数理解析研究所講究録
第 1605 巻 2008 年 81-89

81

Drawing the complex projective structures on
once-punctured tori
Yohei Komori (Osaka City Univ.)

1Introduction

This report is based on my talk at RIMS International Conference on “ Geometry Related to Integrable Systems) organized by Reiko Miyaoka. In my talkI showed many interesting pictures of one-dimensional Teichm\"uller spaces
and related spaces created by Yasushi Yamashita (Nara Women’s Univ.)
which were already appeared in [3]. In this report I would like to explain thebackground of these pictures, which are explained more extensively in [2].
I would like to thank Yasushi Yamashita for his kind assistance with computer graphics, and Yoshihiro Ohnita for his constant encouragement for me
to write this report.

2Definition of $T(X)$Let $X$ be a Riemanm surface of genus with punctures. Here we assume
that $X$ is uniformized by the upper half planein , which implies theinequality $2g-2+n>0$ . The Teichmuller space $T(X)$ of $X$ is the setof equivalent classes of quasi-conformal homeomorphisms from $X$ to otherRiemann surface $Y,$ : $Xarrow Y$ : two maps
: $Xarrow Y_{1}$ and: $Xarrow Y_{2}$:are equivalent if
is homotopic to a conformal map. If weassume : $Xarrow Y$ as a quasi-conformal deformation of $X,$ $T(X)$ can beconsidered as the space of quasi-conformal deformations of $X$ .
We will consider a complex manifold structure on $T(X)$ , embed it holomorphically into complex affine space and try to draw its figure. For thispurpose, we give another characterization of $T(X)$ due to Ahlfors and Bersin the next section.
$g$

$n$

$\mathbb{H}$$f$$f_{2}\circ f_{1}^{-1}$$f$$f_{1}$$Y_{1}arrow Y_{2}$$\mathbb{C}$$f_{2}$

82

3Complex structure on $T(X)$Let
be a Fuchsiaii group uniforniizing. A measurable
function $\nu(z)$ on the Riemann sphere
whose essential $\sup$ norm is lessthan 1 is called a Beltrami differential for if is equal to on the lower
half plane in and satisfies$\Gamma\subset PSL_{2}(\mathbb{R})$$X=\mathbb{H}/\Gamma$$\mathbb{C}P^{1}$$\Gamma$

$0$

$\mu$

$\mathbb{L}$$\mathbb{C}$$\mu(\gamma(z))\cdot\overline{\frac{\gamma’(z)}{\gamma’(z)}}=\mu(z)$for all. This functional equality implies that onandis a lift of $(-1,1)$ form on X. We denote the set of Beltrami differentials by
which has a structure of a unitof complex Banach space. The
measurable Riemann’s mapping theorem due to Ahlfors and Bers guarantees
that for any
there exists a quasi-conformal map:
such thatsatisfies the Beltrami equation$z\in \mathbb{C}P^{1}$$\gamma\in\Gamma$$\mathbb{H}$$\mu$$b\mathfrak{N}$$B_{1}(\Gamma, \mathbb{H})$$\mu\in B_{1}(\Gamma, \mathbb{H})$$\mathbb{C}P^{1}$$f^{\mu}$$\mathbb{C}P^{1}arrow$$f^{\mu}$$\frac{\partial f^{\mu}}{\partial\overline{z}}(z)=\mu(z)\frac{\partial f^{\mu}}{\partial z}(z)$.

Alsois unique up to post-composition by M\"obius transformations.
Here we have two remarks: (i)is conformal on . (ii) The quasiconformal conjugation of byis also a discrete subgroupof.
acting conformally onNow we sayforif. Then $T(X)$
can be identified with the quotient spaceas follows: For any, we have a quasi-conformal deformation of $X$$f^{\mu}$$f^{\mu}$$\Gamma$$f^{\mu},$$PSL_{2}(\mathbb{C})$$\mathbb{L}$$\Gamma^{\mu}=f^{\mu}\Gamma(f^{\mu})^{-1}$$f^{\mu}(\mathbb{H})$$\mu_{1}\sim\mu 2$$\mu_{1},$$\mu_{2}\in B_{1}(\Gamma, \mathbb{H})$$\Gamma^{\mu 1}=\Gamma^{\mu 2}$$B_{1}(\Gamma,\mathbb{H})/\sim$$[\mu]\in B_{1}(\Gamma, \mathbb{H})/\sim$$f^{\mu}:X=\mathbb{H}/\Gammaarrow f^{\mu}(\mathbb{H})/\Gamma^{\mu}$which defines a point of $T(X)$ . $T(X)$ becomes a complex manifold of $dim_{C}T(X)=$. We will embed $T(X)$$3g-3+n$ through the complex structure ofholomorpliically into the complex linear space by means of complex projective structures on , the mirror image of $X$ which will be explained in thenext section.
$B_{1}(\Gamma, \mathbb{H})$$\overline{X}$Complex projective structures on4Let be a surface. A complex projective structure, so calledon is a maximal system of charts with transition maps in
$S$

$S$

$\overline{X}$$\mathbb{C}P^{1}$-structure. Since$PSL_{2}(\mathbb{C})$83

elements ofare holomorphic, any-structure on determines its
underlying complex structure. Suppose we consider a-structure whose
complexunderlyingstructure is equal to, the mirror image ofX. For a local coordinate fumction of this-structure, we can take its
analytic continuation along any curve on
and have a multi-valued locally
univalent holomorphic map from to. This map is lifted to a locallyuiiivalent meromorphic function
called the developing map of
this-structure. It is uniquely determined by the-structure up to
post-composition by M\"obius transformations.
When we take an analytic continuation of a local coordinate function
along a closed curve on
and come back to the initial point, it differsfrom the previous one by a M\"obius transformation since the transition maps
. Consequently we have a homomorphism :are inwliich is called the holonomy representation and satisfies $\chi(\gamma)oW=$. Therefore thefor all-structure on
determines the$(W,\chi)$up to the action ofpair
and vice versa. Here we show themost basic example of-structures on : Let $W$ be the identity map$PSL_{2}(\mathbb{C})$$\mathbb{C}P^{1}$$S$$\mathbb{C}P^{1}$$\overline{X}=\mathbb{L}/\Gamma$$\mathbb{C}P^{1}$$\overline{X}$$\overline{X}$$\mathbb{C}P^{1}$$\mathbb{L}$$W:\mathbb{L}arrow \mathbb{C}P^{1}$$\mathbb{C}P^{1}$$\mathbb{C}P^{1}$$\overline{X}$$PSL_{2}(\mathbb{C})$$\chi$$\Gamma\cong\pi_{1}(\overline{X})arrow$$PSL_{2}(\mathbb{C})$$W\circ\gamma$$\mathbb{C}P^{1}$$\gamma\in\Gamma$$\overline{X}$$PSL_{2}(\mathbb{C})$$\mathbb{C}P^{1}$and$\overline{X}$also be the identity homomorphism :which induces a local coordinate function as a local inverse of the universal. We call thiscovering map-structure the standard-structure$W:\mathbb{L}\hookrightarrow \mathbb{C}P^{1}$$\chi$$\chi$$\mathbb{L}arrow\overline{X}$on$\overline{X}$$\mathbb{C}P^{1}$$\Gammaarrow\rangle PSL_{2}(\mathbb{R})$$\mathbb{C}P^{1}$.

Let $P(\overline{X})=\{(W, \chi)\}/PSL_{2}(\mathbb{C})$ be the set of-structures on . We
will parametrize
by holomorphic quadratic differentials on as follows:A holomorphic fumction on is called a holomorphic quadratic differential
for if it satisfies$\mathbb{C}P^{1}$$\overline{X}$$P(\overline{X})$$\overline{X}$$\mathbb{L}$$\varphi$$\Gamma$$\varphi(\gamma(z))\gamma’(z)^{2}=\varphi(z)$. It is a lift of holomorphic quadratic differentials. Letonbe the set of holomorphic quadratic differentials forwhose hyperbolic $\sup$ norm
is bounded.
has a structure of complex linear space of $dim_{C}Q(\overline{C})=3g-3+n$ which isequal to the dimension of $T(X)$ . We show that there is a canonical bijection
betweenandwhich maps the standard-structure to theorigin: Given a-structures on , take the Schwarzian derivative of $W$
for all$z\in \mathbb{L}$and$\overline{X}=\mathbb{L}/\Gamma$$\gamma\in\Gamma$$Q(\overline{X})$$|| \varphi||=\sup_{z\in L}|\Im z|^{2}|\varphi(z)|$$\Gamma$$P(\overline{X})$$\mathbb{C}P^{1}$$Q(\overline{X})$$\mathbb{C}P^{1}$$Q(\overline{X})$$\overline{X}$$S_{W}:=(f’’/f’)’- \frac{1}{2}(f’’/f’)^{2}$. Conversely given a holomorphic quadratic
which is an element ofdifferential for , solve the differential equation
on . In practice$Q(\overline{X})$$\Gamma$$\varphi$$S_{f}=\varphi$$\mathbb{L}$84

to find the solution , we consider the following linear homogeneous ordinary
differential equation of the second order$f$$2\eta’’+\varphi\eta=0$. Since is simply connected, a unique solution exists on for thegiven initial data $\eta(-i)=a$ and $\eta’(-i)=b$ . Let
andbe the solutionon$\mathbb{L}$$\mathbb{L}$$\mathbb{L}$$\eta$$\eta_{1}$$\eta_{2}$defined by the conditions $\eta_{1}(-i)=0$ and $\eta_{1}’(-i)=1$ , and $\eta_{2}(-i)=1$ and$\eta_{2}’(-i)=0$ . Then the ratio
is a locally univalent meromorphic$f\iota mction$ on
the developing map associated with . A direct computation
also satisfies the above equation hence there isshows thatsuch thata matrix of$f_{\varphi}=\eta_{1}/\eta 2$$\mathbb{L}_{7}$$\varphi$$\eta(\gamma(z))(\gamma’(z))^{-\frac{1}{2}}$$SL_{2}(\mathbb{C})$$(\begin{array}{l}\eta 1(\gamma(z))(\gamma^{/}(z))^{-\frac{1}{2}}\eta 2(\gamma(z))(\gamma^{/}(z))^{-\frac{l}{2}}\end{array})=(\begin{array}{ll}a bc d\end{array})(\begin{array}{l}\eta_{1}\eta_{2}\end{array})$. As a result we have a homomorphism:
for allholonomy representation associated with . We can also consider
monodromy representation of the above differential equation.
$\gamma\in\Gamma$$\chi_{\varphi}$theas the$\Gammaarrow PSL_{2}(\mathbb{C})\}$$\varphi$$\chi_{\varphi}$Bers embedding of $T(X)$5by means of the identification
Now we embed $T(X)$ into
. For each element $[\mu]\in T(X)=B_{1}(\Gamma, \mathbb{H})/\sim,$
is conforis a quasi-fuchsian group. Therefore it determines
mal and
where the developing map is $W=f^{\mu}|_{L}$ and thea-structure on
is defined byholonomy representation :. After$T(X)$ can be embedded into
the identification, whichis called the Bers embedding of $T(X)$ .
We will show not only the picture of $T(X)$ but also other-structures
be the set ofon : Let-structures on
whose holonomy groups. Shiga [4] showed thatare Kleinian groups, discrete subgroups ofthe connected component of the interior ofcontaining the origin co$T(X)$. Shiga and Tanigawa [5] proved that any
incides with-structureof the interior ofhas a quasi-fuchsian holonomy representation. Newith respect to the hyperbolic
hari showed that $T(X)$ is bounded in$||\varphi||=\sup_{z\inL}|\Imz|^{2}|\varphi(z)|$$\sup$ norm
, while Tanigawa proved thatisunbounded.
$Q(\overline{X})\cong \mathbb{C}^{3g-3+n}$$P(\overline{X})\cong Q(\overline{\lambda’}\cdot)$$f^{\mu}|_{L}$$\Gamma^{\mu}=f^{\mu}\Gamma(f^{\mu})^{-1}$$\mathbb{C}\mathbb{P}^{1}$$\mathbb{L}/\Gamma$$\Gammaarrow\Gamma^{\mu}$$\chi(\gamma)=f^{\mu}\gamma(f^{\mu})^{-1}$$\chi$$P(\overline{X})\cong Q(\overline{X}),$$Q(\overline{X})$$\mathbb{C}P^{1}$$\overline{X}$$\mathbb{C}P^{1}$$K(\overline{X})$$\overline{X}$$PSL_{2}(\mathbb{C})$$K(\overline{X})$$\mathbb{C}P^{1}$$K(\overline{X})$$Q(\overline{X})$$K(\overline{X})$85

6Pictures of $T(X)$ and $K(X)$We will show pictures of $T(X)$ and $K(X)_{7}$ all of which depends on the underlying complex structure of . All picture were drawn by Yasushi Yamashita.
Figrrre 1 and figure 2 are the case thathas a hexagonal symmetry. Figure
3 and figure 4 are the case thathas a square symmetry. Black colored
region consists of whose holonomy representation has an indiscrete image.
For both cases, $T(X)$ looks like an isolated planet, wliile $K(X)$ itself looks
like the galaxy: Some planets seem to bump each other... Wlien we takeanti )$mmetric,$ $T(X)$ and $K(X)$ become distorted, which we can see in
figure 5 and $fig\iota ire6$ .
To draw these pictures we need$\overline{X}$$\overline{X}$$\overline{X}$$\varphi$$\overline{X}$$- s$

1. to calculate the holonomy representation

2. to check whether$\chi_{\varphi}(\Gamma)$$\chi_{\varphi}$for$\varphi\in Q(\overline{X})$, andis discrete or not.

First we will explain (1). To deterinine, we must solve $S_{f}=\varphi$ on .
In generalis highly transcendental function on
and it is very
difficult for us to handle it. Here is an idea: If dimcT $(X)=3g-3+n=1$ ,then $(g, n)=(O, 4)$ or (1, 1). Take
, then we can finda basis oflike. Even in this case, it is still
difficult to solve$\mathbb{L}$$\chi_{\varphi}$$\varphi\in Q(\overline{X})$$\mathbb{L}$$\overline{X}=\mathbb{C}\mathbb{P}^{1}-\{0,1, \infty, \lambda\}$$Q(\overline{X})$$Q( \overline{X})=\mathbb{C}\cdot\pi^{*}(\frac{1}{w(w-1)(w-\lambda)})$$S_{f}= \pi^{*}(\frac{t}{w(w-1)(w-\lambda)})$where :
the above equation onto$\pi$$\mathbb{L}arrow \mathbb{C}\mathbb{P}^{1}-\{0,1, \infty, \lambda\}$and$t\in \mathbb{C}\cong Q(\overline{X})$. But we can push down$\overline{X}=\mathbb{C}\mathbb{P}^{1}-\{0,1, \infty, \lambda\}$$S_{f\circ\pi^{-1}}= \frac{t}{w(w-1)(w-\lambda)}+(\frac{1}{2w^{2}(w-1)^{2}}+\frac{1}{2(w-\lambda)^{2}}+\frac{c(\lambda)}{w(w-1)(w-\lambda)})$where
is called the accessory parameter of.
To get the solution we take the ratio of two linearly independent solutionof$\pi:\mathbb{L}arrow\overline{X}$$c(\lambda)$$2y”+( \frac{1}{2w^{2}(w-1)^{2}}+\frac{1}{2(w-\lambda)^{2}}+\frac{t+c(\lambda)}{w(w-1)(w-\lambda)})y=0$

and calculate the monodromy group of this equation with respect to closed
paths of. Since the above ordinary differential equation has
rational coefficients on, we can use computer to get the image of 3$\pi_{1}(\overline{X})\cong F_{3}$$\mathbb{C}P^{1}$

86

numerically. Here we remark that to draw
ingenerators of
the picture of $K(X)$ up to parallel translation, we don’t need to determine
in practice.
the accessory paraineter
is indiscrete,For (2), we apply Shimizu lemma to check whetherand Poincar\’e theorem to construct the Ford fumdamental domain to checkis discrete. This part is so called Jorgensen theory and has
whetherbeen proved recently by Akiyoshi, Sakuma, Wada and Yamashita .
$\pi_{1}(\overline{X})$$PSL_{2}(\mathbb{C})$$c(\lambda)$$\chi_{\varphi}(\Gamma)$$\chi_{\varphi}(\Gamma)$$[1|$

References[1] H. Akiyoslii, M. Sakuma, M. Wada and Y. Yamashita, Punctured Torus
Groups and 2-Bridge Knot Groups I, Springer LNS. 1909.

[2] Y. Imayoslii and M. Taiiiguchi, An Introduction to Teichm\"uller Spaces,Springer (1999).

[3] Y. Komori, T. Sugawa, M. Wada and Y. Yamashita, Drawing Bers embeddings of the Teichm\"uller space of once-punctured tori,Experimental Mathematics, Vol. 15 (2006), 51-60.

[4] H. Shiga, Projective structures on Riemann surfaces and Kleinian groups,
J. Math. Kyoto. Univ. 27:3(1987), 433-438.
[5] H. Shiga and H. Tanigawa, Projective structures with discrete holonomy
representations, Trans. Amer. Math. Soc. 351 (1999), 813-823.

87

Figure 1: $T(X)$ for hexagonal symmetry

Figure 2: $K(X)$ for hexagonal symmetry

88

Figure 3: $T(X)$ for square syimnetry

Figure 4: $K(X)$ for square symmetry

89$Fi_{b^{1}}i_{-}ire5$$Fi_{b^{J}}iire6$: distorted $T(X)$: distorted $K(X)$