数理解析研究所講究録
第 1605 巻 2008 年 120-132

120Real forms of complex surfaces of constant mean
curvature
Shimpei Kobayashi
School of Information Environment
Tokyo Denki University Japan
shimpei@sie.dendai.ac.jp

1

Introduction

This is a summary of the paper [10]. The goal is to give a imified theory for integrable surfaces using real forms of the complex extended framings of complex CMCimmersions and the generalized WeierstraB type representation for complex CMCimmersions.
It is well known that a surface inhas nonzero constant mean curvature (CMC for
short) if and only if there exists a moving frame with spectral parameter, an element
in $SU(2)$ loop group, which satisfies the certain condition (see [5]). Such moving
frame is called the extended framing of a CMC-immersion.
$\mathbb{R}^{3}$

The extended haming of a CMC-immersion in
has a natural complexification,
which is called the complex extended framing ([3]). Moreover in [4], we considereda holomorphic immersion inassociated with the complex extended framing. It
turned out that the holomorphic immersion had nonzero complex constant mean
curvature, which was called a complex CMC-immersion. Then a CMC-immersion in
can be obtained from a real form of the complex extended framing of a complex
CM C-immersion.
$\mathbb{R}^{3}$$\mathbb{C}^{3}$$\mathbb{R}^{3}$

It is known that a CMC-immersion inhas the parallel immersion with constant$GauB$ curvature (CGC for short) $K>0$ in. Similar to the real case, a holomorphic
immersion with complex constant $GaM$ curvature(CGC for short) will beobtained as the parallel immersion of a complex CMC-immersion. Thus a CGCimmersion with $K>0$ in
also can be obtained from a real form of the complex
extended framing. Then it is natural to ask whether other classes of real surfaces
exist from real forms of the complex extended framing of a complex CMC-immersion
or a complex CGC-immersion.
$\mathbb{R}^{3}$$\mathbb{R}^{3}$$K\in \mathbb{C}^{*}$$\mathbb{R}^{3}$

In this surnmary, we show that there are seven classes of surfaces as real forms ofthe complex extended framing, which are called integrable surfaces. These are CGC-

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immersions with $K>0$ $(or K<0)$ in
and their parallel CMC-immersions,spacelike (or timelike) CGC-immersions with $K>0$ $(or K<0)$ in
and their
parallel CMC-immersions, and CMC-immersions with mean curvature $H<1$ in(see Theorem 3. 1 and Corollary 3.2). Some of these classes of surfaces were consideredfrom harmonic maps and integrable systems points of views (see [9], [6], [12], [8] and[1] .
$\mathbb{R}^{3}$$\mathbb{R}^{2,1}$$H^{3}$$)$The generalized Weierstr type representation for complex CMC-immersions is aprocedure to construct complex CMC-immersions in(see Section 4.1 for moredetaik): 1. Define pairs of holomorphic potentials, which are pairs of holomorphic
l-formswithand. Here is the complex
parameter, the so-called “spectral parameter”,
andare diagonal (resp. offdiagonal) holomorphic l-forms depending only on one complex variable if is even(resp. is odd). 2. Solve the pair of ODE’s $d(C, L)=(C, L)\check{\eta}$ with some initial condition $(C(z_{*}), L(w_{*}))$ , and perform the generalized Iwasawa decomposition (Theorem
A. 1) for $(C, L)$ , giving $(C, L)=(F, F)(V_{+}, V_{-})$ . It is known thatis the complex
extended framing of some complex CMC-immersion (Theorem 4.1), where is some-independent diagonal matrix. 3. Form a complex CMC-immersion by the Sym
formula via the complex extended framing(Theorem 2.4).
$\theta$$\mathbb{C}^{3}$$\check{\eta}=(\eta, \tau)$$\tau=\sum_{-\infty}^{j\leq 1}\tau_{j}\lambda^{j}$$\eta=\sum_{j\geq-1}^{\infty}\eta_{j}\lambda^{j}$$\eta_{j}$$\lambda$$\tau_{j}$

$j$

$j$

$F\cdot l$$l$$\lambda$$\Psi$$F\cdot l$

Since each class of integrable surfaces is defined by the real form of a complex extended
haming, there exists a unique semi-linear involution corresponding to each class of
integrable surfaces. Then these semi-linear involutions naturally define the pairs ofsemi-linear involutions on pairs of holomorphic potentials. It follows thatthe generalized Weierstra6 type representation for each class of integrable surfaces
can be formulated by the above construction via a pair of holomorphic potentials
which is invariant under a pair of semi-linear involutions (Theorem 4.2). In this way
we will give a unified theory for all integrable surfaces.
$\rho$$\check{\eta}=(\eta, \tau)$

2

Preliminaries

In this preliminary section, we give a brief review of the basic results for holomorphic
null immersions, complex CMC-immersions and complex CGC-immersions.

Throughout this paper,$\mathbb{C}^{3}$

is identified with st 2,$($$\mathbb{C})$as follows:$(a, b, c)^{t} \in \mathbb{C}^{3}rightarrow-\frac{ia}{2}\sigma_{1}-\frac{ib}{2}\sigma_{2}-\frac{ic}{2}\sigma_{3}\in\epsilon 1(2, \mathbb{C})$,(2.0.1)

where $\sigma_{j}(j=1,2,3)$ are Pauli matrices as follows:$\sigma_{1}=(_{1}^{0}$

$01$

,

$\sigma_{2}=(_{i}^{0}$

$-i0$

and

$\sigma_{3}=(_{0}^{1}$$-10$(2.0.2)

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2.1

Holomorphic null immersions in$\mathbb{C}^{3}$

In this subsection, we show the basic results for holomorphic immersions in. Wegive natural definitions of complex mean curvature (Definition 1) and complex $Ga\theta$
curvature (Definition 2) for a holomorphic immersion analogous to the mean curvatureand the $GaM$ curvature of a surface in . We refer to [4] for more details.
$\mathbb{C}^{3}$$\mathbb{R}^{3}$

be a simply coimected 2-dimensional Stein manifold, and let :be a holomorphic immmersion, i.e. the complex rank ofis two. We consider thefollowing bilinear form on:
Let$\mathcal{M}$$\Psi$$\mathcal{M}arrow \mathfrak{S}[(2, \mathbb{C})$$d\Psi$$5[(2, \mathbb{C})\cong \mathbb{C}^{3}$$\langle a,$$b\rangle=-2Tk$ $ab$,

(2.1.1)

where. We note that the bilinear form (2.1.1) is a -bilinear form on
by the identification (2.0.1). Then it is known that, for a neighborhood, the bilinear form (2.1.1) induces a holomorphic Riemannianaroumd each point, i.e. a holomorphic covariant symmetric 2-tensor (see [11] and [4]).
metric on
From [4], it is also known that there exist special coordinates
suchthat a holomorphic Riemamuian metric can be written as follows:$a,$$b\in \mathfrak{S}[(2, \mathbb{C})$$\mathbb{C}$$\mathbb{C}^{3}$$\tilde{\mathcal{M}}_{p}\subset \mathcal{M}$$p\in \mathcal{M}$$\mathcal{M}_{p}$$g$$(z, w)\in \mathfrak{D}^{2}\subset \mathbb{C}^{2}$$g$$g=e^{u(z,w)}dzdw$ ,(2.1.2)where $u(z, w)$ :is some holomorphic function. The special coordinates defined above are called null coordinates.
now on, we always assume a holomorphic
immersion :has null coordinates. A holomorphic immersion withnull coordinates is ako called the holomo rphic null immersion.
Rom [4], we quote the following theorem:$\mathfrak{D}^{2}arrow \mathbb{C}$$\mathbb{R}om$$\Psi$$\mathcal{M}arrow \mathfrak{S}1(2.\mathbb{C})$

Theorem 2.1 ([4]). Let
Then there exists a$\Psi$:$SL(2, \mathbb{C})$be a holomo rphic null immersion.
matrix $F$ such that the following equations hold:$\mathcal{M}arrow \mathbb{C}^{3}(\cong \mathfrak{S}[(2, \mathbb{C}))$$F_{z}=FU$,(2.1.3)$F_{w}=FV$,

where(2.1.4)$\{\begin{array}{l}U=(Q^{\frac{1}{e4}u_{z}}-u/2 -\frac{1}{-2}He^{u/2}\frac{1}{4}u_{\tilde{4}} ,V=(_{\frac{1}{2}He^{u/2}}-\frac{1}{4}u_{w} -Re^{-u/2}\frac{1}{4}u_{w} ,\end{array}$with$Q:=\langle\Psi_{zz},$$N\rangle,$$R:=\langle\Psi_{ww},$ $N\rangle$

and $H:=2e^{-u}\langle\Psi_{zw},$

We callthe moving frame offor the equations in (2.1.3) is$F:\mathcal{M}arrow SL(2, \mathbb{C})$$\Psi$$N\rangle$.

. Then the compatibility condition$U_{w}-V_{z}+[V)U]=0$ .

(2.1.5)

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A direct computation shows that the equation (2.1.5) can be rephrased as follows:(2.1.6)$\{\begin{array}{l}u_{zw}-2RQe^{-u}+\frac{1}{2}H^{2}e^{u}=0,Q_{w}-\frac{1}{2}H_{z}e^{u}=0,R_{z}-\frac{1}{2}H_{w}e^{u}=0.\end{array}$

The first equation in (2.1.6) will be called the complex $Gau\beta$ equation, and the second
and third equations in (2.1.6) will be $can_{ed}$ the complex Codazzi equations.
We now define a vector$N\in \mathfrak{S}\{(2, \mathbb{C})$as follows:$N:=2ie^{-u}[\Psi_{w}, \Psi_{z}]$.

(2.1.7)$N\rangle=0$ and the
It is easy to verify that$N)=1$ . Thus $N$ is atransversal vector to. Therefore it is natural to call $N$ the $\omega mplexGau\beta$ map of$\langle\Psi_{z},$$N\rangle=\langle\Psi_{w},$$\langle N,$$d\Psi$$\Psi$.

Using the functions $u,$ $Q,$ $R$ and $H$ defined in (2.1.2) and (2.1.4) respectively, thesymmetric quadratic form $II:=-\langle d\Psi,$
can be represented as follows:$dN\rangle$$II$ $:=-\langle d\Psi,$ $dN\rangle=Qdz^{2}+e^{u}Hdzdw+Rdw^{2}$(2.1.8)The symmetric quadratic form II is called the second fundamental form for a holomorphic null immersion . Then the complex mean curvature and the complex $GauB$
curvature for a holomorphic null immersion are defined as follows.
$\Psi$$\Psi$

Definition 1. Let$H=2e^{-u}\langle\Psi_{zw},$$N\rangle$

be a holomorphic null immersion. Then the function
will be called the complex mean curvature of .

$\Psi:\mathcal{M}arrow \mathbb{C}^{3}$$\Psi$

Definition 2. Let (resp. II) be the coefficient matrix of the holomorphic metric(resp. the second fundamental form II). Then the function
will becalled the complex $GauB$ curvature of .
$\tilde{I}$$g$$K=\det(\tilde{I}^{-1}\cdot\tilde{II})$$\Psi$

2.2

Complex CMC and CGC immersions in$\mathbb{C}^{3}$

In this subsection, we give characterizations of complex constant mean curvature immersions via loop groups (see Appendix A for the definitions of loop groups). There isa useful formula representing complex CMC-immersions, which is a generalization of
the Sym formula for CM C-immersions in(see ako [3]). There is also a formula forcomplex CGC-inmersions given by the parallel holomorphic immersions of complex
CM C-immersions with.
$\mathbb{R}^{3}$$H\in \mathbb{C}^{*}$The notions of a complex CMC-immersion and a CGC-immersion are defined analogous to the notions of a CMC-immersion and a CGC-immersion in(see also$\mathbb{R}^{3}$

[4] .
$)$

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Definition 3. Let :be a holomorphic null immersion, and let $H$ (resp.
$K)$ be its complex mean curvature (resp. $Gau\beta$ curvature). Thenis called a complex
constant mean curvature (CMC for short) immersion (resp. a complex constant $Gau\beta$
curvature (CGC for short) immersion) if $H$ (resp. $K$ ) is a complex constant.
$\Psi$$\mathcal{M}arrow \mathbb{C}^{3}$$\Psi$

Remark 2.2. Since we are interested in complexifications of CMC (resp. CGC)
resp. $Gau\beta$ curvaturesurfaces with nonzero mean curvaturefrom
now on, we always assume that the complex mean curvature $H$ (resp. the complex$Gau\beta$ curvature $K$ ) is a nonzero constant.
$H\in \mathbb{R}^{*}$$($$K\in \mathbb{R}^{*})_{f}$From [4], we quote the following characterizations of a complex CMC-immersion:

Lemma 2.3. Let
be a connected 2-dimensional Stein manifold, and let :be a holomorphic null immersion. Further, let $Q,$ $R,$ $H$ and $N$ be thecomplex functions defined in (2.1.4) and themap defined in (2.1.7), respectively.
Then the following statements are equivalent:$\mathcal{M}$$\Psi$$\mathcal{M}arrow$$\mathbb{C}^{3}(\cong \mathfrak{S}1(2, \mathbb{C}))$$Gau\beta$

1. $H$ is a nonzero constant;

2.

$Q$

depends only on$z$

and $R$ depends only on $w$ ;S. $N_{zw}=\rho N$ , for some holomorphic function

4. There exists$\tilde{F}(z, w, \lambda)\in\Lambda SL(2, \mathbb{C})_{\sigma}$$\rho:\mathcal{M}arrow \mathbb{C}$.

such that$\tilde{F}(z, w, \lambda)^{-1}d\tilde{F}(z, w, \lambda)=\tilde{U}dz+\tilde{V}dw$

,

where

$\{\begin{array}{l}\tilde{U}= (1^{\frac{1}{4}u_{z}} -\frac{1}{2}\lambda^{-1}He^{u/2}-\frac{1}{4}u_{z}),\tilde{V}= (_{\frac{1}{2}\lambda He^{u/2}}-\frac{1}{4}u_{w} -\lambda Re^{-u/2}\frac{1}{4}u_{w}),\end{array}$and $\tilde{F}(z, w, \lambda=1)=F(z, w)$ is the moving frame of$\Psi$in (2.1.3).

The
defined in (4) of Lemma 2.3 is called the complex extended framing ofa complex CMC-immersion . From now on, for simplicity, the symbol $F(z, w, \lambda)$
resp. $U(z,$ $w,$
or $V(z,$ $w,$
is used instead of(resp.
or$\tilde{F}(z, w, \lambda)$$\Psi$$($$\lambda)$$\tilde{V}(z, w, \lambda))$$\lambda))$$\tilde{F}(z, w, \lambda)$$\tilde{U}(z\rangle w, \lambda)$.

There is an immersion formula for a complex CMC-immersion using the complex extended franing $F(z, w, \lambda)$ for a complex CMC-immersion
the so-called ‘Sym formula” (see [4]). We show a similar immersion formula for a complex CGC-immersion
using the same complex extended framing $F(z, w, \lambda)$ of a complex CMC-immersion$\Psi_{\rangle}$$\Psi$.

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Theorem 2.4. Let $F(z, w, \lambda)$ be the $\omega mplex$ extended framing of some complex CMCimmersion defined as in Lemma 2.3, and let $H$ be its nonzero complex constant mean
curvature. We set$\{\begin{array}{l}\Psi= -\frac{1}{2H}(i\lambda\partial_{\lambda}F(z, w, \lambda)\cdot F(z, w, \lambda)^{-1}+\frac{i}{2}F(z, w, \lambda)\sigma_{3}F(z, w, \lambda)^{-1}),\Phi = -\frac{1}{2H}(i\lambda\partial_{\lambda}F(z, w, \lambda)\cdot F(z, w, \lambda)^{-1}),\end{array}$(2.2.1)
where
has been defined in (2.0.2). Then
(resp. ) is, for every
plex $\omega nstant$ mean curvature immersion (resp. complex $\omega nstantGau\beta iancun$) $ature$
immersion, possibly degenerate) inwith complex mean curvature(resp.
complex $Gau\beta$ curvatureand the $Gau\beta$ map of (resp. ) can be
described by.
$\Psi$$\sigma_{3}$$\Phi$$\lambda\in \mathbb{C}^{*},$$\mathbb{C}^{3}$$a$$\omega m-$$H\in \mathbb{C}^{*}$$K=4H^{2}\in \mathbb{C}^{*})_{f}$$\Psi$$\Phi$$\frac{i}{2}F(z, w, \lambda)\sigma_{3}F(z, w, \lambda)^{-1}$3Real forms of complex CGC-immersions

In this section, we show that “integrable surfaces” obtained from the real forms ofthe twisted
loop algebra.
$\mathfrak{S}1(2, \mathbb{C})$$\Lambda_{\mathfrak{S}[(2,\mathbb{C})_{\sigma}}$

Integrable surfaces as real forms of complex CGC-immersions

3.1

Let
be the complex extended framing of some complex CGCimmersion . And let $\alpha(z, w, \lambda)=F(z,w, \lambda)^{-1}dF(z, w, \lambda)$ be the Maurer-Cartan
form of $F(z, w, \lambda)$ . From the forms ofand $V$ defined as in Lemma 2.3, we set$\alpha_{i}(i\in\{-1,0,1\})$ as follows:$F(z, w, \lambda)\in\Lambda SL(2, \mathbb{C})_{\sigma}$$\Phi$$U$$\alpha(z, w, \lambda)=F^{-1}dF=Udz+Vdw=\lambda^{-1}\alpha_{-1}+\alpha_{0}+\lambda\alpha_{1}$,(3.1.1)where(3.1.2)$\{\begin{array}{l}\alpha_{-1}= (_{Qe^{-u/2}dz}0 -\frac{1}{2}He^{u/2}dz0’),\alpha_{0}= (^{\frac{1}{4}u_{z}dz\frac{1}{4}u_{w}dw}0 -\frac{1}{4}u_{z}dz+\frac{1}{4}u_{w}dw0),\alpha_{1}=[Matrix].\end{array}$

We denote the space ofvalued l-forms bydefined in (3.1.1) is an element in$\Lambda_{\mathfrak{S}\downarrow(2,\mathbb{C})_{\sigma}}$$\alpha(z, w, \lambda)$$\Omega$$($&1(2, ) . It is clear that$\Omega(\Lambda \mathfrak{s}\mathfrak{l}(2, \mathbb{C})_{\sigma})$$\mathbb{C}$$)$. Then it is also clear that126

the following automorphisms define involutions on$\Omega(\Lambda \mathfrak{S}1(2, \mathbb{C})_{\sigma})$:,$\tilde{c}_{2}:g(\lambda)\mapsto\overline{g(-1/\overline{\lambda})}$$\{\begin{array}{l}\tilde{c}_{1}:g(\lambda)\mapsto-\overline{g(-1/\overline{\lambda})}^{t},\tilde{c}_{3}:g(\lambda)\mapsto-\frac{}{g(1/\overline{\lambda})})\end{array}$$\tilde{c}_{4}:g(\lambda)\mapsto-Ad(^{1/\sqrt{i}0}0\sqrt{}\overline{i})\overline{g(i/\overline{\lambda})}$Then the real form of$\Omega$(Ast(2,$\mathbb{C})_{\sigma}^{(c,j)}$(3.1.3)$\{\tilde{\epsilon}_{2}..g(\lambda)\tilde{\mathfrak{s}}1\cdot...g(\lambda)\tilde{\epsilon}_{3}g(\lambda)\mapsto)\mapsto^{\frac{-g(-\overline{\lambda}}{g_{\frac{(-\overline{\lambda})}{g(\overline{\lambda})}}-}}\mapsto\neg$,

are defined as follows:$\Omega(\Lambda_{\mathfrak{S}}1(2, \mathbb{C})_{\sigma}^{(c,j)})$) $=\{g(\lambda)\in\Omega(Ael(2, \mathbb{C})_{\sigma})|\tilde{c}_{j}\circ g(\lambda)=g(\lambda)\}$ ,(3.1.4)$\Omega(\Lambda_{\mathfrak{S}}1(2, \mathbb{C})_{\sigma}^{(\mathfrak{s},j)})=\{g(\lambda)\in\Omega(\Lambda_{\mathfrak{S}}1(2, \mathbb{C})_{\sigma})|\tilde{\mathfrak{S}}_{j}\circ g(\lambda)=g(\lambda)\}$From now on, for simplicity, we use the s.smbols andtively. We now consider the foUowing conditions on$c_{j}$$\mathfrak{S}_{j}$instead of:$\tilde{c}_{j}$and$\tilde{\mathfrak{S}}_{j}$respec-$\alpha(z, w, \lambda)$$\bullet$Almost Compact cases $(C,j):\alpha(z, w, \lambda)$ is an element in one of the real
forms
for $j\in\{1,2,3,4\}$ .
$\Omega(\Lambda \mathfrak{s}\mathfrak{l}(2\rangle \mathbb{C})_{\sigma}^{(c,j)})$$\bullet$Almost Split cases $(S,j):\alpha(z, w, \lambda)$ is an element in one of the real forms
for $j\in\{1,2,3\}$ .
$\Omega(\Lambda_{\mathfrak{S}[(2,\mathbb{C})_{\sigma}^{(\epsilon,j)})}$

We now set the foUowing formulas
for $j\in\{1,2,3,4\}$ (resp.
analogous to the second formula in (2.2.1):$\Phi^{(c,j)}$$\Phi^{(c,j)}=-\frac{1}{2|H|}(i\lambda\partial_{\lambda}F^{(c,j)}(z,\overline{z}, \lambda)\cdot F^{(c,j)}(z,\overline{z}, \lambda)^{-1})\lambda\in S^{1}$

for $j\in\{1,2,3\}$ )for $j\in\{1,2,3\}$ ,$\Phi^{(c,4)}=\frac{1}{2}(F^{(c,4)}(z,\overline{z}, \lambda)(e^{q/2}00e^{-q/2})(F^{(c,4)}(z,\overline{z}, \lambda))^{*})|_{\lambda\in S^{f}}$$\Phi^{(\epsilon,j)}=-\frac{1}{2|H|}(\lambda\partial_{\lambda}F^{(\mathfrak{g},j)}(x, y, \lambda)\cdot F^{(t,j)}(x, y, \lambda)^{-1})|_{\lambda\in R^{t}}$$\Phi^{(\epsilon,j)}$,(3.1.5)(3.1.6)

for $j\in\{1,2,3\}$ ,(3.1.7)

where $\lambda=\exp(it)\in S^{1}$ or $\lambda=\exp(q/2+it)\in S^{r}$ for (3.1.5) or (3.1.6) (resp.
$\lambda=\pm\exp(t)\in \mathbb{R}^{*}$ for (3.1.7), and wherewithdenotesfor. Then, for each(resp.
or), the formula(resp.
) defines a map into one of the foUowing spaces:$)$$t,$ $q\in \mathbb{R}$$\lambda\in S^{1}$$X\in A’I_{2x2}(\mathbb{C})$$\lambda\in S^{r}$$X^{*}=\overline{X}^{t}$$*$$\Phi^{(c,j)}$$\lambda\in \mathbb{R}^{*}$$\Phi^{(,j)}$$\{\begin{array}{ll}su (1\rangle 1)\cong \mathbb{R}^{1,2} for the (C, 1) and (S, 1) cases,\epsilon 1_{\tau}(2, \mathbb{R})\cong \mathbb{R}^{1,2} for the (C, 2) and (S, 2) cases,\mathfrak{S}u(2)\cong \mathbb{R}^{3} for the (C, 3) and (S, 3) cases,SL(2, \mathbb{C})/SU(2)\cong H^{3} for the (C, 4) case,\end{array}$

where
to

$\epsilon 1_{\tau}(2, \mathbb{R})=\{g\in sl(2, \mathbb{C})|g=(_{c-a}^{ab}), a\in \mathbb{R}, b, c\in i\mathbb{R}\}$$\zeta[(2, \mathbb{R})$. Here$\mathbb{R}^{1,2}$

and$\mathbb{R}^{3}$can be identified with, which is isomorphic$\mathfrak{S}u(1,1),$$\mathfrak{S}1_{*}(2, \mathbb{R})$and$\mathfrak{S}\Downarrow(2)$127

analogous to the identification (2.0.1). Minkowski space
Herm(2) $:=\{X\in M_{2x2}(\mathbb{C})|\overline{X}^{t}=X\}$ via the map$\mathbb{R}^{3,1}$$(x_{1}, x_{2}, x_{3}, x_{0}) \mapsto\frac{1}{2}(_{x_{1}-ix_{2}}^{x_{0}+x_{3}}x_{1}+ix_{2}x_{0}-x_{3}$can be identified with,thencan be identified with Herm(2) with the determinant 1/4. Thenthe inner product forresp.
or
can be$b\rangle=-2$(ab)defined by
Tr
resp.
for
or
su(2) .
The inner product for Herm(2)can be defined byforHerm(2), whereis defined in (2.0.2). From now on, we always assume thatthe spectral parameter is inorfor the almost compact cases and is infor the almost split cases, respectively. Then we have the following theorem:$H^{3}\subset \mathbb{R}^{3,1}$$\mathfrak{s}u(1,1)\cong \mathbb{R}^{1,2}$$($$\mathfrak{s}1_{*}(2,$$a,$ $b\in \mathfrak{S}u(1,1)$$\langle a,$$\mathbb{R})\cong \mathbb{R}^{1,2}$

$($

$a,$

$\cong \mathbb{R}^{3,1}$$a,$ $b\in$$\mathfrak{S}u(2)\cong \mathbb{R}^{3})$$b\in \mathfrak{S}1_{*}(2\rangle \mathbb{R})$$\langle a,$$a,$ $b\in$$)$$b\rangle=-2Tr(a\sigma_{2}b^{t}\sigma_{2})$$\sigma_{2}$$\lambda$$S^{1}$$S^{r}$$\lambda$$\mathbb{R}^{*}$

Theorem 3.1. Let $F(z, w, \lambda)$ be the complex extended framing of some complex
CGC-immersion . Then the following statements hold:$\Phi$), then for eachthe $Sym$ formula in (3.1.5)
If $F^{-1}dF$ is in (Asl(2,.
defines spacelike constant negative GauBian curvature surface in$(C, 2)$ If $F^{-1}dF$ is in, then for eachthe $Sym$ formula in (3.1.5)$GauBian$curvature surface in.
defines timelike constant negative$(C, 3)$ If $F^{-1}dF$ is in, then for eachthe $Sym$ formula in (3.1.5)
positive
constant
GauBiancurvaturesurfacein.
defines$(C,4)$ If $F^{-1}dF$ is in, then for eachthe $Sym$ formula in (3.1.6)
defines constant mean curvature surface with mean curvature $|H^{(c,4)}|<1$ in$(C, 1)$$\Omega$$\lambda\in S^{1}$$\mathbb{C})_{\sigma}^{(c,1)}$$\mathbb{R}^{2,1}$$a$$\Omega(\Lambda \mathfrak{S}[(2, \mathbb{C})_{\sigma}^{(c_{1}2)})$$\lambda\in S^{1}$$\mathbb{R}^{2,1}$$a$$\Omega(\Lambda \mathfrak{S}1(2, \mathbb{C})_{\sigma}^{(c,3)})$$\lambda\in S^{1}$$\mathbb{R}^{3}$$a$$\Omega(\Lambda \mathfrak{S}1(2,\mathbb{C})_{\sigma}^{(c,4)})$$\lambda\in S^{r}$$a$$H^{3}$.

, then for eachthe $Sym$ formula in (3.1.7)
If $F^{-1}dF$ is in.
defines spacelike constant positive GauBian curvature surface in$(S, 2)$then for eachthe $Sym$ formula in (3.1.7)
If $F^{-1}dF$ is in Ael(2,.
defines timelike constant positive GauBian curvature surface in$(S, 3)$ If $F^{-1}dF$ is inAel(2,
then for eachthe $Sym$ formula in (3.1.7)
negative
constant
GaMiancurvature.
surfaceindefines$(S, 1)$$\Omega(\Lambda \mathfrak{S}[(2, \mathbb{C})_{\sigma}^{(z,1)})$$\lambda\in \mathbb{R}^{*}$$\mathbb{R}^{2,1}$$a$$\Omega($$\mathbb{C})_{\sigma}^{(\epsilon,2)})_{f}$$\lambda\in \mathbb{R}^{*}$$\mathbb{R}^{2,1}$$a$$\Omega($$\mathbb{C})_{\sigma}^{(r,3)})_{j}$$\lambda\in \mathbb{R}^{*}$$a$$\mathbb{R}^{3}$

Definition 4. Let
for $j\in\{1,2,3,4\}$ (resp.
for $j\in$$\{1,2,3\})$ be the complex extended framings, which are elements in
resp.
. Thenresp. $F^{(,j)}(x,$is called the extended framing for the immersion(resp.
).
$F^{(c,j)}(z,\overline{z}, \lambda)$$F^{(\epsilon,j)}(x, y, \lambda)$$\Lambda SL(2, \mathbb{C})_{\sigma}^{(c,j)}$$($$\Lambda SL(2,$$\mathbb{C})_{\sigma}^{(\epsilon_{1}j)})$$F^{(c,j)}(z, w, \lambda)$$\Phi^{(c,j)}$

$($

$y,$

$\lambda))$$\Phi^{(B,j)}$

It is known that for three classes of surfaces in the above seven classes, there exist
parallel constant mean curvature surfaces inor(see also [8] and [9]).
$\mathbb{R}^{3}$$\mathbb{R}^{2,1}$

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Table 1: Integrabk surfaces
Corollary 3.2. We retain the assumptions in Theorem S. 1. Then we have the following:$(C, 1M)$

For the $(C, 1)$ case in Theorem S.l, there exists a pamllel spacelike constantmean curvature surface with mean curvaturein.
$H^{(\mathfrak{c},1)}=|H|$$(C, 3M)$$\mathbb{R}^{2,i}$

For the $(C, 3)$ case in Theorem 3.1, there exists a pamllel constant mean curvature surface with mean curvature $H^{(c,3)}=|H|$ in .
$\mathbb{R}^{3}$$(S, 2M)$

For the $(S, 2)$ case in Theorem S. 1, there exists a parallel timelike $\omega nstant$ mean.
curvature surface with mean curvature $H^{(r,2)}=|H|$ in$\mathbb{R}^{2,1}$

Definition 5. The surfaces defined in Theorem S. 1 and Corollary S.2 are called theintegrable surfaces.

Remark 3.3. For the three classes of surfaces in Theorem S. 1, which are spacelike$\omega nstant$ negative $Gau\beta ian$constant positive $Gau\beta ian$ curvature surfaces incurvature surfaces inand timelike $\omega nstant$ negative $Gau\beta ian$ curvature surfaces in, there never exist parallel constant mean curvature surfaces.
$\mathbb{R}^{2,1},$$\mathbb{R}^{3}$$\mathbb{R}^{2,1}$

4

The generalized Weierstra13 type representation
for integrable surfaces

The generalized Weierstra3 type representation for complex CMC-immersions (orequivalently CGC-immersions as the parallel immersions) is the procedure of a construction of complex CMC-immersions from a pair of holomorphic potentiak (see [4]).
In the previous section, we obtained integrabk surfaces according to the real forms of. In this section, we show how all integrable surfaces are obtained from thepairs of holomorphic potentiak in the generalized Weierstr type representation.
$\Lambda \mathfrak{S}[(2, \mathbb{C})_{\sigma}$$\theta$

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4.1

Integrable surfaces via the generalized Weierstrai3 type
representation

The generalized WeierstraB type representation for complex CMC-immersions (orequivalently CGC-immersions as the parallel immersions) is divided into the following
4 steps (see ako [4] for more details):
Stepl Let
forms:$\check{\eta}=(\eta(z, \lambda), \tau(w, \lambda))$be a pair of holomorphic potentiak of the following(4.1.1)$\check{\eta}=(\eta(z, \lambda), \tau(w, \lambda))=(\sum_{k=-1}^{\infty}\eta k(z)\lambda^{k},\sum_{m=-\infty}^{1}\tau_{m}(w)\lambda^{m})$is some holomorphically convex domain in ,$|\lambda|=r(0<r<1)$ , andandare-valued holomorphic
differential l-forms. Moreoverandare diagonal (resp. off-diagonal)
matrices if is even (resp. odd). We also assume that the upper right entry ofand the lower left entry
do not vanish for all
.

where

$(z, w)\in \mathfrak{D}^{2}$and where$\mathfrak{D}^{2}$$\mathbb{C}^{2}$$\lambda\in \mathbb{C}^{*},$$\epsilon 1(2, \mathbb{C})$$\tau_{m}$$\eta k$$\eta_{k}(z)$$\tau_{k}(w)$$k$$\eta_{-1}(z)$$\tau_{1}(w)$Step2 Let $C$ andequations$\cdot$$L$denote the solutions to the following linear ordinary differential$dC=C\eta$where$(z, w)\in \mathfrak{D}^{2}$

and $dL=L\tau$ with $C(z_{*}, \lambda)=L(w_{*}, \lambda)=$ id,$(z_{*}, w_{*})\in \mathfrak{D}^{2}$(4.1.2)is a fixed base point.

Step3 We factorize the pair of matrices $(C, L)$ via the generalized Iwasawa decomposition of Theorem A. 1 as follows:$(C, L)=(F_{7}F)(id, W)(V+’ V_{-})$ ,where$V\pm\in\Lambda^{\pm}SL(2, \mathbb{C})_{\sigma}$(4.1.3).

Theorem 4.1 ([4]). Let $F$ be a-loop defined by the generalized Iwasawa decomposition in (4.1.3). Then there exists a -independent diagonal matrix$l(z, w)\in SL(2, \mathbb{C})$ such thatis a complex extended framing of some $\omega mplex$
CMC-immersion (or equivalently the complex CGC-immersion as the parallel im$\Lambda SL(2, \mathbb{C})_{\sigma}$$\lambda$$F\cdot l$mersion).

Step4 The Sym formula defined in (2.2.1) via $F(z, w, \lambda)l(z, w)$ represents a complex
CMC-immersion and a CGC-immersion in.
$\mathfrak{S}[(2, \mathbb{C})\cong \mathbb{C}^{3}$

Let

for $j\in\{1,2,3,4\}$ and for $j\in\{1,2,3\}$ be the involutions defined in (3.1.3),
respectively. Then we define the following pairs of involutions on$c_{j}$$\mathfrak{S}_{j}$$\dot{\eta}=(\eta, \tau)\in$$\Omega(\Lambda \mathfrak{s}\mathfrak{l}(2, \mathbb{C})_{\sigma})\cross\Omega(\Lambda_{\mathfrak{S}}1(2, \mathbb{C})_{\sigma})$:$\mathfrak{r}_{j}:(\eta, \tau)\mapsto(c_{j}\tau, c_{j}\eta)$

We now prove the following theorem.

and$\Phi_{j}:(\eta, \tau)\mapsto(\epsilon_{j}\eta, \mathfrak{S}_{j}\mathcal{T})$.

(4.1.4)130

be a pair of holomorphic potentials defined
Theorem 4.2. Let
as in (4.1.1), and let for $j\in\{1,2_{i}3,4\}$ and for $j\in\{1,2,3\}$ be the pairs of
involutions defined in (4.1.4). Then the following statements hold:$\eta=(\eta(z, \lambda), \tau(w, \lambda))$$\Phi_{j}$$\mathfrak{r}_{j}$$(C, 1)$, then the resulting immersions given by the generalized $Weierstra\beta$
type representation are spacelike constant negative $Gau\beta ian$ curvature surfaces
.
in

If

$\mathfrak{r}_{1}(\check{\eta})=\check{\eta}$$\mathbb{R}^{2,1}$$(C, 2)$then the resulting immersions given by the genemlized $Weierstra\beta$
representation
type
are timelike constant negative $Gau\beta ian$ curvature surfaces

Ifin$(C, 3)$$\mathfrak{r}_{2}(\check{\eta})=\check{\eta}_{f}$$\mathbb{R}^{2,1}$.

, then the resulting immersions given by the generalized $Weierstra\beta$.
type representation are $\omega nstant$ positive $Gau\beta ian$ curvature surfaces inIf$\mathfrak{r}_{3}(\check{\eta})=\check{\eta}$$\mathbb{R}^{3}$$(C, 4)$If, then the resulting immersions given by the generalized $Weierstra\beta$
representation
are constant mean curvature surfaces with mean curvature
type$|H^{(q4)}|<1$ in.
$\mathfrak{r}_{4}(\check{\eta})=\check{\eta}$$H^{3}$$(S, 1)$If$0_{1}(\check{\eta})=\check{\eta}$, then the resulting immersions given by the generalized $Weierstra\beta$

type representation are spacelike constant positive $Gau\beta ian$ curvature surfaces.
in$\mathbb{R}^{2,1}$$(S, 2)$If$\mathbb{R}^{2,1}type$$(S, 3)$If, then the resulting immersions given by the generalized $Weierstra\beta$
representation are timelike constant positive $Gau\beta ian$ curvature surfaces in$V_{2}(\check{\eta})=\check{\eta}$.

, then the resulting immersions given by the generalized $Weierstra\beta$
type representation are constant negative $Gau\beta ian$ curvature surfaces in .
$b_{3}(\check{\eta})=\check{\eta}$$\mathbb{R}^{3}$

Remark 4.3. From the $f_{07}ms$ of pairs of involutions
for $j\in\{1,2,3,4\}$ definedin (4.1.4), the pairs of holomorphic potentials for $(C,j)$ cases in Theorem 4.2 aregenemted by a single potential, $i.e.\check{\eta}=(\eta, \tau)=(\eta, c_{j}(\eta))$ , where for $j\in\{1,2,3,4\}$are involutions defined in (3.1.4).
$\mathfrak{r}_{j}$$\check{\eta}$$c_{j}$

A

Double loop groups and the generalized Iwasawa decompositions

In this subsection, we give the basic notations and results for double loop groups (see[7] for more detaik). Let $D_{r}:=\{\lambda\in \mathbb{C}||\lambda|<r\}$ be an open disk and denote theby. Ako, let $A_{r}=\{\lambda\in \mathbb{C}|r<|\lambda|<1/r\}$closure of
be an open amulus containing , and denote the closure ofby. Furthermore,
let $E_{r}=\{\lambda\in \mathbb{C}|r<|\lambda|\}\cup\{\infty\}$ be an exterior of the circle.
$D_{r}$$\overline{D_{f}}$$:=\{\lambda\in \mathbb{C}| |\lambda|\leq r\}$$S^{1}$$A_{r}$$C_{r}$$\overline{A_{r}}$131

We recall the definitions of the twisted plus r-loop group and the minus r-loop groupofas follows:$\Lambda SL(2, \mathbb{C})_{\sigma}$

extends holomorph callytoand $W_{+}(0)\in B$ .

$W_{+}(\lambda)$$:=\{W_{+}\in\Lambda_{r}SL(2, \mathbb{C})_{\sigma}$$\Lambda_{r,B}^{+}SL(2, \mathbb{C})_{\sigma}$‘$D_{r}$

extends holomorphicallytoand $W_{-}(\infty)\in B$ .
. If $B=$ {id} we write the subscript instead of$W_{-}(\lambda)$$\Lambda_{r,B}^{-}SL(2, \mathbb{C})_{\sigma}:=\{W_{-}\in\Lambda_{r}SL(2\rangle \mathbb{C})_{\sigma}$‘$E_{r}$

where $B$ is a subgroup of$B$ , if $B=SL(2, \mathbb{C})$ we abbreviate
andby, respectively. Rom now on we will use the subscript $B$ as above
andonly if $B\cap SU(2)=$ {id} holds. When $r=1$ , we always omit the 1.
$SL(2, \mathbb{C})$$*$$\Lambda_{r,B}^{+}SL(2, \mathbb{C})_{\sigma}$$\Lambda_{rB,)}^{-}SL(2, \mathbb{C})_{\sigma}$$\Lambda_{r}^{+}SL(2, \mathbb{C})_{\sigma}$$\Lambda_{r}^{-}SL(2, \mathbb{C})_{\sigma}$

We set the product of two loop groups:$\mathcal{H}=\Lambda_{r}SL(2, \mathbb{C})_{\sigma}\cross\Lambda_{R}SL(2, \mathbb{C})_{\sigma}$

where $0<r<R$ . Moreover we set the subgroups of$\mathcal{H}_{+}=\Lambda_{r}^{+}SL(2, \mathbb{C})_{\sigma}x\Lambda_{R}^{-}SL(2, \mathbb{C})_{\sigma}$$g_{1}$$\mathcal{H}_{-=}\{(g_{1}, g_{2})\in \mathcal{H}|$and$\mathcal{H}$,as follows:,$g_{2}extendho1omorphica11ytoA_{r}andg_{1}|_{A_{r}}=g_{2}|_{A_{r}}\}$,We then quote Theorem 2.6 in [7].

Theorem A.1.
open and dense in$\mathcal{H}_{-}x\mathcal{H}_{+}arrow \mathcal{H}_{-}\mathcal{H}_{+}$$\mathcal{H}$. More precisely

is an a.nalytic diffeomorphism. The image is$\mathcal{H}=\bigcup_{n=0}^{\infty}\mathcal{H}_{-}w_{n}\mathcal{H}_{+}$

where $w_{n}=$ id,$($$(_{0\lambda^{\underline{0}_{n}}}^{\lambda^{n}}))$

if $n=2k$ and id,$($,$(_{-\lambda^{-n}0}0\lambda^{n}$$))$

if $n=2k+1$ .

The proof of the theorem above is abost verbatim the proof given in the basic
decomposition paper [2] (see also [3]).

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