The \mathrm{tG} and \mathrm{rGL} families

Existence of the rhombohedral and tetragonal deformation families of the gyroid

Hao Chen Georg-August-Universität Göttingen, Institut für Numerische und Angewandte Mathematik h.chen@math.uni-goettingen.de
July 20, 2019
Abstract.

We provide an existence proof for two 1-parameter families of triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the rGL family with rhombohedral symmetry that contains the gyroid and the Lidinoid, both discovered numerically in the 1990’s. The existence was previously proved within a neighborhood of the gyroid and the Lidinoid, using Weierstrass data defined on branched rectangular tori. Our main contribution is to extend the technique to branched tori that are not necessarily rectangular. In particular, we make use of the twisted catenoids bounded by curved squares or triangles, which arise naturally from the Weierstrass data. Each twisted catenoid extends infinitely in the space by inversions in its boundaries, but the result is usually not embedded. However, we prove the existence of embedded triply periodic minimal surfaces of genus three in their associate families, and they form the 1-parameter tG and rGL families.

Key words and phrases:
Triply periodic minimal surfaces
2010 Mathematics Subject Classification:
Primary 53A10
H. Chen is supported by Individual Research Grant from Deutsche Forschungsgemeinschaft within the project “Defects in Triply Periodic Minimal Surfaces”, Projektnummer 398759432.

1. Introduction

A triply periodic minimal surface (TPMS) is a minimal surface that is invariant under the action of a 3-dimensional lattice . The quotient surface then lies in the flat 3-torus . The genus of is at least three.

Due to its frequent appearance in nature and science, the study of TPMS enjoys regular contributions from physicists, chemists and crystallographers, whose discoveries of interesting examples often precede the rigorous mathematical treatment. The most famous example would be the discovery of the gyroid in 1970 by Alan Schoen [Sch70], then an engineer at NASA. Unlike other surfaces known at the time, the gyroid does not contain any straight line or planar symmetry curve, hence can not be constructed by the popular conjugate Plateau method [Kar89]. The second TPMS of genus three with this property is discovered only twenty years later in 1990, by chemists Lidin and Larsson [LL90], and known nowadays as the Lidinoid. The embeddedness of the gyroid and the Lidinoid are later proved by mathematicians Große-Brauckmann and Wohlgemuth [GBW96].

By intentionally reducing the symmetry of the gyroid and the Lidinoid, two 1-parameter families of TPMS, known as tG and rGL, were discovered in [FHL93, FH99]; see also [STFH06]. Both families contain the gyroid, and retain respectively the rhombohedral and tetragonal symmetries of the gyroid. Remarkably, none of these surfaces contains straight line or planar symmetry curve. Moreover, tG and rGL families are not contained in the 5-parameter family constructed by Meeks [Mee90]. Today, the only other explicitly known examples outside Meeks’ family are the recently discovered 2-parameter oH (containing Schwarz’ H) and families [CW18].

In these works, periods were closed only numerically, producing convincing images that leave no doubt for the existence. Although the importance of numerical discoveries could never been overestimated, the lack of a formal existence proof (that does not involve any numerics) often indicates room for better mathematical understanding. Indeed, our approach in the current paper do brings new way to visualize tG and rGL surfaces.

An attempt of existence proof for tG and rGL was carried out by Weyhaupt [Wey06, Wey08] using the flat structure technique, with success only in a small neighborhood of the gyroid and the Lidinoid. Unlike [FHL93, FH99] who define Weierstrass parametrization on a branched double cover of the sphere, Weyhaupt defines Weierstrass data on a branched torus, but is only able to deal with rectangular tori. Weyhaupt is aware that, to get away from small neighborhoods, one needs to deal with Weierstrass data defined on non-rectangular tori.

In the current paper, we provide an existence proof for the whole tG and rGL families. More precisely, our main result is

Theorem 1.1.

There are two 1-parameter families of triply periodic minimal surfaces of genus three that fit the descriptions of, respectively, the tG and rGL families.

In particular, in addition to the symmetries, we also recover the following phenomena:

  • The tG family contains the gyroid, joins the tD family on one end, and tends to a 4-fold saddle tower on the other end.

  • The rGL family contains the gyroid and the Lidinoid, joins the rPD family on one end, and tends to a 3-fold saddle tower on the other end.

We will give the Weierstrass data explicitly, in terms of Jacobi function, on branched tori that are not necessarily rectangular. Explicit computations are in general not possible. Meanwhile, we notice the “twisted catenoids” that arise naturally from the Weierstrass data. These are minimal annuli bounded by curved squares or triangles. Thanks to the symmetries encoded in the Weierstrass data, the twisted catenoids can be infinitely extended in the space by inversions in the midpoints of the boundary curves. The result is usually not embedded, but we are able to close the periods for a member in the associate family of each twisted catenoid, which turns out to be embedded and belongs to the tG or rGL family.

The paper is organized as follows.

In Section 2, we describe the symmetries of tG and rGL surfaces. This is done by relaxing a rotational symmetry of the classical tP, H and rPD surfaces to a screw symmetry of the same order. More precisely, we define a family of TPMS with order-4 screw symmetries, which contains the tG family as well as the classical tP, tD and CLP families; we also define a family of TPMS with order-3 screw symmetries, which contains the rGL family as well as the classical H and rPD families.

In Section 3, we deduce the Weierstrass data from the symmetries. We first prove that surfaces with screw symmetries can be represented as branched covers of flat tori. The symmetries then force the branch points at order-2 points of the tori. This allows us to write down the Weierstrass data explicitly in terms of the Jacobi elliptic function . In the end of this section, we establish a convention on the choice of the torus.

In Section 4, we officially enter the unknown territory of non-rectangular tori. In particular, we notice that the Weierstrass data could produce twisted catenoids. It is then possible to adopt the ribbon approach of Grosse-Brauckmann and Wohlgemuth [GBW96]. This allows us to obtain two different formula for the associate angle. The period condition requires that the two formula should be equal.

We then give the existence proof in Section 5. More specifically, we study the behavior of the associate angles on the boundary of a domain of interest, and conclude by continuity the existence of a curve that solves the period condition. Complicated computations are delayed to the Appendix. An uniqueness statement hidden in Weyhaupt’s work [Wey06, Wey08] implies that the curve must contain the gyroid and the Lidinoid, whose embeddedness then ensures the embeddedness of all TPMS on the curve.

In Section 6, we point out that tG and rGL provide bifurcation branches that was missing in [KPS14]. We also conjecture that the known surfaces are the only surfaces in and . First step for proving the conjecture is to confirm a uniqueness statement for every tG and rGL surface.

1.1. Acknowledgement

The author is grateful to Matthias Weber for constant and helpful conversations.

2. Symmetries

2.1. Schwarz’ tP, rPD and H families

The tG and rGL families were discovered by relaxing the symmetries of classical surface [FHL93, FH99]. It is then a good idea to first recall some classical TPMS of genus three, namely the tP, rPD and H surfaces. There surfaces are, remarkably, already known to Schwarz [Sch90].

We recommend the following way to visualize.

tP:

Consider a square catenoid, i.e. a minimal annulus bounded by two horizontal squares related by a vertical translation. Then the order-2 rotations around the edges of the squares generate a tP surface, and any tP surface can be generated in this way.

H:

Consider a triangular catenoid, i.e. a minimal annulus bounded by two horizontal equiangular triangles related by a vertical translation. Then the order-2 rotations around the edges of the triangles generate an H surface, and any H surface can be generated in this way.

rPD:

Similar to the H surfaces, but the two triangles bounding the triangular catenoid are related by a vertical translation followed by an order-2 rotation around the vertical line through the centers of the triangles.

Figure 1. The catenoids that generate tP (left), H (middle) and rPD (right) surfaces.

The catenoids that generate tP, H and rPD surfaces are shown in Figure 1. The 1-parameter families are obtained by vertically “stretching” each catenoid.

These surfaces are invariant under the following transforms.

  • Rotation around the vertical axis through the centers of the bounding squares or triangles. The order of this symmetry is 4 for tP and 3 for H and rPD.

  • Roto-reflection composed of a rotation around the normal vector at a vertex of the bounding squares or triangles, followed by a reflection in the tangent plane at this vertex. The order of such a roto-reflection is 4 for tP and 6 for H and rPD. Note that the vertices have vertical normal vectors, hence are poles and zeros of the Gauss map. Moreover, for H and rPD surfaces, the vertices are also ramification points of the Gauss map.

  • Rotations of order-2 around horizontal axis that swap the bounding squares or triangles.

  • Inversions in the midpoints of the edges. Each inversion can be seen as composed of a rotation around the edge, followed by reflection in the vertical plane that bisects the edge. Note that the midpoints are ramification points of the Gauss map.

2.2. Tetragonal surfaces

The square or triangular catenoids are very similar to the standard catenoid. As one travels along the associate family, the rotational symmetry around the vertical axis of the minimal annulus becomes a screw symmetry of an “infinite gyrating ribbon” [GBW96]. Recall that a screw transform is composed of a rotation and a translation in the rotational axis. In this paper, we count a rotation as a special screw transform.

By interpreting associate families as rotations of tangent spaces, it was showed in [GBW96, Lemma 4] that rotational and roto-reflectional symmetries around the normal vector at a point of a minimal surface are preserved along the associate family of the surface. The gyroid is in the associate family of Schwarz’ P surface. Remarkably, no other member of the tP family contains any other embedded surface in its associate family. Hence the gyroid admits the following symmetries:

  • Screw transforms of order 4 with vertical axis.

  • Roto-reflections of order 4 around the vertical normal vectors at the poles and zeros of the Gauss map.

  • Rotations of order 2 with horizontal axis that (necessarily) swap the poles and zeros of the Gauss map.

  • Inversions in the ramification points of the Gauss map.

We use to denote the set of TPMS of genus three that satisfy the symmetries listed above. Note that not all symmetries are necessary to define . We will see that the inversional symmetries in the ramification points of the Gauss map is in fact a property of TPMS of genus three. Moreover, if we have a screw symmetry of order 2, then the inversions and roto-reflections of order 4 imply a screw symmetry of order 4. Hence only the screw symmetry of order 2 is necessary here.

Schwarz’ tP and the gyroid belong to . The conjugate of tP surfaces, namely Schwarz’ tD surfaces, belong to . We will see that Schwarz’ CLP surfaces also belong to . Our task is to prove the existence of another 1-parameter family in , denoted by tG, that contains the gyroid.

2.3. Rhombohedral surfaces

The P surface is also a member of the rPD family, which is self-conjugate. So the associate family of P contains three embedded surfaces, namely P, G and D. No other surface in the associate family is embedded, and no other member of the rPD family contains any other embedded surface in its associate family. Similarly, the Lidinoid is the only embedded surface in the associate family of a unique member of the H family.

By the same analysis as before, we see that the gyroid and the Lidinoid admit the following symmetries.

  • Screw transform of order 3 with vertical axis.

  • Roto-reflection of order 6 around the vertical normal vectors at the poles and zeros of the Gauss map.

  • Rotation of order 2 with horizontal axis that (necessarily) swap the poles and zeros of the Gauss map.

  • Inversion in the ramification points of the Gauss map.

We use to denote the set of TPMS of genus three that satisfy the symmetries listed above. Again, the inversional symmetries are properties of TPMS of genus three, hence redundant here. By the same argument as in Weyhaupt’s thesis [Wey06, Proposition 3.12], the poles and zeros of the Gauss map must also be ramification points of the Gauss map, hence the roto-reflectional symmetries are also redundant.

It turns out that rPD, H, the gyroid and the Lidinoid are not the only members of . We will prove the existence of another 1-parameter family in , denoted by rGL, that contains the gyroid and the Lidinoid.

3. Weierstrass parametrization

We use [Mee90] for general reference about TPMS.

Let be a TPMS invariant under the lattice . Meeks [Mee90] proved that is of genus three if and only if it is hyperelliptic, meaning that it can be represented as a two-sheeted branched cover over the sphere. In fact, the Gauss map provides such a branched covering. If is of genus three, the Riemann-Hurwitz formula implies eight branch points of . We call ramification points on hyperelliptic points, and denote the stereographic projections of the branch points by . An inversion (in the space) in any of the hyperelliptic points induces an isometry that exchanges the two sheets. This is why the inversional symmetries are redundant for defining and .

Now consider the hyperelliptic Riemann surface of genus three defined by

Then we have the following Weierstrass parametrization for :

(1)

This parametrization has been widely used for constructing TPMS of genus three, ranging from the classical example of Schwarz’ [Sch90] to the tG and rGL families discovered in [FHL93, FH99].

3.1. Weierstrass data on torus

Weyhaupt [Wey06, Wey08] treats as a branched cover of a flat 2-torus, and adopt the following form of Weierstrass parametrization,

(2)

where must all be holomorphic. In particular, the holomorphic differential is called the height differential. By comparing (1) and (2), one sees the correspondence and .

The surfaces we want to construct all admit screw symmetries. The following proposition allows us to follow the approach of Weyhaupt.

Proposition 3.1.

If a TPMS admits a screw symmetry , then is of genus one.

Here we consider rotational symmetries as special screw symmetries, for which the proposition was proved in [Wey08, Proposition 2.8] but with minor flaws. Hence we include a proof here for completeness.

Proof.

The height differential is invariant under , hence descends holomorphically to the quotient . Since there is no holomorphic differential on the sphere, the genus of cannot be .

Recall the Riemann–Hurwitz formula

In our case, is the genus of , is the genus of , is the degree of the quotient map, and is the total branching number. Since the order of a screw symmetry is at least two, we conclude immediately that . It remains the case and , which can not be eliminated by the Riemann–Hurwitz formula.

We then proceed as follows. Without loss of generality, we may assume the screw axis to be vertical. In the case of , we have . In the parametrization (1), since is invariant under , so must the polynomial , hence we can write , where is a polynomial of degree 4. Then defines the quotient surface, whose genus is . So is the only possibility. ∎

A holomorphic 1-form on the torus must be of the form . Varying the modulus only results in a scaling, so we define as the lift of (note the sign!). Varying the argument gives an associate family, so we call the associate angle.

3.2. Locating branch points

By a Theorem in [KK79], we know that the order of the screw symmetry must be , or . If the order is prime, the following formula [FK92] allows us to calculate the number of fixed points:

In particular, a screw symmetry of order 2 fixes exactly four points, and a screw symmetry of order 3 fixes exactly two points. The following lemma follows from the same argument as in [Wey06, Lemmas 3.9, 3.13].

Lemma 3.2.
  • If a TPMS of genus three admits a screw symmetry of order two, then descends to an elliptic function on the torus with two simple zeros and two simple poles.

  • If admits a screw symmetry of order three, then descends to an elliptic function on the torus with a double-order zero and a double-order pole.

We now try to locate the position of the branch points of the covering map. Since the ramification points on are all poles and zeros of the Gauss map, our main tool is naturally Abel’s Theorem, which states that the difference between the sum of poles and the sum of zeros (counting multiplicity) is a lattice point.

If admits a screw symmetry of order two, extra information is needed to determine the branch points. Recall that the surfaces in admit roto-reflectional symmetries of order 4 in the vertical normal vectors at the poles and zeros of the Gauss map. They descend to the quotient torus as inversions in the branch points of the covering map.

Lemma 3.3 (Compare [Wey06, Lemma 3.10]).

Let be a TPMS admitting a screw symmetry of order two. Assume further that , as a branched cover of the quotient torus , is fixed under the inversion in any of the branch point. If one branch point is placed at , then the other branch points must be placed at the three order-2 points of the torus.

Proof.

We may assume that the branch point at correspond to a zero of . If a pole is not at any order-2 points, must be a different pole by assumption. Then Abel’s Theorem forces the other zero to be at a lattice point, which is absurd. So both poles must be placed at order-2 points. Then Abel’s Theorem forces the other zero at the remaining order-2 point. ∎

We have explained that screw symmetries of order 2, roto-reflections of order 4 and inversions imply screw symmetries of order 4. They descend to the quotient torus as the translation that swap the zeros.

If admits a screw symmetry of order three, one may place the zero of at . Then the Abel’s Theorem forces the pole to be at an order-2 point. It is no surprise that no further symmetry is needed here; as we have explained, the roto-reflectional symmetry is redundant for the definition of .

Remark 3.4.

The roto-reflectional symmetry is however not redundant for . The recently discovered surfaces [CW18] have screw symmetries of order 4 with vertical axis, and rotational symmetries of order 2 with horizontal axis, but does not belong to .

3.3. Explicit expression for Gauss map

The locations of poles and zeros determine an elliptic function up to a constant factor. Elliptic functions with poles and zeros at lattice points and order-2 points are famously given by Jacobi elliptic functions. In particular, is an elliptic function with periods and . Its zeros lie at and , and poles at and . Here, is the complete elliptic integral of the first kind with modulus , is the modular lambda function, and .

We use [Law89] as the major reference for elliptic functions. As useful references include [Bow61, BF71, BB98] and NIST’s Digital Library of Mathematical Functions [DLMF].111The website of DLMF is closed due to the US government shutdown during the preparation of this manuscript.

Remark 3.5.

Note that we do not define , and our is half of the traditional definition, but coincides with the definition on [Law89, p. 226].

We may write the Gauss map for a surface as

In particular, the factor on the variable brings the defining torus to , which is more convenient for us. The zeroes of are at and , and the poles at and . The multi-valued function on is given by . We take the branch cuts of the square root along the segments and , compatible with [Wey06]; see Figure 2. None of the branch points is hyperelliptic point. Instead, the symmetry reveals four other inversion centers at , , and , and they are the hyperelliptic points.

Figure 2. Branch cuts in the branched torus for surfaces. Solid circles are zeros; empty circles are poles.

We may write the Gauss map for a surface as

This function is defined on the torus , which is half of the defining torus of . has a double-order zero at and a double-order pole at . The multi-valued function on is given by . We take the branch cuts of the cubic root along the segments , , and , compatible with [Wey06]; see Figure 3. This time, both branch points are hyperelliptic points [Wey06, Proposition 3.12]. We recognise two other inversion centers at and , which are also hyperelliptic points.

Figure 3. Branch cuts in the branched torus for surfaces. Solid circles are zeros; empty circles are poles.

The factor is known as the López–Ros factor [LR91]. Varying its argument only results in a rotation of the surface in the space, hence does not concern us. For surfaces in and , can be determined by the order-2 rotational symmetries that swap the poles and zeros of the Gauss map. More specifically, and for must have the same residue at their respective poles, and and for must have the same principal parts at their poles. By the identity , we deduce that both for and .

Remark 3.6.

Equivalently, we can write the Gauss map for in the form

where is the Jacobi Theta function (actually one of them) for the lattice , and the López–Ros factor . The Gauss map for may have a similar expression. This should help the readers to compare our computation with those in [Wey06, Wey08].

3.4. Conventions on the torus

Note that and are invariant under the transforms and . Let be the congruent subgroup of the modular group generated by these transforms. Then and give the same Weierstrass data if they are in the same orbit under . Hence we have infinitely many choice for for every TPMS. In fact, using Jacobi function already reduces the choice.

Surfaces in [Wey06, Wey08] all admit reflectional symmetries, making it possible to choose to be pure imaginary, so the torus is rectangular (and a different elliptic function should be used). We could not use this convention since, in general, the tG and rGL surfaces do not admit reflectional symmetry, nor do surfaces in their associated families.

A natural choice would be taking in the fundamental domain of . This domain is bounded by the vertical lines and the half circles . Under this convention, our becomes the standard associated elliptic integral of the first kind with modulus ; then we can safely employ the formula in most textbooks. We will use this convention for concrete computations in Appendix, where we use the notation in place of .

This natural choice is, however, not convenient for analysing the tG and rGL families. For instance, the tP and tD families correspond to the same vertical line , and the gyroid also lies on this line. In fact, we will see that, for each , there are two member of tG with . The same happens for rGL. Hence we need another convention.

The symmetries of imply the following property for its members: The poles and zeros of the Gauss map project onto a square lattice in the horizontal plane and, on the vertical line over each lattice point, the zeros and poles of the Gauss map are arranged alternatively and equally spaced; cf. [GBW96]. Similar statement can be made for surfaces in , whose zeros and poles of the Gauss map project onto a triangular lattice. For the sake of a uniform treatment, we make the following convention for both and :

Convention.

We assume that the image of is directly above the image of under the Weierstrass parametrization.

We have explained that and gives the same Weierstrass representation, hence generate the same TPMS. But with different choice of , the branch point corresponds to different point on the surface. Hence our convention can be seen as a marking of a specific pole of the Gauss map. We will see that for the gyroid and the Lidinoid under our convention.

We list in Tabel 1 the Weierstrass data of classical TPMS of genus three under our convention. For each family, we specify the possible for the torus, and the argument of the height differential . We also accompany a diagram, showing the possible (dashed curve), the fundamental parallelogram for a typical example, the poles (empty circles) and the zeros (solid circles) of (whose defining torus could twice the shown quotient torus!), and an arrow indicating by pointing to the direction of increasing height. The bottom-left corner of the parallelogram is always 0, the bottom edge represents 1 for or 1/2 for , and the left edge always represents . The tori in [Wey06, Wey08] are shown as dotted rectangles for comparison.

tP: , ,
tD: , ,
CLP: , ,
H: , ,
rPD: , or ,
Table 1. Weierstrass data for classical TPMS of genus three.

4. Period conditions

Rectangular torus has many convenience. For example, many straight segments in the branched torus correspond to geodesics on the surface, making it possible to compute explicitly [Wey06].

With general tori, we lose all the nice properties. Nevertheless, let us assume for the moment, and study the image of the maps and . An explicit computation is indeed hopeless, but we are still able to say something.

4.1. Twisted catenoids

Let us first look at the lower half of the branched torus of , i.e. the part with . This is topologically an annulus, and lifts to its universal cover , which is a stripe in . By analytic continuation, lifts to a function of period on the stripe. The boundary lines and are then mapped by into periodic or closed curves in .

By the same argument as in the standard proof of the Schwarz–Christoffel formula (see also [Wey06, FW09]), we see that the curves make an angle at each branch point. The interior angle is at the poles of , and at the zeros. By the symmetry , the image of the segments , , are all congruent. And images of adjacent segments only differ by a rotation of around their common vertex. Moreover, by the symmetry , the image of each segment admits an inversional symmetry. The inversion center is the image of the hyperelliptic points, at the midpoint of each segment. The same can be said about the segments , .

Therefore, the boundaries of the stripe are mapped to closed curves with a rotational symmetry of order . The curves look like “curved squares”, obtained by replacing the straight edges of the square by rotational copies of a symmetric curve. Moreover, the two twisted squares share the same rotation center. The stripe is then mapped by to the twisted annulus bounded by the curved squares. See Figure 4.

A similar analysis can be carried out on the annulus . However, the branch cuts ensures that the two annuli continuous into different branches (different sign of square root) when crossing the segments . As a consequence, the boundary curves is turning in the opposite direction. In fact, as one passes through the segment , the images of is extended by an inversion, which reverses orientation.

Because of our choice of the López–Ros parameter, the image of the annulus under is congruent to the image of the annulus under . The only difference is that the inner and the outer boundaries of the annulus are swapped. See Figure 4.

Figure 4. Plot of the annulus under the maps and for , with .

Combining the flat structures and gives us the image under the Weierstrass map. Because of our choice of , we know that the lines and are mapped to two horizontal planar curves, at heights and respectively. The previous analysis tells us that these are two congruent closed curves that look like curved squares. In particular, they admit rotational symmetry of order . The annulus is then mapped to a “twisted square catenoid” bounded by these curves. Moreover, the twisted catenoid admits rotational symmetries of order 2 with horizontal axis that swap its boundaries. See Figure 5.

Figure 5. Plot of the annulus with under the Weierstrass parametrization of , for and , in this order. The red and the blue points indicate the images of and . An increasing “twist angle” is visible. Note that bounding edges are in fact slightly curved, except for and .

Inversion in the midpoint of a curved edge extends the surface with another twisted square catenoid, which is the image of the stripe . Repeated inversions in the midpoints of the curved edges extend the catenoid into a (usually not embedded) periodic minimal surface.

The same argument applies to the Weierstrass data of . The annulus lift to a stripe in of period . We then obtain a twisted triangular catenoid with rotational symmetry of order , and repeated inversions in the midpoints of the curved edges extend the catenoid into a (usually not embedded) periodic minimal surface. See Figures 6 and 7.

Figure 6. Plot of the annulus under the maps and for , with .
Figure 7. Plot of the annulus with under the Weierstrass parametrization of , for and , in this order. The red and the blue points indicate the images of and . An increasing “twist angle” is then visible. Note that bounding edges are in fact slightly curved, except for and .

It is interesting to observe the effect of increasing .

Let us start from a tP surface with . Its catenoid is the standard square catenoid, not twisted. As we increase , the square catenoid becomes “twisted” in two senses: on the one hand, the bounding squares become curved; on the other hand, horizontal projections of the squares form an angle. This “twist angle” seems to increase monotonically with (we are not sure!); see Figure 5. Remarkably, when , reflectional symmetry is restored in the Weierstrass data, and the catenoid is bounded by two straight squares forming a twist angle of . Then the catenoid is again “twisted” as we continue to increase until . During the process, the twist angle increases from at (tP) to at , to at (gyroid), to at , until at (tD). These are the only cases where reflectional symmetry is restored, and the bounding edges are straight. The term “twist angle” is in general ill-defined, but carries a natural meaning in these cases.

Remark 4.1.

The tG surfaces at deserve more attention, as the reflectional symmetries in their Gauss maps seem special.

Similarly, we can start from the triangular catenoid of H with , and increase until . The bounding triangles become curved and form a twisted angle that seems to increase monotonically with ; see Figure 7. In particular, the “twist angle” increases from at (H) to at (gyroid), to at (Lidinoid), until at (rPD). Moreover, these are the only cases where the reflectional symmetry is restored, the bounding edges are straight, and the meaning of “twist angle” is clear.

4.2. Vertical associate angle

The following way of visualizing the gyroid was proposed in [GBW96]: Recall that Schwarz’ P can be obtained from a square catenoid. As one travels from P along the associate family, the bounding squares becomes “square helices”, each running on the boundary of a almost square cylinder. The catenoid then opens up into a ribbon bounded by two helices. Eventually, the two square cylinders coincide and, at the same time, the pitch of each helix doubles the vertical distance between them. This allows ribbons originated from adjacent catenoids to fit exactly in.

The ribbon metaphor can be extrapolated to all surfaces in and (using triangular cylinders). The point that distinguishes tG and rGL surfaces is that “the pitch of the helices doubles the vertical distance between them”.

For tG surfaces, this means that the integral of the height differential from to doubles the integral from to . Or equivalently, the integral from to equals the integral from to . This can be easily achieved by adjusting the associate angle to (compare [Wey08, Definition 4.2])

Similarly, for rGL surfaces, this means that the integral of from to equals the integral from to . Then we deduce that

Remark 4.2.

The ratio of the pitch over the vertical distance of the helices can also take different values, but must be even for adjacent ribbons to fit in. We will see that the ratio is for tP, for tD and for CLP. Other ratios may or may not give new TPMS. We will briefly mention them in Section 6.

4.3. Horizontal associate angle

We now calculate the associate angle in another way, using the fact that the images of and have the same horizontal coordinates.

First note that, by the symmetry , we have the identity

We may place the image of at the origin. First look at the surface with , hence . Then the horizontal coordinates of the image of are

Then we look at the surface with , hence (the conjugate surface). Then the coordinates are

So for the surface with associate angle , the first coordinate is always , while the second coordinate

vanishes when

The period condition that we need to solve is

(3)

We are finally ready to give the existence proof.

5. Existence proof

5.1. tG family

In picture 8, we show for the numerical solutions to (3) with , accompanied by two half-circles representing the CLP family. Our task is to prove the existence of the continuous 1-parameter solution curve that we see in the picture, which we call the tG family.

Figure 8. Solutions for to (3) with . Horizontal and vertical associated angles are marked for the left half-circle (CLP) to help the argument in the proof.
Proposition 5.1.

In the domain , and , there exists a continuous 1-dimensional curve of that solves (3). This curve tends to at one end, and to for some at the other end. Moreover, the triply periodic minimal surfaces of genus three represented by points on the curve are all embedded.

Proof.

We examine the angles and on the boundaries of the domain.

On the vertical line , we see immediately that . Since this line corresponds to the tP family, we know very well that the image of is directly above the image of when . Hence .

The half-circle corresponds to the CLP family. We know very well that the image of is directly above the image of when , so . Then the inequality follows from elementary geometry.

On the half-circle , it follows from elementary geometry that . This half-circle corresponds again to the CLP family. When , the image of is directly above the image of . In this case, we know very well that the flat structure of is as depicted in Figure 9; see [Wey06]. In particular, we have

hence .

Figure 9. Image of rectangle with vertices at , , , , under the map for a CLP surface (the grey area). The twisted square annulus is sketched (exaggeratedly) in dashed curves.

On the vertical line , we see immediately that . Since this line corresponds to the tD family, we know very well that the image of is directly above the image of when , so . Hence this line solves the period condition , but not very helpful for our proof.

As , we see immediately that . Asymptotic behavior of is complicated, so we delay the details to Appendix, where Lemma A.1 states that . Hence for sufficiently large.

Now consider for small . It is immediate that the derivative of with respect to at is , hence tends to as . Meanwhile, the asymptotic behavior of tells us that as . Consequently, there exists two positive numbers and such that for all and .

On the other hand, Lemma A.2 in Appendix claims that as . Consequently, there exists a neighborhood of such that for all .

Finally, by the continuity of the functions and , we conclude that there exists a connected set of in the named area that solves period condition (3), which separates the half-circles from the line and the infinity. Moreover, this set must also separate a neighborhood of from the set . We may extract a continuous curve from the set, which is the tG family. In particular, this curve must tends to the common limit of CLP and tP (a saddle tower of order 4 at ), and intersect the tD family at a finite, positive point.

The proof of [Wey08, Lemma 4.4] implies that the gyroid is the only embedded TPMS on the vertical line that solves the period condition. Hence the tG family must contain the gyroid, whose embeddedness then ensures the embeddedness of all TPMS in the tG family; see [Wey06, Proposition 5.6]. ∎

In Figure 10, we show two adjacent ribbons for some tG surfaces. They are actually fundamental domains for the translational symmetry group.

Figure 10. tG surfaces with , , , , and , in this order.

5.2. rGL family

In picture 8, we show for the numerical solutions to (3) with . The two half-circle represents an order-3 analogue of the CLP surface, termed hCLP in [LL90]. hCLP is not embedded, but also not dense in the space. It is very easy to visualize, and behaves very much like CLP. In particular, it’s Weierstrass data is as shown in Figure 12; compare CLP in Table 1.

Our task is to prove the existence of the continuous 1-parameter solution curve that we see in the picture, which we call the rGL family.

Figure 11. Solutions for to (3) with .
Figure 12. Weierstrass data for hCLP.
Proposition 5.2.

In the domain , and , there exists a continuous 1-dimensional curve of that solves (3). This curve tends to at one end, and to for some at the other end. Moreover, the triply periodic minimal surfaces of genus three represented by points on the curve are all embedded.

Proof.

Part of the proof is very similar to tG, so we just provide a sketch.

The line corresponds to the H family, and we have .

The half-circle corresponds to the hCLP family, and we have .

The line correspond to the rPD family, and we have . This solves the period condition, but not helpful for us.

As , we have . The argument for the asymptotic behavior is very similar as in the proof of Lemma A.1, so we will not repeat it.

By the same argument as for tG, we conclude that for as long as and for some positive constants and .

More care is however needed on the half-circle . It corresponds again to the hCLP family. When , the image of is directly above the image of . In this case, we know very well that the flat structure of is as depicted in Figure 13. We see that

Figure 13. Image of rectangle with vertices at , , , , under the map for a hCLP surface (the grey area). The twisted triangular annulus is sketched (exaggeratedly) in dashed curves.

Let denote the length of the tilted segments (e.g. from to ) in the flat structure, and the length of the vertical segments (e.g. from to ). If follows from an extremal length argument that the ratio increases monotonically as travels along the half-circle with increasing . Then we see that increases monotonically from to .

On the other hand, it follows from elementary geometry that , which decreases monotonically as travels along the half-circle with increasing . Consequently, there is a unique on the half-circle for which , and we know very well that this occurs at . At this point, it is interesting to verify that , hence .

Therefore, by monotonicity, we have on the left quarter of this half-circle.

We then conclude the existence of a continuous curve of , namely the rG family, that solves , and separates the half-circles from the line and the infinity. This curve must tends to the common limit of hCLP and H (a saddle tower of order 3 at ), and intersect rPD at a finite point.

Moreover, the Lidinoid and the gyroid are the unique solution on their respective vertical lines. Hence the rGL family must contain them both. Their embeddedness then ensures the embeddedness of all TPMS in the rGL family. ∎

In Figure 14, we show two adjacent ribbons for some rGL surfaces. They are actually fundamental domains for the translational symmetry group.

Figure 14. rGL surfaces with , , , , and , in this order.

6. Discussion

6.1. Bifurcation

A TPMS of genus three is a bifurcation instance if the same deformation of its lattice could lead to different deformations (bifurcation branches) of the surface. Both the tG–tD and the rGL–rPD intersections are bifurcation instances. Bifurcation instances among classical TPMS are systematically investigated in [KPS14], but some of them had no explicit bifurcation branch at the time.

Two bifurcation instances were discovered in tD [KPS14]. The recently discovered t family provides the missing bifurcation branch to one of them [CW18]. The other bifurcation instance seems to escape the attention, since its conjugate is identified as the the tP surface obtained from the square catenoid of maximum height. However, no bifurcation branch was previously known for itself. Numerics from [KPS14] and [FH99] (see also [STFH06]), which we can confirm with the help of (6) in Appendix, shows that this is exactly the tG–tD intersection with . Hence tG provides the missing bifurcation branch.

Likewise, two bifurcation instances were discovered in rPD [KPS14]. One of them is identified as the rPD surface obtained from the triangular catenoid of maximum height. The other is its conjugate, for which no bifurcation branch was previously known. Numerics from [KPS14] and [FH99] (see also [STFH06]) shows that this is exactly the rGL–rPD intersection, hence rGL provides the missing bifurcation branch.

Therefore, all bifurcation instances discovered in [KPS14] has now an explicit bifurcation branch.

Remark 6.1.

It is curious that both tG–tD and rGL–rPD intersections are conjugate to catenoids of maximum height. In fact, numerics shows that the height of the twisted catenoid seems to increase monotonically with along the tG and rGL families, and reaches maximum at the intersection.

6.2. Uniqueness

We conjecture the following uniqueness statements.

Conjecture 6.2.
  • For every , there is a unique such that solves the period condition (3) for .

  • For every , there is a unique such that solves the period condition (3) for .

The uniqueness has been proved by Weyhaupt [Wey06] for with (gyroid), and for with (gyroid) and (Lidinoid). Our approach leads to a simple proof that also works for with . Here is a sketch: On the one hand, it is immediate that decreases monotonically with . On the other hand, by enlarging the outer square or triangle, and shrinking the inner square or triangle, one easily sees that increases monotonically with . This simple proof is possible because, for these cases, the twist angles take special values and do not vary with . This may not hold for other cases, for which even the meaning of “twist angle” is not clear.

Moreover, we conjecture the following classification statements.

Conjecture 6.3.
  • The tP, tD, CLP and tG families are the only members of .

  • The H, rPD and rGL families are the only members of .

To prove this, one needs to first prove the uniqueness conjecture, then take care of the surfaces with higher pitches.

We have seen that a tG surface consists of ribbons bounded by helices, and the pitch of the helices doubles the vertical distance between them. In general, for any surface in and , let denote the ratio between the pitch of the helices and the vertical distance between them. Then must be even for ribbons to fit into each other.

It is immediate that for tP. We have seen that tD and tG close the period for . It is also not difficult to see that CLP solves the period condition with . In this case, two ribbons occupy the same vertical cylinder, and interlace each other. Their vertical distance equals the vertical width of a single ribbon, allowing adjacent ribbons to fit in.

In fact, tP also provide solutions to the period condition for every , and CLP provide solutions for every , . This can be seen by taking on half-circles of radius . We obtain numerically surfaces with , but they seem to be members of tG. Actually, all surfaces with higher pitch seem to lie outside the fundamental domain of . Hence we strongly believe that they are nothing but another point of view of the known TPMS. The same analysis applies to .

Appendix A Asymptotic behavior of associate angle

We now give the details of the asymptotic analysis of for .

We need to study the shape of the twisted square annulus with more care. For convenience and safety, we switch to where , hence . Correspondingly, the is defined as , which coincides with the usual definition of the associated elliptic integral of the first kind.

Let us first look at

which is a vector pointing from to , hence an straightened edge vector of the inner twisted square. With the change of variable , we have

(4)

Here we used the identities

Then we compute the integral

This is a vector pointing from to , hence from an inner vertex to the nearest outer vertex of the twisted annulus. Then we use the Jacobi imaginary transformation and obtain

(5)

where we changed the variable , and used the identities

The two vectors and can determine the images of all branch points under .

Lemma A.1.

as .

Proof.

As , we have [Law89, (2.1.12)]

So and . Recall that , and that the integral in (4) tends to

which is bounded. Therefore, as . In other words, the size of the inner square tends to .

On the other hand, we have

where we changed the variable . Note again that the integral is bounded, hence . In other words, the size of the outer square grows exponentially with .

Therefore, as , the integral

is dominated by

where we used the convention that so that is analytic. Now we can conclude that

A similar argument applies to rGL, so we omit the proof. The conclusion is that as .

Lemma A.2.

as .

Proof.

By the transformation  [Law89, (9.4.10)], we see that as . Then [Law89, Exercise 8.13]

The standard proof for this also applies to prove that the integral in (4)

One can also quickly convince himself by noting that the integrand differs from the integrand of only by a factor , which can be neglected near , where the divergence occurs. In other words, the integral in (4) is asymptotically equivalent to . So we have as .

On the other hand, the integral in (5) tends to

which is bounded, hence . Therefore, as , we see from the flat structure that , so . Note again that . ∎

Remark A.3.

The integrals arisen from surfaces (e.g. (4) and (5)) can be explicitly expressed in terms of elliptic integrals; see [Bow61, Chapter X] and [BF71, §595 et seq.]. For example, we have

for the inner and outer edge vectors of the twisted annulus, where

We then obtain that

(6)

This facilitates the numeric calculation for the tG–tD intersection, but does not give an explicit expression.

We are not aware of equally explicit expressions for integrals arisen from surfaces.

References

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