On limits of triply periodic minimal surfaces

Abstract

In this paper, we introduce generic limits of triply periodic minimal surfaces and consider the genus-three case. We will prove that generic limits of such minimal surfaces consist of a one-parameter family of Karcher’s saddle towers and Rodríguez’ standard examples.

1 Introduction

It is known that properly immersed triply periodic minimal surfaces in ${\mathbb {R}}^3$ have been used for the description of lipids or synthetic surfactants in physics, chemistry, and so on. Moreover, there are transitions from these membranes to lamellar phases, which are periodic parallel planes by environment conditions. In the previous paper [1], we showed that Rodríguez’ standard examples [16] appear as limits of the real five-dimensional family of triply periodic minimal surfaces given by Meeks [9]. Limits of the standard examples including the Riemann minimal examples look like singly periodic parallel planes similar to lamellar phases. This suggests that limits of triply periodic minimal surfaces might be mathematical description of the transition. This paper continues this work.

Let $f:M\rightarrow {\mathbb {R}}^3/\Lambda$ be a compact oriented minimal surface in a flat three-torus. By the isothermal coordinates, M can be reconsidered as a Riemann surface, and we call f a conformal minimal immersion. The following theorem gives an explicit description for a conformal minimal immersion (see for instance [9]).

Theorem 1.1

(Weierstrass representation formula) Let $f:M \rightarrow {\mathbb {R}}^3/\Lambda$ be a conformal minimal immersion. Then, up to translations, f can be represented by the following path integrals:

$$\begin{aligned} f(p)= \mathfrak {R}\int _{p_0}^p {}^t ( \omega _1, \, \omega _2 ,\, \omega _3 ) \quad \mathrm{mod}\>\>\> \Lambda , \end{aligned}$$

(1.1)

where $p_0$ is a fixed point on M and the $\omega _i$’s are holomorphic differentials on M satisfying the following three conditions:

$$\begin{aligned}&\omega _1^2+\omega _2^2+\omega _3^2=0, \end{aligned}$$

(1.2)

$$\begin{aligned}&\omega _1,\,\omega _2,\,\omega _3 \>\> \mathrm{have} \>\> \mathrm{no} \>\> \mathrm{common} \>\>\mathrm{zeros}, \end{aligned}$$

(1.3)

$$\begin{aligned}&\left\{ \mathfrak {R}\int _{C} {}^t (\omega _1 ,\, \omega _2 ,\, \omega _3 ) \>\bigg |\> C\in H_1(M,\,\mathbb {Z}) \right\} \>\>\mathrm{is}\>\>\mathrm{a}\>\>\mathrm{sublattice}\>\>\mathrm{of} \>\> \Lambda . \end{aligned}$$

(1.4)

Conversely, the real part of path integrals of holomorphic differentials satisfying the above three conditions defines a conformal minimal immersion.

Remark 1.1

We can write out $\omega _1,\,\omega _2,\,\omega _3$ in Theorem 1.1 as $(\omega _1,\,\omega _2,\,\omega _3)= (1-g^2,\,i(1+g^2),\,2g)\omega$ for some holomorphic one-form $\omega$ and meromorphic function g, which is the Gauss map composed with stereographic projection onto $\mathbb {C} \cup \{\infty \}$.

A minimal surface in ${\mathbb {R}}^3$ is said to be periodic if it is connected and invariant under a group $\Gamma$ of isometries of ${\mathbb {R}}^3$ that acts properly discontinuously and freely (see [11]). $\Gamma$ can be chosen to be a rank-three lattice $\Lambda$ in ${\mathbb {R}}^3$ (the triply periodic case), a rank-two lattice $\Lambda \subset {\mathbb {R}}^2 \times \{0\}$ generated by two linearly independent translations (the doubly periodic case), or a cyclic group $\Lambda$ generated by a screw motion symmetry, that is, a rotation around the $x_3$-axis composed with a non-trivial translation by a vector on the $x_3$-axis (the singly periodic case). The geometry of a periodic minimal surface in ${\mathbb {R}}^3$ can usually be described in terms of the geometry of its quotient surface M in the flat three-manifold ${\mathbb {R}}^3/\Lambda$. Hence, a triply periodic minimal surface is a minimal surface in a flat three-torus $\mathbb {T}^3$, a doubly periodic minimal surface is a minimal surface in $\mathbb {T}^2 \times {\mathbb {R}}$ where $\mathbb {T}^2$ is a flat two-torus, and a singly periodic minimal surface is a minimal surface in $S^1 \times {\mathbb {R}}^2$.

We will focus on the genus-three case because of the following motivation in terms of the Morse index of a minimal surface. The Morse index of a compact oriented minimal surface in a flat three-torus is defined as the sum of the dimensions of the eigenspaces corresponding to negative eigenvalues of the Jacobi operator of the area. A minimal surface is said to be stable if it has Morse index zero. It is well known that a compact oriented stable minimal surface in a flat three-torus must be a totally geodesic subtorus in the torus. Thus, a compact oriented minimal surface in a flat three-torus which is not totally geodesic must have Morse index at least one, that is, the Morse index one case is the least one. In 2006, Ros [17] proved that a compact oriented minimal surface in a flat three-torus with Morse index one has genus three. So the genus-three case might be important for natural phenomena. In fact, many one-parameter families of compact oriented minimal surfaces of genus three in flat three-tori have been studied in physics and chemistry (see, for example, [8, 18]).

Recall that a compact oriented minimal surface of genus three in a flat three-torus is hyperelliptic, that is, it can be represented as a two-sheeted branched covering of the sphere (see p. 49 in [13] or Corollary 3.2 in [9]). In this case, the Riemann surface M in Theorem 1.1 can be given by $w^2=\prod _{i=1}^8 (z-a_i)$ for distinct eight complex numbers $a_1,\,\ldots ,\,a_8$ or $w^2=\prod _{i=1}^7 (z-b_i)$ for distinct seven complex numbers $b_1,\,\ldots ,\,b_7$ (see p. 102 in [2] or p. 254 in [3]). We consider the former case since we can apply the similar arguments to the latter case. We define generic limits of triply periodic minimal surfaces as limits of f in Theorem 1.1 for the following three cases: (i) the case $a_2\rightarrow a_1$, (ii) the case $(a_2,\,a_4)\rightarrow (a_1,\,a_3)$, (iii) the case $(a_2,\,a_4,\,a_6) \rightarrow (a_1,\,a_3,\,a_5)$ (see Sect. 2 for the details). Our main result is as follows.

Main Theorem

For a compact oriented embedded minimal surface of genus three in a flat three-torus, generic limits of the minimal surface consist of a one-parameter family of Karcher’s saddle towers and Rodríguez’ standard examples.

The paper is organized as follows. In Sect. 2, we discuss generic limits of triply periodic minimal surfaces and give their properties. In Sect. 3, we prove our main result, and finally in Sect. 4, we introduce some singly periodic examples derived from generic limits of triply periodic minimal surfaces.

2 Generic limits

In this section, we will refer to the details of generic limits of triply periodic minimal surfaces for the genus-three case. The arguments essentially appear in Sect. 3 of [1].

For eight distinct complex numbers $a_1,\,\ldots ,\,a_8$, let M be the hyperelliptic Riemann surface of genus three defined by $w^2=\prod _{i=1}^8 (z-a_i)$. We can write out a basis of holomorphic differentials on M by $\{\mathrm{d}z/w,\,z \mathrm{d}z/w,\,z^2\mathrm{d}z/w \}$ (see p. 255 in [3]). Let $f:M\rightarrow {\mathbb {R}}^3/\Lambda$ be a conformal minimal immersion of M into a flat three-torus, and we can choose $g=z$ and $\omega =\mathrm{d}z/w$ in Theorem 1.1 (see Theorem 3.1 in [9], see also Corollary of Theorem 2 in [13]).

We now consider each behavior of f for the following three cases: (i) the case $a_2\rightarrow a_1$, (ii) the case $(a_2,\,a_4)\rightarrow (a_1,\,a_3)$, (iii) the case $(a_2,\,a_4,\,a_6) \rightarrow (a_1,\,a_3,\,a_5)$. We only treat the case (i) since the similar arguments work for the other two cases.

We first construct M as a two-sheeted branched covering of the sphere by the Gauss map $M\ni (z,\,w)\mapsto z\in \mathbb {C}\cup \{\infty \} \cong S^2$. The branch locus of the Gauss map consists of the following eight points on $S^2$:

$$\begin{aligned} a_1,\> a_2, \> a_3, \> a_4,\> a_5,\> a_6, \> a_7,\> a_8. \end{aligned}$$

We prepare two copies of $\mathbb {C} \cup \{\infty \} \cong S^2$ and take two closed curves passing through the eight points, respectively. So we can divide $S^2$ into two domains and label $``+''$ and $``-''$ (see Fig. 1). Slit thick curves as in the upper pictures in Fig. 1 and glue (i) and (ii) as in the lower pictures in Fig. 1. The thin curves in the upper pictures in Fig. 1 correspond to the thin curves in the lower pictures in Fig. 1.By this procedure, we obtain the hyperelliptic Riemann surface M of genus three.

Fig. 1

The hyperelliptic Riemann surface M of genus three

We now consider the case $a_2 \rightarrow a_1$. Let $\alpha$ be a closed curve enclosing $a_1$ and $a_2$ in the z-plane. Lift $\alpha$ to closed curves in M and name them $\hat{\alpha }$ and $\hat{\alpha }'$. Choosing suitable $\alpha$, we can divide M into two disjoint sets in M with the following properties: One set contains $(a_1,\,0)$ and $(a_2,\,0)$ whose boundary consists of $\hat{\alpha }$ and $\hat{\alpha }'$, and the other set is the remaining one (see Fig. 2). Let $M_{\alpha }$ denote the latter.

Fig. 2

Closed curves $\alpha$, $\hat{\alpha }$, $\hat{\alpha }'$

The Weierstrass integral (1.1) along any closed curve in $M_{\alpha }$ is contained in the lattice of the target torus. So the Weierstrass integral depends only on the endpoint of a path in $M_{\alpha }$. We now assume $p_0 \in M_{\alpha }$. Taking t as a local complex coordinate on $M_{\alpha }$, we can write

$$\begin{aligned} \dfrac{1-z^2}{w}\mathrm{d}z=\varphi _1 (t) \mathrm{d}t , \> \dfrac{i(1+z^2)}{w}\mathrm{d}z=\varphi _2 (t) \mathrm{d}t, \> \dfrac{2 z}{w}\mathrm{d}z=\varphi _3 (t) \mathrm{d}t \end{aligned}$$

for some holomorphic functions $\varphi _1 (t)$, $\varphi _2 (t)$, $\varphi _3 (t)$. Then, the three functions $\varphi _1 (t)$, $\varphi _2 (t)$, $\varphi _3 (t)$ converge uniformly on $M_{\alpha }$ as $a_2 \rightarrow a_1$. Hence, the limits can be moved inside the integrals.

After that, letting $\alpha$ shrink to a point, we define generic limits of f for the case $a_2\rightarrow a_1$ as a limit of f.

For the case $a_2\rightarrow a_1$, M converges to the following Riemann surface with node, denoted by $M'$:

$$\begin{aligned} w^2=(z-a_1)^2 \prod _{j=3}^8 (z-a_j). \end{aligned}$$

Let $M_1$ be the Riemann surface of genus two defined by

$$\begin{aligned} v^2=(z-a_3) (z-a_4) (z-a_5) (z-a_6) (z-a_7) (z-a_8). \end{aligned}$$

Then, there exists the following reparametrization of $M'$:

$$\begin{aligned} M_1 \ni (z,\,v) \mapsto (z,\,(z-a_1) v) \in M'. \end{aligned}$$

By using them, the limit, as $a_2 \rightarrow a_1$, of f is

$$\begin{aligned}&\mathfrak {R}\int _{p_0}^p {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) v} \end{aligned}$$

(2.1)

on $M_1\setminus \{(a_1,\,\pm v(a_1))\}$.

For the case $(a_2,\,a_4)\rightarrow (a_1,\,a_3)$, M converges to the following Riemann surface with nodes, denoted by $M''$:

$$\begin{aligned} w^2=(z-a_1)^2 (z-a_3)^2 \prod _{j=5}^8 (z-a_j). \end{aligned}$$

Let $M_2$ be the torus defined by

$$\begin{aligned} v^2=(z-a_5) (z-a_6) (z-a_7) (z-a_8). \end{aligned}$$

Then, there exists the following reparametrization of $M''$:

$$\begin{aligned} M_2 \ni (z,\,v) \mapsto (z,\,(z-a_1) (z-a_3) v) \in M''. \end{aligned}$$

By using them, the limit, as $(a_2,\,a_4)\rightarrow (a_1,\,a_3)$, of f is

$$\begin{aligned}&\mathfrak {R}\int _{p_0}^p {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) (z-a_3) v} \end{aligned}$$

(2.2)

on $M_2\setminus \{(a_1,\,\pm v(a_1)),\,(a_3,\,\pm v(a_3))\}$.

For the case $(a_2,\,a_4,\,a_6)\rightarrow (a_1,\,a_3,\,a_5)$, M converges to the following Riemann surface with nodes, denoted by $M'''$:

$$\begin{aligned} w^2=(z-a_1)^2 (z-a_3)^2 (z-a_5)^2 (z-a_7) (z-a_8). \end{aligned}$$

Let $M_3$ be the sphere defined by

$$\begin{aligned} v^2=(z-a_7) (z-a_8). \end{aligned}$$

Then, there exists the following reparametrization of $M'''$:

$$\begin{aligned} M_3 \ni (z,\,v) \mapsto (z,\,(z-a_1) (z-a_3) (z-a_5) v) \in M'''. \end{aligned}$$

By using them, the limit, as $(a_2,\,a_4,\,a_6)\rightarrow (a_1,\,a_3,\,a_5)$, of f is

$$\begin{aligned}&\mathfrak {R}\int _{p_0}^p {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) (z-a_3) (z-a_5) v} \end{aligned}$$

(2.3)

on $M_3\setminus \{(a_1,\,\pm v(a_1)),\,(a_3,\,\pm v(a_3)),\,(a_5,\,\pm v (a_5))\}$.

Next, we consider the asymptotic behavior of ends for (2.1), (2.2), (2.3). We set

$$\begin{aligned} \Phi = {\left\{ \begin{array}{ll} {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) v} &{} (\mathrm{for} \>\> (2.1)) \\ {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) (z-a_3) v} &{} (\mathrm{for} \>\>(2.2)) \\ {}^t (1-z^2,\, i (1+z^2),\, 2z) \dfrac{\mathrm{d}z}{(z-a_1) (z-a_3) (z-a_5) v} &{} (\mathrm{for} \>\>(2.3)) \end{array}\right. } \end{aligned}$$

It is straightforward to check that we have the Laurent expansion of $\Phi$ at $(a_1,\,v (a_1))$

$$\begin{aligned} \frac{1}{z}\,\alpha +\beta +\gamma \,z+\cdots , \end{aligned}$$

where

$$\begin{aligned} \alpha = {\left\{ \begin{array}{ll} {}^t (1-a_1^2,\, i (1+a_1^2),\, 2a_1) \frac{1}{v(a_1)} &{} (\mathrm{for} \>\> (2.1)) \\ {}^t (1-a_1^2,\, i (1+a_1^2),\, 2a_1) \frac{1}{(a_1-a_3) v(a_1)} &{} (\mathrm{for} \>\> (2.2)) \\ {}^t (1-a_1^2,\, i (1+a_1^2),\, 2a_1) \frac{1}{(a_1-a_3)(a_1-a_5) v(a_1)} &{} (\mathrm{for} \>\> (2.3)) \end{array}\right. } \end{aligned}$$

Let $\langle \cdot ,\,\cdot \rangle$ be the complex bilinear inner product. Then, we have $\langle \alpha ,\,\alpha \rangle =0$. Thus, $\alpha$ is not real nor pure imaginary, and so $\mathfrak {R}(2\pi i\alpha )\ne {0}$. We call the end Scherk-type end with period $\mathfrak {R}(2\pi i\alpha )$ (see, for instance, Theorem 5 in [11]).

Note that there exists the hyperelliptic involution $(z,\,v) \mapsto (z,\,-v)$ on each $M_j$ ($1\le j\le 3$). Hence, we obtain a pair of parallel Scherk-type ends given by $\alpha /z+\beta +\gamma \,z+\cdots$ and $-(\alpha /z+\beta +\gamma \,z+\cdots )$ with nonzero periods $\pm \mathfrak {R}(2\pi i \alpha )$.

Lemma 2.1

For (2.1), (2.2), and (2.3), ends consist of pairs of parallel Scherk-type ends. For each pair of parallel Scherk-type ends, up to translation in ${\mathbb {R}}^3$, one end can be transformed into the other end by a suitable inversion. In particular, the Gaussian curvature at each end converges to zero.

We conclude this section with the following proposition.

Proposition 2.1

Generic limits of triply periodic embedded minimal surfaces of genus three consist of singly periodic or doubly periodic properly embedded minimal surfaces in ${\mathbb {R}}^3$ with pairs of Scherk-type ends. For each pair of Scherk-type ends, the Gauss image at one end coincides with the Gauss image at the other end.

Proof

We use the notation as above. Let $f_j:M_j\setminus E_j \rightarrow {\mathbb {R}}^3$ ($1\le j\le 3$) be an element of generic limits of triply periodic embedded minimal surfaces of genus three defined by (2.1), (2.2), (2.3), where $E_j$ is a set of ends for each $f_j$. Note that $f_j$ might be multivalued in general. Thus, by taking a suitable covering space $\tilde{M_j} \rightarrow M_j\setminus E_j$, there exists a single-valued minimal immersion $\tilde{f_j}:\tilde{M_j} \rightarrow {\mathbb {R}}^3$. By similar arguments to the proof of Theorem 7.1 in [9], $\hat{M_j}=\tilde{f_j} (\tilde{M_j})=f_j (M_j\setminus E_j)$ is embedded.

It is straightforward to check that each $\hat{M_j}$ is complete from (2.1), (2.2), (2.3). By Lemma 2.1, $\hat{M_j}$ has only Scherk-type ends, and thus it has bounded Gaussian curvature. Combining the two facts yields $\hat{M_j}$ is proper (see [12]), and in particular, $\hat{M_j}$ is orientable.

Recall that each $\tilde{f_j}:\tilde{M_j} \rightarrow {\mathbb {R}}^3$ is derived from a triply periodic minimal surface $f:M\rightarrow {\mathbb {R}}^3/\Lambda$. To obtain $\tilde{f_j}$, we collapse some one cycles on M. By (1.4), $\tilde{f_j}$ must be triply periodic or doubly periodic or singly periodic or non-periodic minimal embedding. Since $\tilde{f_j}$ has Scherk-type ends with nonzero periods and is proper, it cannot be triply periodic nor be non-periodic minimal embedding. Therefore, $\tilde{f_j}$ is singly periodic or doubly periodic.

It follows that the minimal embedding $\tilde{f_j}:\tilde{M_j} \rightarrow {\mathbb {R}}^3$ gives rise to a conformal minimal embedding $\overline{f_j}$ of a Riemann surface $\overline{M_j}$ into an $S^1 \times {\mathbb {R}}^2$ or a $\mathbb {T}^2 \times {\mathbb {R}}$. Also, $f_j$ is a conformal minimal immersion of $M_j\setminus E_j$ into the $S^1 \times {\mathbb {R}}^2$ or the $\mathbb {T}^2 \times {\mathbb {R}}$, and there exists a covering map $\pi :M_j\setminus E_j \rightarrow \overline{M_j}$ such that $f_j = \overline{f_j} \circ \pi$. Hence, $\overline{f_j}$ has finite total curvature, and so $\overline{M_j}$ is biholomorphic to a compact Riemann surface $\overline{N_j}$ with a finite number of points removed. Moreover, $\pi$ extends to a covering map $\pi :M_j \rightarrow \overline{N_j}$.

Let G and $\overline{G}$ be the Gauss map of $f_j$ and $\overline{f_j}$, respectively. Note that G and $\overline{G}$ can be extended to holomorphic maps $M_j \rightarrow S^2$ and $\overline{N_j} \rightarrow S^2$, respectively. Then, we have $G=\overline{G} \circ \pi$. Note that G can be obtained by $(z,\,v)\mapsto z$, and so $\deg (G)=2$. To show $\deg (\pi )=1$, we now assume $\deg (\overline{G})=1$, and thus $\overline{G}$ is biholomorphic.

Fig. 3

A singly periodic minimal surface with a pair of Scherk-type ends

For each $k=1,\,3,\,5$, the equation $G(a_k,\,v (a_k))=G(a_k,\,-v (a_k))$ implies that $\overline{f_j}$ is a singly periodic or doubly periodic properly minimal embedding of genus zero with at most three ends. However, for each case, $\overline{f_j}$ has at least four ends ([10, 15]), and it leads to a contradiction.

Therefore, we have $\deg (\pi )=1$, and so $f_j$ coincides with $\overline{f_j}$. Again, by the equation $G(a_k,\,v (a_k))=G(a_k,\,-v (a_k))$, the Gauss image at one end coincides with the Gauss image at the other end for each pair of ends. $\square$

3 Proof of main theorem

In this section, we will prove Main Theorem in Introduction. We use the notation as in the previous section.

We first consider $f_1$ defined by (2.1), and then $f_1$ is a properly embedded singly or doubly periodic minimal surface with a pair of Scherk-type ends with periods $\pm \mathfrak {R}(2\pi i \alpha )$. Suppose that $f_1$ is singly periodic. Let $\{E_1,\, E'_1\}$ be the pair of Scherk-type ends and $G(E_1)$, $G(E'_1)$ their Gauss images. In this case, there exists a plane $\Pi$ such that its normal vector is parallel to $G(E_1)$, and by the strong halfspace theorem [4], it intersects the minimal surface at an interior point (see Fig. 3). However, it contradicts the maximum principle for minimal surfaces. Thus, $f_1$ is doubly periodic, but it contradicts Meeks–Rosenberg’s result [10]. Hence, this case is excluded.

We next consider $f_2$ defined by (2.2), and in this case, $f_2$ gives a properly embedded singly periodic or doubly periodic minimal surface of genus one with two pairs of Scherk-type ends. If $f_2$ is singly periodic, then by Proposition 2.1, there are two essential cases as in Fig. 4. For both cases, there exists a closed curve which intersects the minimal surface once (see the curves in Fig. 4), and it contradicts the classical result in surface theory. Suppose that $f_2$ is doubly periodic. If the ends are not parallel, then we can apply the same arguments as in Fig. 4, and it contradicts. Hence, the ends are parallel, and by the result of Pérez–Rodríguez–Traizet [14], it must be Rodríguez’ standard example.

Fig. 4

Two pairs of Scherk-type ends $\{E_1,\,E'_1\}$ and $\{E_2,\,E'_2\}$

We finally consider $f_3$ defined by (2.3), and then $f_3$ is a singly or doubly periodic minimal embedding of genus zero with six Scherk-type ends. Combining Lazard-Holly and Meeks’ result [7] and Pérez–Traizet’s result [15] yields that $f_3$ must be a singly periodic Scherk surface or Karcher saddle tower or doubly periodic Scherk surface. It has six ends, and thus, it must be the Karcher saddle tower.

Remark 3.1

There exists a doubly periodic Scherk surface with a handle [5]. It has two pairs of Scherk-type ends, and its ends are not parallel. However, for each pair of ends, the Gauss image at one end is distinct from the Gauss image at the other end. By Proposition 2.1, this surface cannot be contained in generic limits of triply periodic minimal surfaces of genus three.

4 Appendix (related examples)

We now consider singly periodic minimal surfaces given by (2.3). By the period condition for singly periodic minimal surfaces, it can be reduced to two cases essentially. One is given by

$$\begin{aligned} a_1= & {} -e^{i\pi /4},\qquad a_3=e^{i(\theta +\pi /4)},\qquad a_5=e^{i(-\theta +\pi /4)},\nonumber \\ a_7= & {} e^{i(\varphi +\pi /4)},\qquad a_8=e^{i(-\varphi +\pi /4)} \end{aligned}$$

(4.1)

in (2.3), where $\theta \in (\pi /2,\pi )$ and

$$\begin{aligned} \varphi =\arccos \left( \frac{2\sin ^2\theta }{1-2\cos \theta }-1\right) \end{aligned}$$

are constants (see the left-hand side of Fig. 5). One can see that none of this family is embedded, and thus, it cannot be included in generic limits of embedded triply periodic minimal surfaces of genus three. The other is embedded (see the right-hand side of Fig. 5); more precisely, these singly periodic minimal surfaces were found by Karcher. See Sect. 2.5.1 in [6].

Fig. 5

Left: Singly periodic minimal surface given by (4.1) with $\theta =3\pi /4$. Right: Singly periodic minimal surface given by Sect. 2.5.1 in [6] with $r=1/2$

It is known that the symmetric Karcher saddle tower (Sect. 2.5.1 in [6] with $r=0$) is a limit of a family of embedded triply periodic minimal surfaces of genus three, called Schwarz H family. However, a family of embedded triply periodic minimal surfaces of genus three whose limit is non-symmetric Karcher saddle tower is yet to be found, leading us to the following open problem:

Problem

Whether there exists a family of embedded triply periodic minimal surfaces of genus three whose limit is non-symmetric Karcher saddle tower (Sect. 2.5.1 in [6] with $r\ne 0$) or not?