHiggs field Φ := res λ =0 ξ is a meromorphic section of the bundle(4.24) K ⊗ End ( E ) → C P where K = O ( −
2) is the canonical bundle of C P and End ( E ) denotes the trace-freeendomorphisms of E . Moreover, Φ is nilpotent as the immersion is conformal, has atmost simple poles at z , . . . , z by construction and does not vanish on C P as f is animmersion. Using the decomposition E = O ( d ) ⊕ O ( − d ) the Higgs field is of the form(4.25) Φ = (cid:20) a bc − a (cid:21) where a, b, c are meromorphic sections in O ( − O ( − d ) , O ( − − d ) , respectively,with at most simple poles at z , . . . , z . Hence c = 0. As Φ is nilpotent a = 0 as well.For d > − d > b would have a zero contradicting the fact that Φ isnowhere vanishing. (cid:3) Theorem 4.7.
Let f be a complete CMC surface without Delaunay ends in S or R .If f is built from a CMC quadrilateral in a fundamental tetrahedron of a tessellationof the ambient space then it is obtained from a Fuchsian DPW potential (3.1) on the4-punctured sphere.Proof. We give a proof by contradiction. Assume that the bundle type at λ = 0 is(4.26) E = O ( − ⊕ O (1) . MC SURFACES BASED ON FUNDAMENTAL QUADRILATERALS 21
By the proof of proposition 4.2 the nilpotent λ − -part Φ = ξ − of the meromorphicpotential has no zeros, and poles of order 1 at the 4 branch points z , . . . , z . Thus,with respect to (4.26), it must be of the form(4.27) Φ = (cid:20) s − D (cid:21) where s − D ∈ M ( C P , O ( − D = z + · · · + z . Moreover, the positive eigenvalues ν i of the residuesof the connections are contained in the respective kernels of the residues of ξ − . Thisequips E with a parabolic structure (see for example [21, 26, 23, 11, 3] for definitionsand further references) which is unstable. We denote the parabolic bundle also by E .The pair ( E , Φ) is a stable strongly parabolic Higgs pair. Note that(4.28) (cid:88) i ν i < . It is easy to see (compare with [11]) that ( E , Φ) is the only stable strongly parabolicHiggs pair with nilpotent Higgs field on the 4-punctured sphere with unstable under-lying parabolic bundle. Consider the compact Riemann surface X → C P on whichthe rotational symmetry is trivial. Its Fuchsian monodromy (given by uniformization)corresponds by the Hitchin-Kobayashi correspondence to a stable nilpotent Higgs pair(4.29) (cid:18) S ∗ ⊕ S , (cid:20) (cid:21)(cid:19) , where S = K X . Its underlying holomorphic structure is unstable. As the rotationalsymmetries act on X we obtain, in the same manner as for f , an strongly parabolicnilpotent Higgs pair with underlying parabolic structure. As the holomorphic structureis unstable, the parabolic structure must be unstable as well; see [2], and hence itmust be ( E , Φ). Thus, the holomorphic Higgs pair of f , i.e., ( ∂ ∇ , Φ) would be gaugeequivalent to ( S ∗ ⊕ S, (cid:2) (cid:3) ). This is only possible if the Hopf differential of the minimal(respectively CMC) surface vanishes (compare with [14, sections 2 and 3]), which givesa contradiction. (cid:3) Finally, we show under which conditions the Fuchsian DPW potential ξ can begauged into the form (3.2).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 residuesare contained in (0 , ). Corollary 4.8.
Assume that the potential of theorem 4.7 has equal pairs of eigenval-ues. Then, there exist a coordinate change and a gauge such that the potential is ofthe form (3.2) .Proof.
We only sketch the proof. Assume that the eigenvalues at z and z , respectively z and z are equal. First, apply a so-called flip gauge which flips the eigenvalues at z and z by adding ∓ . This can be achieved by conjugating the potential by aDPW gauge which is constant in z such that the residues at z and z are lowerrespectively upper triangular, and then gauge with diag( (cid:113) z − z z − z , (cid:113) z − z z − z ). Denote theresidues of the potential ˜ ξ obtained in this way by R k , and find T such that R = T D − T − R T DT − . Then, T − ˜ ξT turns out to be of the form (3.2). (cid:3) References
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