A New Look at Equivariant Minimal Lagrangian Surfaces in ${\mathbb {C}} P^2$

Abstract

In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.

1 Introduction

Minimal Lagrangian surfaces in the complex projective plane $\mathbb {C}P^2$ endowed with the Fubini-Study metric are of great interest from the point of view of differential geometry, symplectic geometry and mathematical physics [2, 5, 6, 9, 10, 13]. They give rise to local models of singular special Lagrangian 3-folds in Calabi-Yau 3-folds, hence play an important role in the development of mirror symmetry [7]. The Gauss-Codazzi equations for minimal Lagrangian surfaces in $\mathbb {C}P^2$ are given by

$$\begin{aligned} u_{z\bar{z}}=e^{-2u}|\psi |^2-e^{u},\quad \psi _{\bar{z}}=0, \end{aligned}$$

where $g=2e^{u}dzd\bar{z}$ is the Riemannian metric of a Riemann surface and $\psi dz^3$ is a holomorphic cubic differential defined on the surface. Since any minimal Lagrangian surface of genus zero in $\mathbb {C}P^2$ is totally geodesic, it is the standard immersion of $S^2$ in $\mathbb {C}P^2$ [11, 14]. In a nice paper [2] by Castro and Urbano, they reduced the PDE above to an ODE and constructed translationally equivariant minimal Lagrangian tori in $\mathbb {C}P^2$. Later on it was shown that any minimal Lagrangian immersed surface of genus one in $\mathbb {C}P^2$ can be constructed in terms of algebraically completely integrable systems [9, 10, 13]. Recently, a loop group method introduced by Dorfmeister, Pedit and Wu [3] has proven to be efficient in constructing surfaces related to a family of flat connections with nontrivial topology. As a preparation for the construction of minimal Lagrangian surfaces with “ends”  in $\mathbb {C}P^2$, we would like to present a new look at translationally equivariant minimal Lagrangian surfaces in $\mathbb {C}P^2$ via the loop group method in this note.

The paper is organized as follows: In Sect. 2, we recall the basic set-up for minimal Lagrangian surfaces in $\mathbb {C}P^2$. In Sect. 3, we explain the definition of equivariant minimal Lagrangian surfaces in $\mathbb {C}P^2$. In Sect. 4, we show that every translationally equivariant minimal Lagrangian surface in $\mathbb {C}P^2$ is generated by a degree one constant potential. In Sect. 5, we present an explicit Iwasawa decomposition for any translationally equivariant minimal Lagrangian surface. In Sect. 6, we discuss the periodicity condition for translationally equivariant minimal Lagrangian cylinders and tori. Finally, we compare our loop group approach to the work of Castro-Urbano [2].

2 Minimal Lagrangian Surfaces in $\mathbb {C}P^2$

We recall briefly the basic set-up for minimal Lagrangian surfaces in $\mathbb {C}P^2$. For details we refer to [9] and references therein.

Let $\mathbb C P^2$ be the complex projective plane endowed with the Fubini-Study metric of constant holomorphic sectional curvature 4. Let $f:M\rightarrow {\mathbb C}P^2$ be a Lagrangian immersion of an oriented surface. The induced metric on M generates a conformal structure with respect to which the metric is $g=2e^{u} dzd{\bar{z}}$, and where $z=x+iy$ is a local conformal coordinate on M and u is a real-valued function defined on M locally. For any Lagrangian immersion f, there exists a local horizontal lift $F: \mathrm{U}\rightarrow S^5(1) =\{Z\in \mathbb {C}^3 | \, Z\cdot \bar{Z}=1\}$, where $Z\cdot \overline{W}=\sum _{k=1}^{3} z_{k} \overline{w_{k}}$ denotes the Hermitian inner product for any $Z=(z_1,z_2,z_3)$ and $W=(w_1,w_2,w_3)\in \mathbb {C}^3$. In fact, choose any local lift F. Then $dF\cdot \bar{F}$ is a closed one-form. Hence there exists a real function $\eta \in C^\infty (\mathrm{U})$ locally such that $i d\eta = dF\cdot \bar{F}$. Then $\tilde{F}=e^{-i\eta } F$ is a local horizontal lift of f to $S^5(1)$. We can therefore assume

$$\begin{aligned} F_z \cdot {\overline{F}}=F_{\bar{z}}\cdot {\overline{F}} =0. \end{aligned}$$

(1)

The fact that the metric g is conformal is equivalent to

$$\begin{aligned}&F_z\cdot \overline{F_z}=F_{\bar{z}}\cdot \overline{F_{\bar{z}}}=e^{u},\nonumber \\&F_z\cdot \overline{F_{\bar{z}}}=0. \end{aligned}$$

(2)

Thus $\mathscr {F}=(e^{-\frac{u}{2}}F_z, e^{-\frac{u}{2}}F_{\bar{z}}, F)$ is a Hermitian orthonormal moving frame globally defined on the universal cover of M. Furthermore, let us assume that f is minimal now. It follows from (1) and (2) and the minimality of f that $\mathscr {F}$ satisfies the frame equations

$$\begin{aligned} \mathscr {F}_{z}=\mathscr {F} {\mathscr {U}}, \quad \mathscr {F}_{\bar{z}}=\mathscr {F} {\mathscr {V}}, \end{aligned}$$

(3)

where

$$\begin{aligned} {\mathscr {U}}=\left( \begin{array}{ccc} \frac{u_z}{2} &{} 0 &{} e^{\frac{u}{2}} \\ e^{-u}\psi &{}-\frac{u_z}{2} &{} 0 \\ 0 &{} -e^{\frac{u}{2}} &{}0 \\ \end{array} \right) , \quad {\mathscr {V}}=\left( \begin{array}{ccc} -\frac{u_{\bar{z}}}{2} &{} - e^{-u}\bar{\psi }&{}0 \\ 0&{} \frac{u_{\bar{z}}}{2} &{}e^{\frac{u}{2}} \\ -e^{\frac{u}{2}} &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

(4)

with

$$\begin{aligned} \psi =F_{zz}\cdot \overline{F_{\bar{z}}}. \end{aligned}$$

(5)

The cubic differential $\varPsi =\psi dz^3$ is globally defined on M and independent of the choice of the local horizontal lift. The differential $\varPsi$ is called the Hopf differential of f.

The compatibility condition of the equations (3) is ${\mathscr {U}}_{\bar{z}}-{\mathscr {V}}_z =[{\mathscr {U}},{\mathscr {V}}]$, and using (4) this turns out to be equivalent to

$$\begin{aligned} u_{z\bar{z}}+e^u-e^{-2u}|\psi |^2&=0,\end{aligned}$$

(6)

$$\begin{aligned} \psi _{\bar{z}}&=0. \end{aligned}$$

(7)

Notice that the integrability conditions (6)–(7) are invariant under the transformation $\psi \rightarrow \nu \psi$ for any $\nu \in S^1$.

This implies that after replacing $\psi$ in (4) by $\psi ^\nu =\nu \psi$ the Eq. (3) are still integrable. Therefore, the solution $\mathscr {F}(z,\bar{z}, \nu )$ to this changed system is a frame of some minimal Lagrangian surface $f^\nu$.

It turns out to be convenient to consider in place of the frames $\mathscr {F}(z,\bar{z}, \nu )$ the gauged frames

$$\begin{aligned} \mathbb {F}(\lambda )=\mathscr {F}(\nu )\left( \begin{array}{ccc} -i\lambda &{} 0 &{} 0 \\ 0 &{} \frac{1}{i\lambda } &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right) , \end{aligned}$$

where $i\lambda ^3 \nu =1$.

For these frames we obtain the equations

$$\begin{aligned} \mathbb {F}^{-1}\mathbb {F}_z&= \frac{1}{\lambda }\left( \begin{array}{ccc} 0 &{} 0 &{} i e^{\frac{u}{2}} \\ -i \psi e^{-u} &{} 0 &{} 0 \\ 0 &{} i e^{\frac{u}{2}} &{} 0 \\ \end{array} \right) +\left( \begin{array}{ccc} \frac{u_z}{2} &{} &{} \\ &{} -\frac{u_z}{2} &{} \\ &{} &{} 0 \\ \end{array} \right) \nonumber \\&:=\lambda ^{-1}U_{-1}+U_0,\nonumber \\ \mathbb {F}^{-1}\mathbb {F}_{\bar{z}}&=\lambda \left( \begin{array}{ccc} 0 &{} -i\bar{\psi } e^{-u} &{} 0 \\ 0&{} 0 &{} i e^{\frac{u}{2}} \\ i e^{\frac{u}{2}} &{}0 &{} 0 \\ \end{array} \right) +\left( \begin{array}{ccc} -\frac{u_{\bar{z}}}{2} &{} &{} \\ &{} \frac{u_{\bar{z}}}{2} &{} \\ &{} &{} 0 \\ \end{array} \right) \\&:=\lambda V_{1}+V_0.\nonumber \end{aligned}$$

(8)

Proposition 1

Let M be a Riemann surface and U a simply-connected open subset of M. Let $\mathbb {F}(z, \bar{z},\lambda ): U\rightarrow SU(3)$, $\lambda \in S^1, z \in U,$ be a solution to the system (8). Then $[\mathbb {F}(z, \bar{z}, \lambda )e_3]$ gives a minimal Lagrangian surface defined on U with values in $\mathbb {C}P^2$ and with the metric $g=2e^{u}dzd\bar{z}$ and the Hopf differential $\varPsi ^{\nu }=\nu \psi dz^3$.

Conversely, suppose $f^{\nu }:M\rightarrow \mathbb {C}P^2$ is a conformal parametrization of a minimal Lagrangian surface in $\mathbb {C}P^2$ with the metric $g=2e^{u}dzd\bar{z}$ and Hopf differential $\varPsi ^{\nu }=\nu \psi dz^3$. Then for any open, simply-connected subset U of M there exists a unique frame $\mathbb {F}: U \rightarrow SU(3)$ satisfying (8) and $[\mathbb {F}(z, \bar{z},\lambda )e_3] = f$.

Remark 1

• 1. In general, the notion of a “frame”only denotes maps $\mathbb {F}: U \rightarrow SU(3)$ such that $[\mathbb {F}(z, \bar{z}, \lambda )e_3]$ is a minimal Lagrangian surface. Then two such frames $\mathbb {F}$ and $\hat{\mathbb {F}}$ are in the relation $\hat{\mathbb {F}} = W \mathbb {F}k$ with $W\in SU(3)$ and k a map $k:U \rightarrow U(1)$.

• 2. Note that in this paper U(1) acts by diagonal matrices of the form $\mathrm {diag}(a,a^{-1},1)$ on the right. In particular, any gauge k for $\mathbb {F}$ is of this form.

2.1 The Loop Group Method for Minimal Lagrangian Surfaces

Let $\sigma$ denote the automorphism of $SL(3,\mathbb {C})$ of order 6 defined by

$$\begin{aligned} \sigma : g\mapsto P (g^t)^{-1} P^{-1}, \quad P=\left( \begin{array}{ccc} 0 &{} \alpha &{} 0 \\ \alpha ^2 &{} 0 &{} 0 \\ 0 &{}0 &{} 1\\ \end{array} \right) , \quad \alpha =e^{2\pi i/3}, \end{aligned}$$

Let $\tau$ denote the anti-holomorphic involution of $SL(3,\mathbb {C})$ which defines the real form SU(3),

$$\tau (g):=(\bar{g}^t)^{-1}.$$

Then the corresponding automorphism $\sigma$ of order 6 and the anti-holomorphic automorphism $\tau$ of $sl(3,\mathbb {C})$ are

$$\begin{aligned} \sigma : \xi \mapsto -P\xi ^t P^{-1}, \quad \tau : \xi \mapsto -\bar{\xi }^t. \end{aligned}$$

By $\mathfrak {g}_l$ we denote the $\varepsilon ^l$-eigenspace of $\sigma$ in $\mathfrak {g}^{\mathbb C}$, where $\varepsilon =e^{\pi i/3}$. Explicitly these eigenspaces are given as follows

Remark that the automorphism $\sigma$ gives a 6-symmetric space SU(3) / U(1) and any minimal Lagrangian surface in $\mathbb {C}P^2$ frames a primitive map $\mathbb {F}|_{\lambda =1}: M\rightarrow SU(3)/U(1)$.

Using loop group terminology, we can state (refer to [9]):

Proposition 2

Let $f: \mathbb {D} \rightarrow \mathbb {C}P^2$ be a conformal parametrization of a contractible Riemann surface. Then the following statements are equivalent:

• 1. f is minimal Lagrangian.

• 2. The moving frame $\mathbb {F}|_{\lambda =1}=(-ie^{-\frac{u}{2}} F_z, -ie^{-\frac{u}{2}}F_{\bar{z}}, F): \mathbb {D} \rightarrow SU(3)/U(1)$ is primitive.

• 3. $\mathbb {F}^{-1}d\mathbb {F} =(\lambda ^{-1} U_{-1}+U_0)dz+(\lambda V_1+V_0)d\bar{z} \subset \varLambda su(3)_{\sigma }$ is a one-parameter family of flat connections.

Set

$$\begin{aligned} \varLambda SL(3,\mathbb {C})_{\sigma }= & {} \{ g: S^1\rightarrow SL(3, {\mathbb C}) | g \text { has finite Wiener norm},\, g(\varepsilon \lambda )=\sigma g(\lambda )\}, \\ \varLambda ^{+} SL(3,\mathbb {C})_{\sigma }= & {} \{g\in \varLambda SL(3,\mathbb {C})_{\sigma } | \, g \text { extends holomorphically to } \mathbf {D}, g(0)\in U(1)^{\mathbb C}\},\\ \varLambda SU(3,\mathbb {C})_{\sigma }= & {} \{g\in \varLambda SL(3,\mathbb {C})_{\sigma } | \tau (g(\frac{1}{\bar{\lambda }}))=g(\lambda ) \}, \end{aligned}$$

where $\varepsilon =e^{\frac{\pi i}{3}}$ and $\mathbf {D}$ denotes the interior of the unit disk. Note that by $U(1)^{\mathbb C}$ we denote here the group of $\lambda$-independent matrices $\mathrm {diag}(a,a^{-1},1)$ with $a \in \mathbb {C}^*$.

The general Iwasawa decomposition theorem [12] takes in our case, i.e. for the groups $\varLambda SL(3,\mathbb {C})_{\sigma }$ and $\varLambda SU(3)_{\sigma }$, the following explicit form:

Theorem 1

(Iwasawa Decomposition theorem of $\varLambda SL(3,\mathbb {C})_{\sigma }$) Multiplication $\varLambda SU(3)_{\sigma }\times \varLambda ^{+} SL(3, \mathbb {C})_{\sigma } \rightarrow \varLambda SL(3, \mathbb {C})_{\sigma }$ is a diffeomorphism onto. Explicitly, every element $g\in \varLambda SL(3,\mathbb {C})_{\sigma }$ can be represented in the form $g=h V_{+}$ with $h\in \varLambda SU(3)_{\sigma }$ and $V_{+}\in \varLambda ^{+} SL(3,\mathbb {C})_{\sigma }$. One can assume without loss of generality that $V_{+} (\lambda =0)$ has only positive diagonal entries. In this case the decomposition is unique.

3 Equivariant Minimal Lagrangian Surfaces

In this section we will investigate minimal Lagrangian immersions for which there exists a one-parameter family $(\gamma _t, R_t) \in (Aut(M), Iso(\mathbb {C}P^2))$ of symmetries.

Definition 1

Let M be any connected Riemann surface and $f:M \rightarrow \mathbb {C}P^2$ an immersion. Then f is called equivariant, relative to the one-parameter group $(\gamma _t, R(t)) \in (Aut(M), Iso(\mathbb {C}P^2))$, if

$$f(\gamma _t \cdot p)=R(t) f(p)$$

for all $p\in M$ and all $t\in \mathbb {R}$.

By the definition above, any Riemann surface M admitting an equivariant minimal Lagrangian immersion admits a one-parameter group of (biholomorphic) automorphisms. Fortunately, the classification of such surfaces is very simple:

Theorem 2

(Classification of Riemann surfaces admitting one-parameter groups of automorphisms, e.g. [4])

• 1. $S^2$,

• 2. $\mathbb {C}$, $\mathbf {D}$,

• 3. $\mathbb {C}^*$,

• 4. $\mathbf {D}^*, \mathbf {D}_r$,

• 5. $T=\mathbb {C}/\varLambda _{\tau }$,

where the superscript “$^*$” denotes deletion of the point 0, the subscript “r” denotes the open annulus between $0< r < 1/r$ and $\varLambda _{\tau }$ is the free group generated by the two translations $z\mapsto z+1$, $z\mapsto z+\tau$, $\mathrm {Im}\tau >0$.

Looking at this classification, one sees that after some composition with some holomorphic transformation one obtains the following picture, including the groups of translations:

Theorem 3

(Classification of Riemann surfaces admitting one-parameter groups of automorphisms and representatives for the one-parameter groups, e.g. [4])

• 1. $S^2$, group of rotations about the z-axis,

• 2.

  • a. $\mathbb {C}$, group of all real translations,

  • b. $\mathbb {C}$, group of all rotations about the origin 0,

  • c. $\mathbf {D}$, group of all rotations about the origin 0,

  • d. $\mathbf {D} \cong \mathbb {H}$, group of all real translations,

  • e. $\mathbf {D} \cong \mathbb {H} \cong \log \mathbb {H} =\mathbb {S}$, the strip between $y=0$ and $y = \pi$, group of all real translations,

• 3. $\mathbb {C}^*$, group of all rotations about 0,

• 4. $\mathbf {D}^*, \mathbf {D}_r,$ group of all rotations about 0,

• 5. T, group of all real translations.

For later purposes we state the following

Definition 2

Let $f: M \rightarrow \mathbb {C}P^2$ be an equivariant minimal Lagrangian immersion, then f is called “translationally equivariant”, if the group of automorphisms acts by (all real) translations. It is called “rotationally equivariant”, if the group acts by all rotations about 0.

Remark 2

Since we know that any minimal Lagrangian immersion f from a sphere is totally geodesic and it is the standard immersion of $S^2$ into $\mathbb {C}P^2$ [14], we will exclude the case $S^2$ from the discussions in this paper.

4 Translationally Equivariant Minimal Lagrangian Immersions

By what was said just above, we will assume throughout this section that the surface M is a strip $\mathbb {S}$ in $\mathbb {C}$ parallel to the x-axis. Actually, by applying a translation in y–direction we can assume that the real axis is contained in $\mathbb {S}$ and in particular $0 \in \mathbb {S}$.

We thus consider minimal Lagrangian immersions $f:\mathbb {S}\rightarrow \mathbb {C} P^2$ for which there exists a one-parameter subgroup R(t) of SU(3) such that

$$f(t + z, t+\bar{z})=R(t) f(z,\bar{z})$$

for all $z\in \mathbb {S}$.

Let $\mathbb {F}:\mathbb {S}\rightarrow SU(3)$ be a frame of f satisfying $\mathbb {F}(0)=I$. Since f is translationally equivariant we obtain that the frame $\mathbb {F}$ of f is translationally equivariant in the sense that

$$\begin{aligned} \mathbb {F}(t + z)=R(t) \mathbb {F}(z, \bar{z}) \mathscr {K}(t,z) \end{aligned}$$

(9)

holds, where $\mathscr {K}(t,z)$ is a crossed homomorphism with values in U(1). This means that $\mathscr {K}$ can be chosen such that $\mathbb {F}(0) = I$ and satisfies the following cocycle condition:

$$\begin{aligned} \mathscr {K}(t+s,z)=\mathscr {K}(s,z)\mathscr {K}(t,s+z). \end{aligned}$$

(10)

In fact,

Theorem 4

$\mathscr {K}(t,z)$ is a coboundary. More precisely, for the matrix function $h(z)=\mathscr {K}(x,iy)^{-1}$ we have

$$\begin{aligned} \mathscr {K}(t,z) = h(z)h(t+z)^{-1}. \end{aligned}$$

(11)

Replacing h by $\hat{h} = h(0)^{-1} h$ if necessary we can even assume without loss of generality that the coboundary equation above holds with some matrix function h also satisfying $h(0) = I$.

Proof

Setting $z=iy$ in (10), we get

$$\begin{aligned} \mathscr {K}(t+s, iy)=\mathscr {K}(s,iy)\mathscr {K}(t,s+iy). \end{aligned}$$

(12)

Take $h(z)=\mathscr {K}(x,iy)^{-1}$, where $z=x+iy$. Then putting $s=x$ in (12), we obtain

$$\begin{aligned} h(z)h(t+z)^{-1}=\mathscr {K}(x,iy)^{-1}\mathscr {K}(t+x,iy)=\mathscr {K}(t,z), \end{aligned}$$

which completes the proof of (11). The last statement is trivial.

This implies the important

Theorem 5

For any translationally equivariant minimal Lagrangian immersion, the frame $\mathbb {F}$ can be chosen such that $\mathbb {F}(0) = I$ and

$$\begin{aligned} \mathbb {F}(t+z)=\chi (t)\mathbb {F}(z), \end{aligned}$$

holds, where $\chi (t)$ is a one-parameter group in SU(3).

Proof

Choosing $h \in U(1)$ as in the theorem above, satisfying $h(0) = I$ and replacing $\mathbb {F}$ by $\hat{\mathbb {F}}(z):= \mathbb {F}(z)h(z)$, we obtain from (9) and (11)

$$\begin{aligned} \hat{\mathbb {F}}(t + z)= & {} \mathbb {F}(t + z) h(t +z)=R(t) \mathbb {F}(z)\mathscr {K}(t,z)h(t +z)\nonumber \\= & {} R(t) \mathbb {F}(z)h(z) =R(t) \hat{\mathbb {F}}(z). \end{aligned}$$

Thus $\hat{\mathbb {F}}$ satisfies the claim.

Let’s now consider the frame $\mathbb {F}$ obtained in the theorem above. It satisfies the Maurer-Cartan equation $\alpha = \mathbb {F}^{-1}d\mathbb {F}=U_{-1}dz+U_{0}dz+V_{0}d\bar{z}+V_{1}d\bar{z}$. Now we introduce $\lambda \in S^1$ as usual. We set

$$\begin{aligned} \alpha _\lambda =\lambda ^{-1}U_{-1}dz+U_{0}dz+V_{0}d\bar{z}+\lambda V_{1}d\bar{z}. \end{aligned}$$

(13)

Then $\alpha (z,\lambda )$ is integrable, since the Maurer-Cartan form of the original frame (used in the theorem above) is integrable.

Let $\mathbb {F}(z,\lambda )$ denote the solution to

$$\begin{aligned} \mathbb {F}(z, {\lambda })^{-1}d\mathbb {F}(z, {\lambda }) = \alpha _\lambda , \end{aligned}$$

also satisfying $\mathbb {F}(0,\lambda ) = I$. Then $\mathbb {F}$ satisfies

$$\begin{aligned} \mathbb {F}(t+z, \lambda )=\chi (t,\lambda )\mathbb {F}(z,\lambda ), \end{aligned}$$

(14)

for any $z\in M$ and a one-parameter group $\chi (t,\lambda )=e^{tD(\lambda )}$ for some $D(\lambda )\in \varLambda su(3)_{\sigma }$. This is the equivariance condition on the extended frame $\mathbb {F}(z,\lambda )$ assumed in [1].

But in the context of equivariant minimal Lagrangian immersions it is obvious that the coefficient matrices of (8) are independent of x. Therefore, the solution $\mathbb {F}_0$ to the differential equations (8) with initial condition $\mathbb {F}_0(0,\lambda ) = I$ also satisfies (14). It is easy to see that two frames satisfying (14) only differ by some gauge in U(1) which is independent of x. Thus we obtain

Theorem 6

For the extended frame $\mathbb {F}$ of any translationally equivariant minimal Lagrangian immersion we can assume without loss of generality $\mathbb {F}(0,\lambda )=I$ and

$$\begin{aligned} \mathbb {F}(t+z, \lambda )=\chi (t,\lambda )\mathbb {F}(z,\lambda ), \end{aligned}$$

with $\chi (t,\lambda )=e^{tD(\lambda )}$ for some $D(\lambda )\in \varLambda su(3)_{\sigma }$. Moreover, we can also assume that $\mathbb {F}$ satisfies (8).

Any two frames satisfying (14) only differ by some gauge in U(1) which only depends on y.

4.1 Application of a Result by Burstall and Kilian for Translationally Equivariant Minimal Lagrangian Immersions

In this section we assume that the frame is chosen as in Theorem 6. Then, following Burstall-Kilian [1] and setting $t=-x$ and $z=x+it$, we derive from (14),

$$\begin{aligned} \mathbb {F}(z,\lambda )= & {} e^{x D(\lambda )}\mathbb {F}(iy,\lambda ). \end{aligned}$$

(15)

We also assume as before $\mathbb {F}(0,\lambda ) = I$. Then

$$\begin{aligned} \alpha _{\lambda }= \mathbb {F}^{-1}(z,\lambda )d\mathbb {F}(z,\lambda )=\mathbb {F}(iy, \lambda )^{-1} D\mathbb {F}(iy, \lambda )dx+\mathbb {F}(iy,\lambda )^{-1}\frac{d}{dy}\mathbb {F}(iy,\lambda )dy. \end{aligned}$$

So

$$\begin{aligned} A_\lambda (y):=\mathbb {F}(iy, \lambda )^{-1} D(\lambda )\mathbb {F}(iy, \lambda ),\quad B_\lambda (y):=\mathbb {F}(iy, \lambda )^{-1}\frac{d}{dy}\mathbb {F}(iy, \lambda ) \end{aligned}$$

(16)

depend only on y. Comparing to (13), we infer

$$\begin{aligned} A_{\lambda }(y)&=\lambda ^{-1}U_{-1}+U_{0}+V_{0}+\lambda V_{1},\nonumber \\ B_{\lambda }(y)&=i(\lambda ^{-1}U_{-1}+U_{0}-V_{0}-\lambda V_{1}), \end{aligned}$$

(17)

where we have used

$$\alpha _{\lambda }=\lambda ^{-1} U_{-1}dz+U_0 dz+V_0d\bar{z}+\lambda V_1 d\bar{z}.$$

Hence $A_{\lambda }(y), B_{\lambda }(y), D(\lambda )\subset \varLambda _1 su(3)_{\sigma }$ and $\alpha _{\lambda }(\partial /\partial \bar{z})$ is holomorphic in $\lambda$. Set $b_{\lambda }(y):=e^{-iyD(\lambda )} \mathbb {F} (iy, \lambda )$. Then its Maurer-Cartan form is given by $b_{\lambda }^{-1} db_{\lambda }=-2i \alpha _{\lambda }(\frac{\partial }{\partial \bar{z}})dy$.

Therefore, with the initial condition $\mathbb {F}(0,\lambda ) = I$, we know that $b_{\lambda }$ is holomorphic in $\lambda$ and $b_{\lambda }\in \varLambda ^{+}SL(3, \mathbb {C})_{\sigma }$. It follows that

$$\mathbb {F}(z, {\lambda })=e^{xD(\lambda )}\mathbb {F}(iy, {\lambda })=e^{zD(\lambda )}b_{\lambda }(y)$$

is an Iwasawa decomposition of $\mathbb {F}$. This means that $\mathbb {F}$ is generated by the degree one constant potential $D(\lambda )=A_{\lambda }(0) \in \varLambda su(3)_{\sigma }$.

Conversely, for any constant degree one potential $D(\lambda )\in \varLambda su(3)_{\sigma }$, we have the solution $C(z,\lambda ):=e^{zD(\lambda )}=e^{xD(\lambda )} e^{iyD(\lambda )}$ to $dC = CD(\lambda )dz$, $C(0,\lambda )=I$. Assume that an Iwasawa decomposition of $e^{iyD(\lambda )}$ is given by

$$\begin{aligned} e^{iyD(\lambda )}=U(y,\lambda )U_{+}(y,\lambda ), \end{aligned}$$

(18)

where $U(y,\lambda ): M \rightarrow \varLambda SU(3)_{\sigma }$ and $U_{+}(y,\lambda )\in \varLambda ^{+} SL(3,\mathbb {C})_{\sigma }$. Because $e^{xD(\lambda )}\in \varLambda SU(3)_{\sigma }$ for all $x\in \mathbb {R}$, we conclude that an Iwasawa decomposition of $e^{zD(\lambda )}$ is given by $e^{zD(\lambda )}=\mathbb {F}(z,\lambda )U_{+}(y,\lambda )$, where

$$\begin{aligned} \mathbb {F}(z,\lambda )=e^{xD(\lambda )}U(y,\lambda ). \end{aligned}$$

(19)

Hence, $\mathbb {F}(z,\lambda )$ is translationally equivariant.

Thus we conclude

Proposition 3

A minimal Lagrangian surface in $\mathbb {C}P^2$ is translationally equivariant if and only it is generated by a degree one constant potential $D(\lambda )dz$. In this case the immersion can be defined without loss of generality on all of $\mathbb {C}$. The potential function $D(\lambda )$ can be obtained from the extended frame $\mathbb {F}$ satisfying (14) and $\mathbb {F}(0,\lambda )=I$ by the equation

$$D(\lambda )=\mathbb {F}(z,\lambda )^{-1}\partial _x\mathbb {F}(z,\lambda )|_{z=0}.$$

Remark 3

Since any two frames satisfying (14) and attaining I at $z =0$ also satisfy Eq. (15), it is easy to see that $D(\lambda )$ is uniquely determined. From this it also follows again that two such frames only differ by some gauge $k(iy) \in U(1)$.

5 Explicit Iwasawa Decomposition for Translationally Equivariant Minimal Lagrangian Immersions

5.1 The Basic Set-Up

We have seen above in Sect. 4.1 that every translationally equivariant minimal Lagrangian immersion can be obtained from some potential of the form

$$\begin{aligned} \eta = D(\lambda ) dz, \end{aligned}$$

where

$$\begin{aligned} D(\lambda ) = \lambda ^{-1} D_{-1} + D_0 + \lambda D_1 \in \varLambda su(3)_\sigma . \end{aligned}$$

(20)

The general loop group approach requires to consider the solution to $dC = C \eta , C(0,\lambda ) = I$. This is easily achieved by $C(z,\lambda ) = \exp (zD(\lambda )).$

Next one needs to perform an Iwasawa splitting. In general this is very complicated and difficult to carry out explicitly. But, for translationally equivariant minimal Lagrangian surfaces in $\mathbb {C}P^2$, one is able to carry out an explicit Iwasawa decomposition of $\exp (zD(\lambda ))$.

From (15), (18) and (19), we get

$$\begin{aligned} F(iy,\lambda )=U(y,\lambda )=e^{iyD(\lambda )} U_{+}(y,\lambda )^{-1}. \end{aligned}$$

(21)

Substituting (21) into (16), we obtain

$$\begin{aligned} A_{\lambda }(y)&=U_{+}(y,\lambda )D(\lambda )U_{+}(y,\lambda )^{-1},\nonumber \\ B_{\lambda }(y)&=U_{+}(y,\lambda )i D(\lambda )U_{+}(y,\lambda )^{-1}- \frac{d}{dy}U_{+}(y,\lambda ) U_{+}(y,\lambda )^{-1}. \end{aligned}$$

(22)

Comparing this to (17), we obtain the equations

$$\begin{aligned} U_{+}(y,\lambda )D(\lambda )U_{+}^{-1}(y,\lambda )= \lambda ^{-1}U_{-1}+U_0+\lambda V_1+V_0 =:\varOmega , \end{aligned}$$

(23)

$$\begin{aligned} \frac{d}{dy}U_{+}(y,\lambda ) U_{+}(y,\lambda )^{-1}= 2i(\lambda V_1+V_0). \end{aligned}$$

(24)

It is important to note that because $U_{+}$ only depends on y and $\mathbb {F}$ satisfies (8), the matrix $\varOmega$ is of the form

$$\begin{aligned} \varOmega =\begin{pmatrix} \frac{u_z-u_{\bar{z}}}{2}&{}-i\lambda \bar{\psi }e^{-u}&{}i\lambda ^{-1}e^{\frac{u}{2}}\\ -i\lambda ^{-1}\psi e^{-u}&{}-\frac{u_z-u_{\bar{z}}}{2}&{}i\lambda e^{\frac{u}{2}} \\ i\lambda e^{\frac{u}{2}}&{}i\lambda ^{-1} e^{\frac{u}{2}}&{}0 \end{pmatrix}, \end{aligned}$$

where u only depends on y and $\psi$ is constant.

The Eqs. (23) and (24) are the basis for an explicit computation of the Iwasawa decomposition of $\exp (z D(\lambda ))$. There will be two steps:

Step 1: Solve Eq. (23) by some matrix Q. Then $U_+$ and Q satisfy

$$\begin{aligned} U_+ = Q E, \quad \text { where} \quad E \quad \text {commutes with} \quad D. \end{aligned}$$

Step 2: Solve Eq. (24). This will generally only mean to carry out two integrations in one variable.

5.2 Evaluation of the Characteristic Polynomial Equations

Step 1 mentioned above actually consists of two sub-steps. First of all one determines $\varOmega$ from D and then one computes a solution Q the Eq. (23).

In this section we will discuss the first sub-step. In our case we observe that D and $\varOmega$ are conjugate and therefore have the same characteristic polynomials. Using the explicit form of $\varOmega$ stated just above and writing D in the form

$$\begin{aligned} D=\begin{pmatrix} \alpha &{}-\lambda \bar{b} &{} \lambda ^{-1} a \\ \lambda ^{-1} b &{}-\alpha &{}-\lambda \bar{a}\\ -\lambda \bar{a} &{}\lambda ^{-1} a &{}0 \end{pmatrix}\in \varLambda _1 \subset \varLambda su(3)_{\sigma }, \end{aligned}$$

where $\alpha , a$ and b are constants, (23) leads to

$$\begin{aligned}&2e^{u}+|\psi |^2e^{-2u}+\frac{1}{4}(u^\prime )^2=-\alpha ^2+2|a|^2+|b|^2=:\beta ,\end{aligned}$$

(25)

$$\begin{aligned}&\psi =-ia^2b, \end{aligned}$$

(26)

where $\alpha , a, b$ and $\psi$ are constants.

Remark 4

We have seen that if $\psi \equiv 0,$ then the surface is totally geodesic, hence the image of f is an open portion of the real projective plane. We will ignore this case from now on and will assume that $\psi \not \equiv 0.$

5.3 Explicit Solutions for Metric and Cubic Form

If $u^{\prime }\equiv 0$, then u is constant and the surface is flat. It is well known that the only flat minimal Lagrangian surface in $\mathbb {C}P^2$ is an open subset of the Clifford torus up to isometries of $\mathbb {C}P^2$ [8]. In the following we will assume $u^{\prime }\not \equiv 0$.

Notice that (25) is a first integral of the Gauss equation

$$\begin{aligned} \frac{1}{4}u^{\prime \prime }+e^u-|\psi |^2 e^{-2u}=0. \end{aligned}$$

(27)

Making the change of variables $w=e^{u}$ in (25), we obtain equivalently

$$\begin{aligned} (w^{\prime })^2+8w^3-4 \beta w^2 +4|\psi |^2=0. \end{aligned}$$

(28)

Since we assume $\psi \ne 0$, the solutions to (28) are given in terms of bounded Jacobi elliptic functions. Since all Jacobi elliptic functions are periodic, there exists a point, where the derivative of u vanishes. Choosing this point as the origin, we can always assume $u^{\prime }(0)=0$. For our loop group setting this has an important consequence:

Theorem 7

By choosing the coordinates such that the metric for a given translationally equivariant minimal Lagrangian immersion has a vanishing derivative at $z = 0$, we obtain that the generating matrix D in (20) satisfies $D_0 = 0$.

We will therefore always assume this condition. This convention in combination with (25) implies

$$\begin{aligned} 2a_1+\frac{|\psi |^2}{a_1^2}=\beta , \end{aligned}$$

(29)

where $a_1:=e^{u(0)}>0$.

The following computations are very similar to the ones given in [2]. We include them for the convenience of the reader.

Using (29) it is easy to verify that (28) can be rewritten in the form

$$\begin{aligned} (w^\prime )^2+8(w-a_1)(w-a_2)(w+a_3)=0, \end{aligned}$$

(30)

where

$$\begin{aligned} a_1= & {} e^{u(0)}>0,\\ a_2= & {} \frac{\frac{\beta }{2}-a_1+\sqrt{(\frac{\beta }{2}-a_1)^2+4a_1 (\frac{\beta }{2}-a_1)}}{2}>0,\\ a_3= & {} \frac{-(\frac{\beta }{2}-a_1)+\sqrt{(\frac{\beta }{2}-a_1)^2+4a_1(\frac{\beta }{2}-a_1)} }{2}>0. \end{aligned}$$

Since $\frac{\beta }{2}-a_1=\frac{|\psi |^2}{2a_1^2}>0$, we know $a_2> a_3$. Notice that $a_1=a_2$ if and only if $\beta =3|\psi |^{2/3}$, then (27) has the unique solution $u \equiv \frac{2}{3} \log |\psi |$, which conflicts our starting assumption $u^{\prime }\not \equiv 0$. Therefore we can assume without loss of generality that $a_1>a_2$ holds.

Then with $a_2<w<a_1$, (30) leads to

$$dy=\frac{1}{\sqrt{-8(w-a_1)(w-a_2)(w+a_3)}}dw.$$

Integrating gives

$$\begin{aligned} y=-\frac{1}{\sqrt{2(a_1+a_3)}}J(\arcsin \sqrt{\frac{a_1-w}{a_1-a_2}}, k), \end{aligned}$$

where

$$\begin{aligned} k^2=\frac{a_1-a_2}{a_1+a_3} \end{aligned}$$

(31)

and J denotes the elliptic integral of the first kind

$$J(\theta ,k)=\int _0^{\theta }\frac{d\alpha }{\sqrt{1-k^2\sin ^2\alpha }}, $$

for any $0\le k\le 1$. Thus the solution to (28) is

$$\begin{aligned} w(y)=e^{u(y)}=a_1(1-q^2\mathrm {sn}^2(ry,k)), \end{aligned}$$

(32)

where

$$\begin{aligned} q^2=\frac{a_1-a_2}{a_1}, \quad r=\sqrt{2(a_1+a_3)}. \end{aligned}$$

(33)

It is easy to see that the solution u(y) is an even periodic function with period 2T, where $T=\frac{K}{r}$ and $K=J(\frac{\pi }{2}, k)$ is the complete elliptic integral of the first kind. Thus for any (in x–direction) translationally equivariant minimal Lagrangian surface in $\mathbb {C}P^2$, its metric conformal factor $e^u$ is given by (32) in terms of a Jacobi elliptic function and its cubic Hopf differential is constant and given by (26).

5.4 Explicit Iwasawa Decompositions

Recall that for translationally equivariant minimal Lagrangian surfaces, the potential matrix $D(\lambda )$ coincides with $A_{\lambda }(0)=\varOmega |_{y=0}$ in (22), so we have (including the convention above about the origin)

$$\begin{aligned} D(\lambda )=\begin{pmatrix} 0&{}-i\lambda \bar{\psi }e^{-u(0)}&{}i\lambda ^{-1}e^{\frac{u(0)}{2}}\\ -i\lambda ^{-1}\psi e^{-u(0)}&{}0&{}i\lambda e^{\frac{u(0)}{2}}\\ i\lambda e^{\frac{u(0)}{2}}&{}i\lambda ^{-1} e^{\frac{u(0)}{2}}&{}0 \end{pmatrix}, \end{aligned}$$

where $\alpha =-\frac{iu^{\prime }(0)}{2}=0$, $a=ie^{\frac{u(0)}{2}}$ and $b=-i\psi e^{-u(0)}$. We may summarize the following proposition:

Proposition 4

Up to isometries in $\mathbb {C}P^2$, any translationally equivariant minimal Lagrangian surface can be generated by a potential of the form

$$\begin{aligned} \begin{pmatrix} 0&{}-\lambda \bar{b}&{}\lambda ^{-1}a\\ \lambda ^{-1}b&{}0&{}-\lambda \bar{a}\\ -\lambda \bar{a}&{}\lambda ^{-1} a&{}0 \end{pmatrix} dz, \end{aligned}$$

(34)

where a is nonzero purely imaginary and $a, b=\frac{i\psi }{a^2}$ are constants.

Thus the characteristic polynomial of $D(\lambda )$ in (34) is given by

$$\begin{aligned} \det (\mu I-D(\lambda ))=\mu ^3+\beta \mu -2i\mathrm {Re}(\lambda ^{-3}\psi ), \end{aligned}$$

(35)

where

$$\begin{aligned} \beta =2|a|^2+|b|^2=2e^{u(0)}+|\psi |^2 e^{-2u(0)}:=2a_1+\frac{|\psi |^2}{a_1^2}. \end{aligned}$$

(36)

Lemma 1

With the notation introduced above we have

• 1. The characteristic polynomial (35) of $D(\lambda )$ has purely imaginary roots which depend on $\lambda$, but not on z.

• 2. For any non-flat minimal Lagrangian surface in $\mathbb {C}P^2$, the characteristic polynomial (35) of $D(\lambda )$ has three distinct roots for any choice of $\lambda \in S^1$. Moreover, the root 0 occurs if and only if $\lambda ^{-3}\psi$ is purely imaginary. This case can only happen for six different values of $\lambda$.

Proof

Claim 1 simply follows from the observation that entries of $D(\lambda )\in su(3)$ only depend on $\lambda \in S^1$. The characteristic polynomial (35) of $D(\lambda )$ has three distinct purely imaginary roots if and only if its discriminant satisfies

$$(\frac{\beta }{3})^3-[\mathrm {Re}(\lambda ^{-3}\psi )]^2>0.$$

Regarding $\beta$ as a function of $a_1\in (0,\infty )$ it is easy to see that $\beta$ attains the minimum value $3|\psi |^{\frac{2}{3}}$ when $a_1=|\psi |^{\frac{2}{3}}$. In this case $D(\lambda )$ has multiple eigenvalues and $a_2=a_1=|\psi |^{\frac{2}{3}}$, $a_3=|\psi |^{\frac{2}{3}}/2$. It follows from (30) that the corresponding surface is flat. We have excluded this case. Therefore we have $\beta >3|\psi |^{\frac{2}{3}}$, which completes the proof of Claim 2.

Now take

$$\begin{aligned} Q_0=\mathrm {diag}(i a^{-1} e^{\frac{u}{2}}, -i a e^{-\frac{u}{2}},1), \end{aligned}$$

(37)

such that

$$\hat{\varOmega }=Q_0^{-1}\varOmega Q_0=\begin{pmatrix} -\frac{iu^{\prime }}{2}&{}i\lambda \bar{\psi }a^2 e^{-2u}&{}\lambda ^{-1}a\\ \lambda ^{-1}b&{}\frac{iu^{\prime }}{2}&{}-\lambda a^{-1}e^u\\ -\lambda a^{-1}e^u&{}\lambda ^{-1}a&{}0 \end{pmatrix}$$

has the same coefficients at $\lambda ^{-1}$ as $D(\lambda )$.

Then by a straightforward computation, we solve $\tilde{Q}D(\lambda )\tilde{Q}^{-1}= \hat{\varOmega }$ by the following matrix

$$\begin{aligned} \tilde{Q}=\frac{\lambda ^3}{\kappa }\begin{pmatrix} \check{p}&{}\check{q}&{}\check{v}_1\\ \check{s}&{}\check{t}&{}\check{v}_2\\ 0&{}0&{}\check{c} \end{pmatrix}, \end{aligned}$$

(38)

where

$$\begin{aligned} \check{p}&= -|a|^2 \frac{u^{\prime }}{2}+\lambda ^3 \bar{\psi }|a|^2 e^{-u}-\lambda ^{-3}\psi ,\nonumber \\ \check{q}&= \frac{\lambda ^{-2}a}{\bar{a}}[\frac{u^{\prime }}{2} |a|^2 -\lambda ^3\bar{\psi }e^{-u}(|a|^2-e^u)],\nonumber \\ \check{s}&=\frac{\lambda ^2}{a^2}[|a|^2 \frac{u^{\prime }}{2}e^u+\lambda ^{-3} \psi (|a|^2- e^u)],\nonumber \\ \check{t}&=\frac{1}{|a|^2} (-|a|^2 \frac{u^{\prime }}{2}e^u+\lambda ^3 \bar{\psi }e^u -\lambda ^{-3}\psi |a|^2),\\ \check{v}_1&=-2i\lambda ^{-1}a (|a|^2-e^u),\nonumber \\ \check{v}_2&=-2i\lambda a^{-1} e^u (|a|^2-e^u),\nonumber \\ \check{c}&=\lambda ^3 \bar{\psi }-\lambda ^{-3}\psi -e^u u^{\prime },\nonumber \\ \kappa&=(\lambda ^6 \bar{\psi }-\psi -\lambda ^3e^u u^{\prime })^{2/3}( \lambda ^6 \bar{\psi }-\psi )^{1/3}.\nonumber \end{aligned}$$

(39)

Moreover, $\det \tilde{Q}=1$ and $\tilde{Q}(0,\lambda )=Q_0(0,\lambda )=I$ due to $a=ie^{\frac{u(0)}{2}}$. If $\lambda$ is small, the denominator of the coefficient of $\tilde{Q}$ is single-valued. Altogether we have found a solution to equation (23) by $Q=Q_0 \tilde{Q}$.

Since also $U_+$ has the same properties, we obtain that $E = Q^{-1} U_+$ has determinant 1, attains the value I for $z=0$, is holomorphic for all small $\lambda$ and satisfies $[Q^{-1}U_{+}, D]=0.$

By Lemma 1 we can assume without loss of generality that $D=D(\lambda )$ is regular semi-simple for all but finitely many values of $\lambda$. Therefore, for all small z and small $\lambda$ we can write $E = \exp ( \mathscr {E} )$, where $[\mathscr {E}, D]=0.$

Since, in the computation of Q, we did not worry about the twisting condition, the matrix $\mathscr {E}$ is possibly an untwisted loop matrix in $SL(3,\mathbb {C}).$ But since $SL(3,\mathbb {C})$ has rank 2, for any regular semi-simple matrix $D=D(\lambda ),$ the commutant of $D(\lambda )$ is spanned by $D(\lambda )$ and one other matrix.

Lemma 2

Every element in the commutant $\{X\in \varLambda sl(3,\mathbb {C})_{\sigma }: [X,D]=0\}$ of $D(\lambda )$ has the form $X(\lambda )=\kappa _1(\lambda ) D(\lambda )+\kappa _2(\lambda )L_0(\lambda )$ with $\kappa _1(\varepsilon \lambda )=\kappa _1(\lambda )$, $\kappa _2(\varepsilon \lambda )=-\kappa _2 (\lambda )$, where $L_0=D^2(\lambda ) -\frac{1}{3} \mathrm {tr}(D^2) I$.

Corollary 1

The matrix $Q^{-1} U_+$ has the form

$$\begin{aligned} Q^{-1}U_{+}=\exp (\beta _1 D+\beta _2 L_0), \end{aligned}$$

where $\beta _1$ and $\beta _2$ are functions of y and $\lambda$ near 0.

With this description of $U_+$ Eq. (24) leads to the following two equations:

$$\begin{aligned}&-\beta _1^{\prime }\lambda \bar{a} +\beta _2^{\prime }\lambda ^{-2}ab=-\frac{2i\lambda e^u}{a}\frac{\check{p}}{\check{c}},\\&\beta _1^{\prime } \lambda ^{-1}a+\beta _2^{\prime }\lambda ^2 \bar{a}\bar{b}=-\frac{2i\lambda e^u}{a}\frac{\check{q}}{\check{c}}. \end{aligned}$$

Integrating then yields

$$\begin{aligned} \beta _1(y)&= \int _0^y \frac{2i\lambda ^3\bar{\psi }- i u^{\prime } e^u}{\lambda ^3\bar{\psi }-\lambda ^{-3}\psi -e^u u^{\prime }}ds, \nonumber \\ \beta _2(y)&= \int _0^y \frac{2e^u}{\lambda ^3\bar{\psi }-\lambda ^{-3}\psi -e^u u^{\prime }}ds. \end{aligned}$$

(40)

Putting everything together we obtain

Theorem 8

(Explicit Iwasawa decomposition) The extended frame for the translationally equivariant minimal Lagrangian surface in $\mathbb {C}P^2$ generated by the potential $D(\lambda )dz$ with vanishing diagonal satisfying $ab \ne 0$ is given by

$$\begin{aligned} \mathbb {F}(z,\lambda )=\exp (zD-\beta _1(y,\lambda ) D-\beta _2(y,\lambda ) L_0) Q^{-1}(y,\lambda ), \end{aligned}$$

(41)

with $\beta _1, \beta _2$ as in (40) and $Q=Q_0\tilde{Q}$ as in (37)–(39) and u as in (32).

Remark 5

In the proof of the last theorem we have derived the equation $U_{+}=Q\exp (\beta _1 D+\beta _2 L_0).$ In this equation each separate term is only defined for small $\lambda$ and a restricted set of $y'$s. However, due to the globality and the uniqueness of the Iwasawa splitting, the matrix $U_+$ is defined for all $\lambda$ in $\mathbb {C}^*$ and all $z \in \mathbb {C}$.

5.5 Explicit Expressions for Minimal Lagrangian Immersions

To make formula (41) explicit we need to know how the exponential factor acts on $Q^{-1}e_3$. This can be done in two ways: Since the exponential factor commutes with D, one can express it in terms of a linear combination of the matrices $I,D,D^2$. Once the coefficients are known, the horizontal lift F is given explicitly. The second way is to diagonalize D and to expand $Q^{-1}e_3$ relative to an eigenvector basis of D. It turns out that this second approach can be carried out quite easily and yields a straightforward comparison with the work of Castro-Urbano [2] which we will discuss in the next section. We would like to point out that in these computations we ignore any “twisting”.

We start by computing an eigenvector basis for D. Let $\mu$ be an eigenvalue of D. Then by (35) and (36) it is easy to verify that the vector

$$\begin{aligned} s_\mu = \begin{pmatrix} |a|^2 + \mu ^2\\ \lambda ^{-1}b \mu + \lambda ^2 \bar{a}^2\\ \lambda ^{-2} ab - \lambda \bar{a} \mu \end{pmatrix} \end{aligned}$$

is an eigenvector for D for the eigenvalue $\mu$. We know from Lemma 1 that for any non-flat minimal Lagrangian surface, up to possibly six values of $\lambda$ the matrix D has three different nonzero eigenvalues. Since D is skew-Hermitian, we also know that the corresponding eigenvectors are automatically perpendicular. Therefore there exists a unitary matrix L such that $D=L\mathrm {diag}(\mu _1,\mu _2,\mu _3) L^{-1}$, where, as before, $\mu _j \, (j=1,2,3)$ denote eigenvalues of D. As a consequence, for the extended horizontal lift F we thus obtain

$$ F = \mathbb {F}e_3=L\exp (z\varLambda -\beta _1\varLambda -\beta _2(\varLambda ^2-\frac{\mathrm {tr}\varLambda ^2}{3}I))L^{-1}Q^{-1}e_3,$$

where $\varLambda =\mathrm {diag}(\mu _1,\mu _2,\mu _3)$.

From (37)–(39), it is easy to derive

$$Q^{-1}e_3= \frac{1}{\kappa } \begin{pmatrix}2i\lambda ^{-1}a(|a|^2-e^u)\\ 2i \lambda \bar{a}(|a|^2-e^u)\\ \lambda ^3\bar{\psi }-\lambda ^{-3}\psi \end{pmatrix},$$

where $\kappa = (\lambda ^3\bar{\psi }-\lambda ^{-3}\psi -e^u u^{\prime })^{1/3}(\lambda ^3\bar{\psi }-\lambda ^{-3}\psi )^{2/3}$. Since we will eventually project to $\mathbb {C}P^2$, the factor $\kappa$ is actually irrelevant.

Setting $l_j = \frac{s_j}{||s_j||}$, where we put $s_\mu = s_j$ if $\mu = \mu _j$, we obtain

$$ L= ( l_1,l_2,l_3) \text{ and } L^{-1} = \bar{L}^t. $$

Altogether we have shown

Theorem 9

Every translationally equivariant minimal Lagrangian immersion generated by the potential $D(\lambda ) dz$ has a canonical horizontal lift $F = F(z,\lambda )$ of the form

$$\begin{aligned} F(z,\lambda )= e^{g_1( z,\lambda )} \langle l_1, Q^{-1}e_3\rangle l_1 + e^{g_2(z,\lambda )} \langle l_2, Q^{-1}e_3\rangle l_2 + e^{g_3(z,\lambda )} \langle l_3, Q^{-1}e_3 \rangle l_3, \end{aligned}$$

where

$$\begin{aligned} g_j (z,\lambda ) = z\mu _j(\lambda ) - \beta _1(y, \lambda ) \mu _j(\lambda ) - \beta _2(y,\lambda )(\mu _j(\lambda )^2 - \frac{1}{3}( \mu _1^2 + \mu _2^2 + \mu _3^2) ). \end{aligned}$$

6 Equivariant Cylinders and Tori

6.1 Translationally Equivariant Minimal Lagrangian Cylinders

Based on the description of the frames of (real) translationally equivariant minimal Lagrangian surfaces, in this section we will investigate for which (generally complex) periods such an immersion is periodic.

Definition 3

Let $f:\mathbb {D} \rightarrow \mathbb {C}P^2$ be a (relative to translations by real numbers) translationally equivariant minimal Lagrangian surface. Then f is called an equivariant cylinder, if there exists some complex number $\omega$ such that $f(z + \omega ) = f(z)$ for all $z \in \mathbb {D}$. In this case, $\omega$ is called a period of f. If f satisfies this equation for two (over $\mathbb {R}$) linearly independent periods, then f will be called an equivariant torus.

Clearly, every period $\omega$ of some translationally equivariant minimal Lagrangian immersion leaves the metric invariant. Since the metric is periodic with (smallest) period 2T, it follows that the imaginary part of $\omega$ is an integer multiple of 2T. Hence we will only consider translations of the form

$$z\mapsto z+p+m2Ti, \quad \text { with } p \in \mathbb {R}, m\in \mathbb {Z}.$$

From (39) we derive by inspection that Q is invariant under the above translation by $p+m2Ti$. Therefore, in view of formula (41) for the extended frame we obtain that the monodromy matrix is determined completely by its exponential factor.

From the properties of u we derive the following properties of $\beta _1$ and $\beta _2$:

Lemma 3

• 1. $\beta _j(y+m2T,\lambda )=\beta _j(y,\lambda )+m\beta _j(2T,\lambda )$ for $m\in \mathbb {Z}$ and $j=1,2$.

• 2. $\beta _1(2T,\lambda )-\overline{\beta _1(2T,\lambda )}=4iT$, $\beta _2(2T,\lambda )+\overline{\beta _2(2T,\lambda )}=0$.

• 3. $\mathrm {Re}\beta _1(2T,\varepsilon \lambda )=\mathrm {Re}\beta _1(2T,\lambda )$, $\mathrm {Im}\beta _2(2T,\varepsilon \lambda )=-\mathrm {Im}\beta _2(2T,\lambda )$, where $\varepsilon =e^{\pi i/3}$ is a sixth root of unity as in the definition of the twisted loop group.

As a consequence, the monodromy matrix of the extended frame $\mathbb {F}(z, \lambda )$ for the translation by $\omega = p + m2Ti$ is

$$\begin{aligned} \mathbb {F}(z+p+m2Ti,\lambda )=M(\lambda )\mathbb {F}(z, \lambda ), \end{aligned}$$

where

$$\begin{aligned} M(\lambda )=\exp (pD(\lambda )-m \mathrm {Re}\beta _1(2T,\lambda ) D(\lambda )-im\mathrm {Im}\beta _2(2T,\lambda )L_0(\lambda )). \end{aligned}$$

Moreover, $M(\lambda )\in \varLambda SU(3)_\sigma$ for any $\lambda \in S^1$.

Thus every translation $\omega =p+m2Ti$, $p\in \mathbb {R}$, $m\in \mathbb {Z}$, induces a symmetry of the translationally equivariant minimal Lagrangian surface constructed from $D(\lambda )$.

Let $id_1(\lambda ), id_2(\lambda ), id_3(\lambda )$ denote the eigenvalues of D. Recalling $\beta$ from (36), we see that the monodromy $M(\lambda )$ of the translation $\omega = p + m2Ti$ has the eigenvalues

$$\begin{aligned} i \lbrace pd_j(\lambda ) -m[\mathrm {Re}\beta _1(2T,\lambda ) d_j(\lambda )+\mathrm {Im}\beta _2(2T,\lambda )(-d_j(\lambda )^2+\frac{2\beta }{3})] \rbrace \end{aligned}$$

(42)

for $j=1,2,3$.

As a consequence it is easy to obtain

Theorem 10

For $\lambda = \lambda _0$ the following statements are equivalent.

• 1. The minimal Lagrangian immersion $f(z, \lambda _0)$ is an equivariant minimal Lagrangian cylinder relative to translation by $\omega =p + m2Ti.$

• 2. The monodromy matrix $M(\lambda )$ of the translation by $\omega =p + m2Ti$ satisfies for $\lambda = \lambda _0$ the equation $M(\lambda _0) = I$.

• 3. For the eigenvalues of the monodromy matrix $M(\lambda )$ of the translation by $\omega =p + m2Ti$, the following relation holds for $j = 1,2$ and $\lambda = \lambda _0$ and integers $l_1, l_2$

  $$\begin{aligned} pd_j(\lambda _0)-m[\mathrm {Re}\beta _1(2T,\lambda _0) d_j(\lambda _0)+\mathrm {Im}\beta _2(2T, \lambda _0) (-d_j(\lambda )^2+\frac{2\beta }{3})]=2l_j \pi . \end{aligned}$$

  (43)

• 4. In addition we note: If $d_1\ne d_2$ and $\lambda = \lambda _0,$ the following relations, for appropriate integers $l_1$ and $l_2$, are equivalent with the relations above

  $$\begin{aligned}&m \mathrm {Im}\beta _2(2T, \lambda _0)[d_1(\lambda _0) - d_2(\lambda _0)] \lbrace d_1(\lambda _0)d_2(\lambda _0)+\frac{2\beta }{3}\rbrace =2\pi (l_1 d_2(\lambda _0) -l_2 d_1(\lambda _0)),\nonumber \\&(d_1-d_2)\{p-m \mathrm {Re}\beta _1(2T, \lambda _0) +m \mathrm {Im}\beta _2(2T, \lambda _0)(d_1(\lambda _0)+d_2(\lambda _0))\} =2(l_1-l_2)\pi . \end{aligned}$$

  (44)

There are two particularly simple choices of translations $\omega =p + m2Ti$, namely purely real and purely imaginary translations. Consequently we obtain:

Corollary 2

Retaining the assumptions and the notation of Theorem 10 for the translation $\omega =p + m2Ti$ and the fixed value $\lambda = \lambda _0$, we obtain two natural cases:

• 1. Real translations: If $m = 0$, then $f(z,\lambda _0)$ is an equivariant cylinder if and only if $d_1(\lambda _0)/d_2(\lambda _0)$ is rational.

• 2. Purely imaginary translations: If $p = 0$, then $f(z,\lambda _0)$ is an equivariant cylinder if and only if

  $$\begin{aligned} \mathrm {Re}\beta _1(2T,\lambda _0) d_j(\lambda _0)+\mathrm {Im}\beta _2(2T,\lambda _0)(-d_j(\lambda _0)^2+\frac{2\beta }{3}) = 2 \pi r_j, \end{aligned}$$

  where $r_j$ $(j = 1,2)$ are rational numbers.

Examples for the above two cases will be presented later in Sects. 7.2 and 7.3.

6.2 Translationally Equivariant Minimal Lagrangian Tori

6.2.1 Basic Discussion of Possible Tori

By definition, a minimal Lagrangian torus $\mathbb {T}$ is a minimal Lagrangian surface which admit for some $\lambda = \lambda _0$ two over $\mathbb {R}$ linearly independent periods $\omega _1 = p_1+ im_12 T$ and $\omega _2 = p_2+ i m_2 2 T,$ with real numbers $p_1,p_2$ and integers $m_1,m_2$. Hence $\mathbb {T}$ is of the form $\mathbb {T}= \mathbb {C}/\mathscr {L}$, where $\mathscr {L}$ is a rank 2 lattice. Then $\hat{p} = m_2 \omega _1 - m_1 \omega _2 \in \mathscr {L}$ is a real period of f. Since $\omega _1$ and $\omega _2$ are linearly independent, it follows that $\hat{p}$ is not 0, i.e., $\hat{p}$ is a nonzero real period of f. Therefore, by Corollary 2 we obtain that $r(\lambda _0) = d_1(\lambda _0) / d_2(\lambda _0)$ is a rational number. Thus every translationally equivariant minimal Lagrangian torus admits a real period and a non-real period.

Next we consider the period lattice

$$\mathscr {L}(f) = \lbrace p+m2Ti \in \mathbb {C}; f(z+p+m2Ti) = f(z) \text{ for } \text{ all } z \in \mathbb {C}\rbrace $$

associated with a translationally equivariant minimal Lagrangian surface f. Note that $\mathscr {L}(f)$ is indeed a lattice.

For a general minimal Lagrangian surface the period lattice will be empty. For some such surfaces it will be of the form $\omega \mathbb {Z}$. Our goal in this section is to understand better the case where the period lattice is a lattice of rank 2. Clearly, if $\mathbb {T} = \mathbb {C}/ \mathscr {L}$ is a translationally equivariant minimal Lagrangian torus, then $\mathscr {L} \subset \mathscr {L}(f)$ holds and also $\mathbb {T}(f) = \mathbb {C}/ \mathscr {L}(f)$ is a translationally equivariant minimal Lagrangian torus.

More precisely,

Proposition 5

Assume the translationally equivariant minimal Lagrangian surface f defined on $\mathbb {C}$ descends to some torus $\hat{\mathbb {T}}$, then this torus is induced by some sub-lattice $\hat{\mathscr {L}}$ of $\mathscr {L}$ and there exists a covering $\hat{\pi }: \hat{\mathbb {T}} \rightarrow \mathbb {T}$ with fiber $\mathscr {L}/\hat{\mathscr {L}}$. In particular, if f descends to some torus, it can be injective only if the torus is the one defined by the period lattice. In particular, an embedding of a translationally equivariant minimal Lagrangian torus is only possible, if the torus is defined by the period lattice.

6.2.2 The Period Lattice

In the case under consideration it is fortunately possible to give a fairly precise description of the period lattice.

Theorem 11

The period lattice $\mathscr {L}(f)$ of any translationally equivariant minimal Lagrangian torus f is of the form

$$\begin{aligned} \mathscr {L}(f) = p_f \mathbb {Z} + \omega _f \mathbb {Z}, \end{aligned}$$

where $p_f$ is the smallest (real) positive period and $\omega _f$ the period with smallest positive imaginary part.

Proof

We have seen above that any translationally equivariant minimal Lagrangian torus has a non-zero real period. Let $p_f$ denote the smallest positive real period of f. Assume p is any other positive period. Then $0< p_f < p$. If p is not an integer multiple of $p_f$, then we can substract an integer multiple from p such that $0< p - k p_f < p_f$. This is a contradiction. Let’s consider next all non-real periods of f and let’s choose any such period $\omega _f = q + m_f i 2T$ for which $m_f$ is positive and minimal. Now choose any other period $\omega = a + b i2T$, with $a \in \mathbb {R}$ and b an integer. We can assume that b is positive. If b is not an integer multiple of $m_f$, then one can subtract an integer multiple of $\omega _f$ from $\omega$ such that $\omega - k \omega _f = (a-kq) +(b - km_f)i 2T$ and $0< b-km_f < m_f$. This is a contradiction. Therefore $b = m m_f$ with an integer m.

Moreover $\omega - m \omega _f = a - m q$ is a real period. But we have seen above that all real periods are an integer multiple of $p_f$. Hence $a-m q = np_f$ and $\omega = np_f + m\omega _f$ follows.

Since the two generating periods for the period lattice $\mathscr {L}$ above are determined by some minimality condition, to find all translationally equivariant minimal Lagrangian tori it basically suffices to find a real period and a non-real period. The existence of such periods can be rephrased as follows

Theorem 12

Let f be a translationally equivariant minimal Lagrangian immersion. Then f descends to a torus if and only if

• (1) The eigenvalues $d_1(\lambda _0)$ and $d_2(\lambda _0)$ have a rational quotient.

• (2) Either the eigenvalues $d_1(\lambda _0)$ and $d_2(\lambda _0)$ equal or $\frac{1}{2\pi }\mathrm {Im}\beta _2(2T, \lambda _0)\lbrace d_1(\lambda _0)d_2(\lambda _0)+\frac{2\beta }{3}\rbrace$ is rational.

Proof

We know that (1) is equivalent with the existence of a real period and (2) follows for a non-real period by (44). It thus remains to show that (1) and (2) together imply the existence of a non-real period. First, if $d_1$ and $d_2$ are equal (for a fixed $\lambda = \lambda _0$), then (43) actually is only one equation and one can compute p for $m=1$. Actually we see from Lemma 1 that $D(\lambda )$ having multiple eigenvalues implies that the minimal Lagrangian surface is flat and needs to be a part of the Clifford torus. Assume now $d_1 \ne d_2$. Then we can compute $m \ne 0,$ $l_1$ and $l_2$ from the first equation in (44) and then p from the second equation in (44).

6.2.3 The Case of a Real Cubic Form $\lambda ^{-3}\psi$

We know that in the case of a real cubic form $\lambda ^{-3}\psi$, the canonical lift F is invariant under translations by $\omega = 4Ti$ (see Sect. 7.3).

From Theorem 11 we know that in the case under consideration the period lattice is spanned by a real period and a non-real period with smallest positive imaginary part. This non-real period is thus either 4Ti or of the form $b+m2Ti$ with $m = 2k +1$. Then we can assume that this second period is of the form $b + 2Ti$.

Moreover, with $\omega$ also $2\omega = 2b + 4Ti$ is a period, whence 2b is a period. Since we can assume that either $b=0$ or $0< b < p_f$, we obtain $b= \frac{1}{2} p_f$.

At any rate, the quotient of the eigenvalues $d_1$ and $d_2$ of D is rational.

Proposition 6

Keeping the definitions and the notation introduced for translationally equivariant minimal Lagrangian surfaces we obtain in the case of a real cubic form $\lambda ^{-3}\psi$ the possible period lattices $\mathscr {L}(f) = p_f\mathbb {Z} + 4Ti\mathbb {Z}$ and $\mathscr {L}(f) =p_f \mathbb {Z}+ (\frac{1}{2}p_f + 2Ti)\mathbb {Z}.$ In both cases, the quotient of the eigenvalues $d_1$ and $d_2$ of D is rational. Conversely, if the cubic form $\lambda ^{-3}\psi$ is real and $d_1/d_2$ is rational, then the corresponding translationally equivariant minimal Lagrangian surface descends to some torus which is defined by a lattice of the type given above.

7 Comparison with the Work of Castro-Urbano

In this section, we will show how our approach relates to the one of Castro-Urbano [2]. As before, also in this section we will consider the whole associated family.

To simplify notation, in this section we will (usually) not indicate dependence on variables like z, $\bar{z}$ or $\lambda$.

Let again $f :\mathbb {C}\rightarrow \mathbb {C}P^2$ denote the associated family of translationally equivariant minimal Lagrangian immersions with horizontal conformal lift F and frame $\mathbb {F}$.

Then f is generated by some matrix D and

$$\mathbb {F}(x,y)=e^{xD}\mathbb {F}(y)$$

holds.

The characteristic polynomial of D is given by (35), therefore we immediately obtain

$$\begin{aligned} \partial _x^3 \mathbb {F}+\beta \partial _x \mathbb {F}-2i\mathrm {Re}(\lambda ^{-3}\psi ) \mathbb {F}=0. \end{aligned}$$

(45)

Remark 6

We would like to point out that instead of using the third order equation above, in [2] the authors prove the existence of a sixth order equation due to the real orthogonal frames they used. So from here on our computations are usually somewhat simpler, but follow a very similar idea.

Equation (45) holds, of course, for each column of $\mathbb {F}$ separately. In particular, we know that the immersion f(x, y) is given by

$$f(x,y)=[\mathbb {F}(x,y) e_3]=[e^{xD}\mathbb {F}(y)e_3].$$

Thus the horizontal conformal lift $F(x,y) = \mathbb {F}(x,y) e_3$ satisfies

$$\begin{aligned} F(x,y) = \exp (xD) F(y) \end{aligned}$$

(46)

and therefore also

$$\begin{aligned} \partial _x^3 F +\beta \partial _x F- 2i \mathrm {Re}(\lambda ^{-3}\psi ) F=0. \end{aligned}$$

Recall that for a minimal Lagrangian immersion $f: M\rightarrow \mathbb {C}P^2$ with induced metric $g=2e^udzd\bar{z}$, its horizontal lift $F: U\rightarrow S^5(1)\subset \mathbb {C}^3$ satisfying the Eqs. (2)–(4) with $\psi$ defined by (5) gives an associated family of minimal Lagrangian surfaces with the cubic differential $-i\lambda ^{-3}\psi$. Explicitly, the associated extended frame $F(z,\bar{z}, \lambda )$ satisfies

$$\begin{aligned} F_{zz}&=u_zF_z-e^{-u}i\lambda ^{-3}\psi F_{\bar{z}},\nonumber \\ F_{z\bar{z}}&=-e^u F,\\ F_{\bar{z}\bar{z}}&=-e^{-u}i\lambda ^{3}\bar{\psi }F_{z}+u_{\bar{z}}F_{\bar{z}}.\nonumber \end{aligned}$$

(47)

It is straightforward to rewrite the equations (47) involving derivatives for z and $\bar{z}$ into

$$\begin{aligned}&F_{xx}=-ie^{-u}\mathrm {Re}(\lambda ^{-3}\psi )F_x -\frac{u^{\prime }-2i e^{-u}\mathrm {Im}(\lambda ^{-3}\psi )}{2}F_y -2e^u F,&\end{aligned}$$

(48)

$$\begin{aligned}&F_{xy}=\frac{u^{\prime }+2i e^{-u} \mathrm {Im}(\lambda ^{-3}\psi )}{2} F_x+i e^{-u} \mathrm {Re}(\lambda ^{-3}\psi )F_y,&\end{aligned}$$

(49)

$$\begin{aligned}&F_{yy}=i e^{-u} \mathrm {Re}(\lambda ^{-3}\psi ) F_x+\frac{u^{\prime }- 2 i e^{-u} \mathrm {Im}(\lambda ^{-3}\psi )}{2} F_y-2e^u F.&\end{aligned}$$

(50)

We want to evaluate (46) by writing F as a linear combination of eigenvectors of D.

It follows from (35) that the eigenvalues $\mu _1=id_1$, $\mu _2=id_2$, $\mu _3=id_3$ satisfy

$$\begin{aligned} d_1+d_2+d_3=0, \quad d_1 d_2+d_2 d_3+d_3 d_1=-\beta , \quad d_1 d_2 d_3=-2\mathrm {Re}(\lambda ^{-3}\psi ). \end{aligned}$$

Let $l_1, l_2$ and $l_3$ denote an orthonormal basis of eigenvectors of $D(\lambda )$ for the eigenvalues $i d_1, \, id_2, \, id_3$, respectively. Then there exist scalar functions $p_j(y)$ such that

$$\begin{aligned} F(y)=p_1(y) l_1+p_2(y)l_2+p_3(y)l_3 \end{aligned}$$

holds. As a consequence, for $F(x,y) = \exp (xD) F(y)$ we obtain

$$\begin{aligned} F(x,y)=p_1(y)e^{i d_1 x}l_1+p_2(y)e^{i d_2 x}l_2+p_3(y) e^{i d_3 x}l_3. \end{aligned}$$

(51)

Next we evaluate Eq. (49) and obtain for $j = 1,2,3$ the scalar equations

$$\begin{aligned} (d_j e^u -\mathrm {Re}(\lambda ^{-3}\psi ) )p_j^{\prime }= \frac{u^{\prime } e^u +2i \mathrm {Im}(\lambda ^{-3}\psi )}{2}d_j p_j. \end{aligned}$$

(52)

Remark 7

• 1. The Eq. (47) lead to three real differential equations and via (51) yield three scalar differential equations for the coefficient functions $p_j(y)$. Two of these three differential equations are of first order and of the form $A_j p_j^\prime = B_j p_j$ and the third one is a second order equation with leading coefficient 1. Since the two first order equations describe the same function $p_j$ we obtain for the equivalence of these two equations the identity $A_1 B_2 = A_2 B_1$ which turns out to be

  $$\begin{aligned}{}[d_j e^u-\mathrm {Re}(\lambda ^{-3}\psi )][d_j^2 e^u +\mathrm {Re}(\lambda ^{-3}\psi )d_j-2e^{2u}] =[\frac{1}{4} (u^{\prime })^2e^{2u}+(\mathrm {Im}(\lambda ^{-3}\psi ))^2] d_j. \end{aligned}$$

  (53)

• 2. There are several cases that need to be distinguished:

  • a. The first case is, where the matrix $D(\lambda _0)$ is not invertible. In this case $\lambda _0^{-3}\psi$ is purely imaginary and one eigenvalue vanishes, say $i d_1(\lambda _0) = 0$, and the other two eigenvalues are $id_\pm (\lambda _0) = \pm i \sqrt{\beta }$. This case will be discussed separately. Therefore, in the rest of this remark we will always assume that all eigenvalues are non-zero at all values of $\lambda$ considered.

  • b. Assuming now that no eigenvalue $d_j(\lambda _0)$ vanishes, it can happen that two eigenvalues coalesce. In this case we know from Lemma 1 that the minimal Lagrangian surface is flat, a case which is no longer considered at this point. Therefore, from now on we will assume that all eigenvalues are different and non-zero at all values of $\lambda$ considered.

• 3. There are two more cases to distinguish. Namely the cases where $\lambda _0^{-3}\psi$ is real and non-real and non-purely-imaginary. These two cases will also be treated separately below.

• 4. In view of (53) it turns out to be useful to note that if $d_j(\lambda _0) \ne 0$, and if $d_j (\lambda _0) e^{u(y_0)} -\mathrm {Re}(\lambda _0^{-3}\psi ) = 0$, then $\lambda _0^{-3}\psi$ is real and $u^\prime (y_0) = 0$.

• 5. One could evaluate the remaining two Eqs. (48) and (50) in an analogous manner. However, it turns out that these two equations do not produce any new information, if $d_j(\lambda _0) \ne 0$ and $d_j (\lambda _0)e^u -\mathrm {Re}(\lambda _0^{-3}\psi ) \ne 0$.

7.1 The Case of Non-invertible $D(\lambda )$

In view of (35) it is clear that $\mathrm {Re}(\lambda _0^{-3}\psi )=0$ is equivalent to that $D(\lambda _0)$ is not invertible and to $\lambda _0^{-3}\psi$ being purely imaginary.

Let’s assume now that $d_1(\lambda _0) = 0$. Then, fixing $\lambda = \lambda _0$, the eigenvalues of D are, without loss of generality, $id_1 = 0, id_2 = i \sqrt{\beta }$ and $id_3 = -i\sqrt{\beta }$.

We note that, in full generality, the Eq. (48) translates, in view of (51), to

$$\begin{aligned} ( u^\prime e^u - 2 i \mathrm {Im}(\lambda ^{-3} \psi )) p_j^\prime = 2( d_j^2 e^u + \mathrm {Re}(\lambda ^{-3}\psi )d_j - 2 e^{2u}) p_j. \end{aligned}$$

(54)

Note that here the coefficient of $p_j^{\prime }$ on the left side does not vanish in the case under consideration, where $\mathrm {Re}(\lambda ^{-3}\psi ) =0$). Writing out the three equations of (54) it is easy to observe that the differential equations for $p_2$ and $p_3$ are equal. Therefore, the solutions $p_2$ and $p_3$ of these differential equations only differ by some constant. But then, say $p_3 = \alpha p_2$, we obtain $|\alpha | = 1$, since F has length 1. As a consequence, up to some isometry of $\mathbb {C}P^2$ the surface only takes value in some hyperplane.

7.2 The Case of Non-real $\lambda ^{-3}\psi$

Now let’s assume that $\lambda ^{-3}\psi$ is not real. Then $d_j e^u -\mathrm {Re}(\lambda ^{-3}\psi )\ne 0$. We obtain

$$\begin{aligned} p_j(y)= \rho _j (d_je^u-\mathrm {Re}(\lambda ^{-3}\psi ))^{\frac{1}{2}} e^{i\int _0^y \frac{d_j \mathrm {Im}(\lambda ^{-3}\psi )}{d_j e^u-\mathrm {Re}(\lambda ^{-3}\psi )}ds}, \end{aligned}$$

where $\rho _1, \rho _2, \rho _3$ are independent of z.

To determine the coefficients $\rho _j$, we recall that the lift F is conformal and horizontal, whence we have

$$\begin{aligned}&F\cdot \bar{F}=1,\,\, F_x\cdot \bar{F}=F_y\cdot \bar{F}=F_x\cdot \overline{F_y}=0,\\&F_x\cdot \overline{F_x}=F_y\cdot \overline{F_y}=2e^u. \end{aligned}$$

These equations lead to the following 3 equations:

• 1. $\sum _{j=1}^3 |\rho _j|^2(d_je^u-\mathrm {Re} (\lambda ^{-3}\psi ))=1,$

• 2. $\sum _{j=1}^3 d_j |\rho _j|^2(d_je^u-\mathrm {Re} (\lambda ^{-3}\psi ))=0,$

• 3. $\sum _{j=1}^3 d_j^2 |\rho _j|^2(d_je^u-\mathrm {Re} (\lambda ^{-3}\psi ))=2e^u.$

Since the Vandermonde matrix built from distinct $d_1, d_2,d_3$ is invertible, if the surface is not flat (see Lemma 1), this system of equations can be solved for $|\rho _j|^2 (d_j e^u - \mathrm {Re}(\lambda ^{-3}\psi ))$.

Since $d_je^u-\mathrm {Re}(\lambda ^{-3}\psi )\ne 0$, we obtain

$$|\rho _j|^2=\frac{1}{d_j^3-\mathrm {Re}(\lambda ^{-3}\psi )}, \quad j=1,2,3.$$

Choose $\rho _j=\frac{1}{(d_j^3-\mathrm {Re}(\lambda ^{-3}\psi ))^{\frac{1}{2}}}$, then we have

$$\begin{aligned} F(x,y, \lambda )= h_1(y)e^{id_1 x +iG_1(y)} l_1 + h_2(y)e^{id_2 x +iG_2(y)} l_2 + h_3(y)e^{id_3 x +iG_3(y)}l_3, \end{aligned}$$

(55)

where

$$\begin{aligned} h_j(y)=\left( \frac{d_je^u-\mathrm {Re}(\lambda ^{-3}\psi )}{d_j^3-\mathrm {Re}(\lambda ^{-3}\psi )}\right) ^{\frac{1}{2}}, \quad \quad G_j(y)=\int _0^y \frac{d_j \mathrm {Im}(\lambda ^{-3}\psi )}{d_j e^u-\mathrm {Re}(\lambda ^{-3}\psi )}ds. \end{aligned}$$

(56)

Note that also the eigenvalues $d_j$ depend on $\lambda$. In terms of the orthonormal basis of eigenvectors of $D(\lambda )$ chosen above, we can assume, by the discussion just above, that the phase factor of the $l_j$ is chosen such that $h_j$ is positive and real. We will therefore continue to denote this basis by the letters $l_1,l_2$ and $l_3$.

Next we want to consider $F(x+p, y+ m2T, \lambda )$. At one hand we obtain

$$\begin{aligned} F(x+p,y+m2T,\lambda )=\sum _{j=1}^3 h_j(y + m2T,\lambda )e^{id_j(\lambda ) (x+p) +iG_j(y + m2T,\lambda ))}l_j(\lambda ), \end{aligned}$$

(57)

and on the other hand we obtain

$$\begin{aligned} F(x+p,y+m2T,\lambda ) = M(\lambda ) F(x,y,\lambda ). \end{aligned}$$

Using the simple equations $h_j(y + m2T,\lambda ) = h_j(y,\lambda )$, since $e^u$ is $2T-$periodic, and the obvious identity $G_j(y + m2T,\lambda ) = G_j(y,\lambda )+m G_j(2T,\lambda )$, we see that the coefficient for $l_j(\lambda )$ in the Eq. (57) actually is of the form

$$\begin{aligned} e^{id_j(\lambda )p + i mG_j(2T,\lambda ))} \cdot h_j(y,\lambda ) e^{i(d_j(\lambda )x + G_j(y,\lambda ))}. \end{aligned}$$

(58)

Since the $l_j(\lambda )$ are eigenvectors of $M(\lambda )$, the left factors of these expressions in (58) are exactly the eigenvalues of $M(\lambda )$. Hence comparing with (42), we obtain

Theorem 13

Retaining the notation used so far we obtain for every translation $p + m2Ti$ and $j = 1,2,3$ the equation

$$\begin{aligned} p d_j(\lambda )&+ mG_j(2T,\lambda ) \equiv pd_j(\lambda )-m[\mathrm {Re}\beta _1(2T, \lambda ) d_j(\lambda )\\&\quad \quad +{\mathrm {Im}}\beta _2(2T,\lambda )(-d_j(\lambda )^2+\frac{2\beta }{3})]\mod 2 \pi \mathbb {Z}. \end{aligned}$$

Actually we can directly show that $G_j(2T,\lambda )+\mathrm {Re}\beta _1(2T, \lambda ) d_j(\lambda ) +\mathrm {Im}\beta _2(2T,\lambda )[-d_j(\lambda )^2+\frac{2\beta }{3}]=0$ for each j. Note that by summing up the three equations we obtain

$$\begin{aligned} G_1(2T,\lambda ) + G_2(2T, \lambda ) + G_3(2T,\lambda ) = 0 \end{aligned}$$

(59)

for all $\lambda$.

From the argument in Sect. 6.2, if f descends to a torus, then $d_1/d_2$ is rational. Moreover, if there exists some $p\in \mathbb {R}$ and $m\in \mathbb {Z}$ such that $f(x+p, y+m2T)=f(x,y)$, then

$$p d_j(\lambda )+m G_j(2T,\lambda )\in 2\pi \mathbb {Z}$$

for $j=1,2,3$. From $d_1+d_2+d_3=0$ and (59), we can easily obtain that

$$\frac{1}{2\pi } (\frac{d_2}{d_1} G_1(2T,\lambda )-G_2(2T,\lambda ))$$

is rational. The converse obviously also holds. So we have

Theorem 14

([2]) If the cubic form $\lambda ^{-3}\psi$ of a translationally equivariant minimal Lagrangian surface f is not real, then the canonical horizontal lift F of f has the form of (55). In this case, f descends to a torus if and only if both $d_1/d_2$ and $\frac{1}{2\pi } (\frac{d_2}{d_1} G_1(2T)-G_2(2T))$ are rational.

7.3 The Case of Real $\lambda ^{-3}\psi$

Retaining the notation used so far, we assume in this section that $\lambda ^{-3}\psi$ is real. In this case, $d_j e^u -\mathrm {Re}(\lambda ^{-3}\psi )$ can vanish at some points.

Recall that $a_1, a_2, -a_3$ are the roots of $w^3-\frac{\beta }{2}w^2+\frac{|\psi |^2}{2}=0$ in Sect. 5.3. We see that the roots of the characteristic polynomial (35) of $D(\lambda )$ can be given by

$$\begin{aligned} \mu _1=id_1=\frac{i \lambda ^{-3}\psi }{a_1}, \quad \mu _2=id_2=\frac{i \lambda ^{-3}\psi }{a_2} , \quad \mu _3=id_3=-\frac{i \lambda ^{-3}\psi }{a_3}. \end{aligned}$$

Recall the following properties of the Jacobi elliptic functions

$$\begin{aligned}&\mathrm {sn}^2 z+\mathrm {cn}^2 z=1, \, k^2 \mathrm {sn}^2z+\mathrm {dn}^2 z=1,\\&\frac{d}{dz} \mathrm {sn}z=\mathrm {cn}z \mathrm {dn}z, \, \frac{d}{dz} \mathrm {cn}z=-\mathrm {sn}z \mathrm {dn}z,\, \frac{d}{dz} \mathrm {dn}z=-k^2 \mathrm {sn}z \mathrm {cn}z. \end{aligned}$$

Taking into account formulas (31)–(33), we can rewrite (52) as

$$\begin{aligned} p_1^{\prime }(y)\mathrm {sn}(ry, k)= & {} p_1(y) \frac{d}{dy}\mathrm {sn}(ry, k),\\ p_2^{\prime }(y)\mathrm {cn}(ry, k)= & {} p_2(y) \frac{d}{dy}\mathrm {cn}(ry, k),\\ p_3^{\prime }(y)\mathrm {dn}(ry, k)= & {} p_3(y) \frac{d}{dy}\mathrm {dn}(ry, k). \end{aligned}$$

Integration gives

$$p_1(y)=c_1 \mathrm {sn}(ry, k), \, p_2(y)=c_2 \mathrm {cn}(ry, k), \, p_3(y)=c_3 \mathrm {dn}(ry, k),$$

where $c_1$, $c_2$ and $c_3$ are constant complex numbers.

Thus by using an analogous argument involving the Vandermonde matrix given in Sect. 7.2, we obtain the constants $c_1, c_2, c_3$ from (56):

$$\begin{aligned} c_1=a_1 \sqrt{\frac{a_1-a_2}{a_1^3-|\psi |^2}},\quad c_2= a_2\sqrt{\frac{a_1-a_2}{|\psi |^2-a_2^3}},\quad c_3= a_3\sqrt{\frac{a_1+a_3}{|\psi |^2+a_3^3}}. \end{aligned}$$

This yields the canonical horizontal lift of the associated minimal Lagrangian surface

$$\begin{aligned} F(x,y,\lambda )=c_1 \mathrm {sn}(ry, k) e^{id_1x} l_1 + c_2 \mathrm {cn}(ry, k) e^{id_2x}l_2 + c_3 \mathrm {dn}(ry, k) e^{id_3x} l_3, \end{aligned}$$

(60)

where $l_1, l_2, l_3$ is an orthonormal basis of eigenvectors of D for the eigenvalues $\mu _1, \, \mu _2, \, \mu _3$, respectively.

Based on our discussion in Sect. 6, we immediately obtain

Theorem 15

If the cubic form $\lambda ^{-3}\psi$ of a translationally equivariant minimal Lagrangian surface f is real, then the canonical horizontal lift F of f has the form of (60) and satisfies $F(x, y+4T)=F(x,y)$. In particular, f is defined on the cylinder $\mathscr {C} = \mathbb {C}/ 4Ti \mathbb {Z}$.

If there also exists some $\tau \in \mathbb {R}$ such that $f(x+\tau , y)=f(x,y)$, then $e^{id_1 \tau }=e^{id_2 \tau }=e^{id_3\tau }$, which implies $d_1/d_2$ is rational. In this case, f descends to the torus $\mathbb {T} = \mathbb {C}/\mathscr {L}$, where $\mathscr {L}$ denotes the lattice $\mathscr {L} = 4Ti \mathbb {Z} + \tau \mathbb {Z}$.

Conversely, if $d_1/d_2$ is rational, then there exists some $\tau \in \mathbb {R}$ such that $f(x+\tau , y)=f(x,y)$ holds.