数理解析研究所講究録
第 1777 巻 2012 年 50-56

50

A SURVEY ON THE EXTREMAL LENGTH GEOMETRY ON
TEICHM\"ULLER SPACEHIDEKI MIYACHI

1. INTRODUCTION
By the extremal length geometry, we naively mean the geometry on the Teichm\"uller space studied via the extremal length on measured foliations. From the
Kerckhoff’s formula on the Teichm\"uller distance, the geometry on the Teichm\"uller
distance is naturally in the category of the extremal length geometry.
In [12], S. Kerckhoff developed the study of the “end“ of the Teichm\"uller spaceby using the extremal length. In [6], F. Gardiner and H. Masur formulated theextremal geometry of Teichm\"uller space and defined the compactification, which
we recently call the Gardiner-Masur boundary, in terms of the extremal length
geometry.
The aim of this paper is to give a survey of the author’s resent progress in theextremal length geometry on Teichm\"uller space.
Acknowledgement. The author would like to express his sincere appreciation tothe organizers, Professor Teruaki Kitano and Professor Takayuki Morifuji, for giving
him an opportunity to talk the conference “Geometric and analytic approaches torepresentations of a group and representation spaces”.

2. TEICHM\"ULLER THEORY

2.1. Teichmilller space and Measured foliations. Let $X$ be a Riemann surface of analytically finite type $(g,n)$ with $2g-2+2>0$ . The Teichmtiller space
and$T(X)$ is the set of equivalence classes of pairs $(Y, f)$ of Riemann surfaces
are equivalentandquasiconformal mapping : $Xarrow$ Y. Two pairs
to . Let $x_{0}=(X, id)$if
is homotopic to a conformal mapping ffom
be the base point.
between
For $y_{1}=(Y_{1}, f_{1}),$ $y_{2}=(Y_{2}, f_{2})\in T(X)$ , the Teichmuller distanceis defined by
and$Y$$(Y_{2}, f_{2})$$(Y_{1}, f_{1})$$f$$Y_{1}$$f_{2}\circ f_{1}^{-1}$$Y_{2}$$d_{T}$$y_{1}$$y_{2}$$d_{T}(y_{1},y_{2})= \frac{1}{2}\log\inf_{h}K(h)$homotopic towhere runs over all quasiconformal mappings
and $K(h)$ is the maximal dilatation of . A metric space $(T(X), d_{T})$ is known
to be complete and a uniquely geodesic space (cf. [9]). However, to the author’s
knowledge, there is no nice characterization of the metric space $(T(X),d_{T})$ , andseveral sad news are known. For instance, it is known that $(T(X), d_{T})$ is neither aCAT(0)-space or a Gromov hyperbolic space (cf. [17], [25], [18], and [19]).
Let be the set of homotopy classes of non-peripheral and non-trivial simple
be the space of non-negative functions on whichclosed curves on $X$ . Let
equipped with the topology of the pointwise convergence, and set$Y_{1}arrow Y_{2}$$h$$f_{2}of_{1}^{-1}$

$h$

$S$

$\mathbb{R}_{+}^{S}$$S$$P\mathbb{R}_{+}^{S};=(\mathbb{R}_{+}^{S}-$51
HIDEKI MIYACHI

the projective space. The spaceof the embedded image of the mapping$\{0\})/\mathbb{R}_{>0}$$\Lambda t\mathcal{F}$of measured foliations is the closure$\mathbb{R}_{+}\otimes S\ni t\alpha\mapsto[S\ni\beta\mapsto ti(\beta, \alpha)]\in \mathbb{R}_{+}^{S}$

is the set of formal products
, and
of
and
is thegeometric intersection number between simple closed curves. It is known that the$\mathbb{R}+\otimes S$$t\alpha$$\alpha\in S$$t\in \mathbb{R}+$$i(\cdot,$$\cdot)$intersection number$(\mathbb{R}_{+}\otimes S)\cross(\mathbb{R}_{+}\otimes S)\ni(t\alpha, s\beta)\mapsto i(t\alpha, s\beta):=tsi(\alpha, \beta)$

extends continuously on$\Lambda t\mathcal{F}\cross \mathcal{M}\mathcal{F}$(cf. [1] and [26]). The projective space$\mathcal{P}\mathcal{M}\mathcal{F}=(\mathcal{M}\mathcal{F}-\{0\})/\mathbb{R}_{>0}\subset P\mathbb{R}_{+}^{S}$is called the space of projective measured foliations.

2.2. Extremal length. For
and $y=(Y, f)\in T(X)$ , the extremal lengthof on is the reciprocal of the supremum of the modulus of annuli
whose cores are homotopic toin Y. S. Kerckhoff showed that when we set$Ext_{y}(t\alpha)=t^{2}Ext_{y}(\alpha)$ for, the extremal length
extends continuously on(cf. [12]).
It is known that the Teichm\"uller distance has a geometric description$\alpha\in S$$Ext_{y}(\alpha)$$\alpha$$y$$f(\alpha)$$t\alpha\in \mathbb{R}_{+}\otimes S$$Ext_{y}(\cdot):\mathbb{R}_{+}\otimes Sarrow \mathbb{R}$$\mathcal{M}\mathcal{F}$$d_{T}(y_{1}, y_{2})= \frac{1}{2}\log\sup_{\alpha\in S}\frac{Ext_{y_{1}}(\alpha)}{Ext_{ya}(\alpha)}$for$y_{1},$$y_{2}\in T(X)$, which we call Kerckho$ff^{f}s$formula (cf. [12]). We define$\mathcal{M}\mathcal{F}_{1}=\{F\in \mathcal{M}\mathcal{F}|Ext_{xo}(F)=1\}$.

2.3. Gardiner-Masur closure. In [6], F. Gardiner and H. Masur observe thatthe mapping$\Phi_{GM}:T(X)\ni y\mapsto[S\ni\alpha\mapsto Ext_{y}(\alpha)^{1/2}]\in P\mathbb{R}_{+}^{S}$

is embedding and the image is relatively compact. The mappingis called theGardiner-Masur embedding. The closure $c1_{GM}(T(X))$ is said to be the GardinerMasur compactification and the complement $\partial_{GM}T(X)$ of the image from the closure is called the Gardiner-Masur boundary. In [6], Gardiner and Masur observed
the following (see also [20] and [21]).
$\Phi_{GM}$

Theorem 2.1 (Gardiner and Masur). We have $P$ ル tF
GMT(X) in general. If$X$ isneither a four punctured sphere or once punctured torus,
is a proper$\subset\partial$

subset of$\partial_{GM}T(X)$$a$.

$\mathcal{P}\lambda 4\mathcal{F}$Hence, we have the following topological observation.

Corollary 2.1. If $X$ is neither a four punctured sphere or once punctured torws,
the Gardiner-Masur boundary is not homeomorphic to the sphere of dimension $6g-$$a$$7+2n$.

Proof. Otherwise, ffom Borsuk-Ulam theorem (cf. [16]), the inclusion$\mathcal{P}\mathcal{M}\mathcal{F}arrow$should be surjective, because PMF is homeomorphic to the sphere ofdimension $6g-7+2n$ (cf. [3]).
$\partial_{GM}T(X)$$\square |$On the other hand, if $X$ is either a four punctured sphere or a once punctured
torus, $\partial_{GM}T(X)$ coincides with, and hence $\partial_{GM}T(X)$ is homeomorphic toa circle (i.e. the one-dimensional sphere). For instance, see [20] for the proof.
$\mathcal{P}\mathcal{M}\mathcal{F}$52
3. THE INTERSECTION NUMBER
3.1. Motivation. In [25], we develop the extremal length geometry on Teichm\"uUerspace via intersection number (cf. Theorem 3.2). This study is motivated from thecomparison with the Thurston compactification. Namely, to define the Thurston
and $y\in T(X)$ is recognized as thecompactification, the hyperbolic length of“intersection number” between a marked Riemann surface and a simple closed(cf. [3]). With this recognition, any point of $T(X)$ is thought of an
curve
of functions on the set of simple closed curves. The
element of the space
Thurston compactification is defined by taking the closure of the image of $T(X)$. Thurston‘s celebrated theorem telk us thatofin the projective space. This setting is also
the boundary defined by this closure coincides with(cf. [1]).
currents
geodesic
work
on
the
Bonahonwell-understood from
The main goal here is to unify the geometric structures (or geometric quantities)
on a surface via “intersection number”. Flom our observation (Theorem 3.2), we
can define the intersection number in the category of the extremal length geometry.
Indeed, in this category, we observe that the intersection number between $z\in$$T(X)$ (with respect to the base point) is equal to $\exp(-2(y|z\rangle_{x0})$ , wherewith respect tois the Gromov product between and with the base point
Teichm\"uller
theofgeometrythelinks
Teichm\"uller
observationThis
distance.
thedistance (an analytic aspect in Teichm\"uller theory) and the geometry of measured
foliations via intersection number (an topological aspect in Teichm\"uller theory).
$\alpha\in S$$y$$\alpha\in S$$S$$\mathbb{R}_{+}^{S}$$P\mathbb{R}_{+}^{S}$$\mathbb{R}_{+}^{S}$$PM\mathcal{F}$

$s$

$y,$

$\langle y|z\rangle_{xo}$

$z$

$y$

$x_{0}$

3.2. Thurston theory for the extremal length geometry. For $y\in T(X)$ , weondefine a continuous function$\mathcal{M}\mathcal{F}$$\mathcal{E}_{y}$(3.1)$\mathcal{E}_{y}(F)=\{\frac{Ext_{y}(F)}{K_{y}}\}^{1/2}$

the intersection number between
where $K_{y}=\exp(2d_{T}(x0,y))$ . We will thuink
depends
theorem,the function
following$y\in T(X)$ andthein. Notice.
on the choice of the base point since so does$\mathcal{E}_{y}(F)$$F\in \mathcal{M}\mathcal{F}$$\mathcal{E}_{y}$$x_{0}$$K_{y}$

Theorem 3.1 (cf. [21] and [25]). For any $p\in c1_{GM}(T(X))$ , there is a uniquewith the following properties.
oncontinuous function
represents .
(1) The function
tends to $p\in c1_{GM}(T(X))$ , the fiunctions(2) For a sequence.
uniformly on any compact set of
converges to$\mathcal{M}\mathcal{F}$$\mathcal{E}_{p}$$[S\ni\alpha\mapsto \mathcal{E}_{p}(\alpha)]\in \mathbb{R}_{+}^{S}$$p$$\{y_{n}\}_{n}\subset T(X)$$\mathcal{E}_{y_{n}}$$\mathcal{M}\mathcal{F}$$\mathcal{E}_{p}$(3)(4) For$\max_{F\in \mathcal{M}F_{1}}\mathcal{E}_{p}(F)=1$$[G]\in \mathcal{P}\mathcal{M}\mathcal{F}$.

,$\mathcal{E}_{[G]}(F)=\frac{i(F,G)}{Ext_{x_{0}}(G)^{1/2}}$

for$F\in M\mathcal{F}$.

Consider the mapping$\Psi_{GM}$: cl$GM(T(X))\ni p\mapsto[S\ni\alpha\mapsto \mathcal{E}_{p}(\alpha)]\in \mathbb{R}_{+}^{S}$

From (3.1), the mapping
Namely,$\Psi_{GM}$.

is a lift of the Gardiner-Masur embedding$\Phi_{GM}(y)=projo\Psi_{GM}(y)$$\Phi_{GM}$.

53
HIDEKI MIYACHI

for $y\in T(X)$ , where proj:$\mathbb{R}_{+}^{S}arrow P\mathbb{R}_{+}^{S}$

is the projection. Let$C_{GM}=proj^{-1}(c1_{GM}(T(X)))\cup\{0\}\subset \mathbb{R}_{+}^{S}$

Notice $homPM\mathcal{F}\subset\partial_{GM}T(X)$ that
becauseis a lift of.

$C_{GM}$$\Psi_{GM}$

Theorem 3.2 (uous function$\mathcal{E}_{p}$$M\mathcal{F}\subset C_{GM}$.

. Furthermore, $\Psi_{GM}(clGM(T(X)))\subset$$\Phi_{GM}$is an intersection number (cf. [25])). There$i(\cdot,$$\dot{u}$a unique contin-$\cdot):C_{GM}\cross C_{GM}arrow \mathbb{R}$with the following properties.
(i) For any $y\in T(X)$ , the projective class of the function
is exactly the image $ofy$ under the Gardiner-Masur embedding. In addition,$S\ni\alpha\mapsto i(\Psi_{x_{O}}(y), \alpha)$$i(\Psi_{GM}(p), F)=\mathcal{E}_{p}(F)$for $p\in c1_{GM}(T(X))$ and$F\in\lambda 4\mathcal{F}$.

(ii) For $b\in C_{GM},$ (a, b) $=i(b, a)$ .
(iii) For $b\in C_{GM}$ and $s\geq 0,$ $i(ta, sb)=tsi(a, b)$ .
(iv) For any $y,$ $z\in T(X)_{f}$
$i$

$a,$

$t,$

$a,$

$i(\Psi_{xo}(y), \Psi_{x0}(z))=\exp(-2\langle y|z\rangle_{x0})$

where
is the Gromov product of andrespect to the Teichmuller distance , that is:$\langle y|z\rangle_{x_{0}}$

$y$

$z$

.
with base point$x_{0}$

with$d_{T}$$\langle y|z\rangle_{xo}=\frac{1}{2}(d_{T}(x_{0}, y)+d_{T}(x_{0}, z)-d_{T}(y, z))$(v) For $F,$.

, the value $i(F, G)$ is equal to the original geometricintersection number between $F$ and .
$G\in\lambda 4\mathcal{F}\subset C_{GM}$$G$

As a corollary, we obtain an alternate approach to the characterization of theisometry group of $(T(X), d_{T})$ (cf. [25]). Namely, we can see that with few exception,
the isometry group of $(T(X), d_{T})$ is canonically isomorphic to the extended mappingclass group. This type of the characterization was already given by Royden [28],Earle-Kra [4], Earle-Markovic [5], and Ivanov [11].

4. BUSEMANN POINTS
Let be an unbounded set inwith $0\in T$ . A mapping : $Tarrow T(X)$
is said to be an almost geodesic ray if for any
there is an $N>0$ such that$\gamma(0)=x_{0}$ and$T$$[0, \infty)$$\gamma$$\epsilon>0$$|d_{T}(\gamma(t),\gamma(s))+d_{T}(\gamma(s),\gamma(0))-t|<\epsilon$

for all $t>s>N$ . By definition, any geodesic ray emanating
is an almost
geodesic ray.
In [14], L. Liu and W. Su observed that the Gardiner-Masur compactification is
canonically identified with the horofunction compactification of $T(X)$ with respect(cf. Gromov [7]). Combining Rieffel result in [27], they showed that any
toalmost geodesic ray has the limit in the Gardiner-Masur boundary (see also [24] fora proof from Teichm\"uller theory).
The boundary point $p\in\partial_{GM}T(X)$ is called a Busemann point if it is the limit
point of some almost geodesic ray.
$x_{0}$$d_{T}$$s$

Theorem 4.1 (cf. [24]). The Gardiner-Masur boundary contains a point which is
not a Busemann point.

54
Since the horofunction boundary of any CAT(0)-space consists of Busemannpoints (cf. [2]), we deduce the following corollary, which was first observed by H.
Masur in [17].
Corollary 4.1. Teichmuller space equipped with the Teichmuller distance is not a
CAT (0) $-spaoe$ .

5. LIPSCHITZ ALGEBRA
Lipschitz functions on a metric space are basic functions for investigating thegeometry of the metric space. In [22], we develop an algebraic structure of the
Lipschitz algebra on $(T(X), d_{T})$ and give a relation between the Gardiner-Masur
compactification and the compactification, which we call Q-compactification, defined with a subset of the Lipschitz algebra.
$Q$

5.1. Lipschitz algebra. Let $[F]$ be a projective measured foliation. Consider thefunction.

$\ell_{F}(y)=\frac{1}{4}(\log Ext_{y}(F)-\log Ext_{x0}(F)-2d_{T}(x_{0}, y))$depends only on the projective class of
Notice that $P_{F}(x_{0})=0$ for all , and. The function is a non-positive l-Lipschitz function on $T(X)$ with respect tois not bounded below, we consider a truncation
Teichmmler distance. Since
$F$

$F$

$\ell_{F}$$\ell_{F}$$\ell_{F}$$\ell_{F:a}=\ell_{F}\vee a=\sup\{l_{F}, a\}$for $a<0$ to obtain a bounded Lipschitz function.
of projective measured foliations and a set
For a subset in the spacein (-00, , we define a family$\mathcal{L}_{0}(\Sigma,T_{0})=\{\ell_{F:a}|[F]\in\Sigma, a\in T_{0}\}$ ,$\mathcal{P}\mathcal{M}\mathcal{F}$$\Sigma$$T_{0}$$0]$

We first study the algebraic structure of the Lipschitz algebra. Indeed, in [22],we show a version of the Stone-Weierstrass theorem for the space $BL_{0}(T(X), F)$of bounded F-valued Lipschitz functions on $T(X)$ which vanish at , where is$F$$x_{0}$

either$\mathbb{R}$or C.

Theorem 5.1 (Stone-Weierstrass theorem for $BL_{0}(T(X),F)$ (cf. [22])). Let
be a self-adjoint, norm-closed and order-complete subalgebra in $BL_{0}(T(X), F)$ . If(-00, such thatand an unbounded set
there are a dense subset in$\mathcal{A}=BL_{0}(T(X),F)$, then.
$\mathcal{A}$$\Sigma$$\mathcal{P}\Lambda t\mathcal{F}$$0]$$T_{0}\subset$$\mathcal{L}_{0}(\Sigma,T_{0})\subset \mathcal{A}$is said to be
be a subspace of either Lip $(T(X), F)$ or $BL_{0}(T(X),F)$ .
. A self-adjoint
is infor any
self-adjoint if the complex conjugate
is, by definition, order-complete if every norm-bounded directed net
subspaceof real valued functions in has a least upper bound in , to which it converges
in $A$
pointwise. Finally,is said to be norm-closed if whenever a sequence$g\inA$(cf. e.g. [29] and [30]).
converges to in norm, thenLet$\mathcal{A}$$\mathcal{A}$$\overline{f}$$f\in \mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\{f_{n}\}_{n}$$\mathcal{A}$$g$

5.2. Q-compactification. A Hausdorff compactification of a Hausdorff space $M$
is a Hausdorff compact space which contains, as a dense subset, the image of $M$
Y. We always identiwith its image
under a fixed homeomorphism :$M$$f(M)$ , and we say thattheas a dense subset. We denote bycontains
closure of $Y-M$ (cf. [15]).
Let $M$ be a non-compact Hausdorff space, and let be a nonvoid set of continuous functions on $M$ with each $f\in Q$ having its range contained in a compact
$Y$

$f$

$Mrightarrow$$\theta M$$Y$$\Delta Y$

$Q$

55

HIDEKI MIYACHI

Hausdorff space . Let
be a product space. The evaluation map$e:Marrow S_{Q}$ is defined by $e(x)(f)=f(x)$ for all $f\in Q$ . Set$S_{Q}= \prod_{f\in Q}S_{f}$$S_{f}$$\triangle^{Q}M=\cap${ $\overline{e(X-K)}|K$ compact,

and let $c1_{GM}(M)^{Q}$ be the disjoint unioncompact set $K\subset M$ , we set$M\cup\triangle$$K\subset M$}. Given an open set$U_{K}=(U\cap\Delta)\cup(e^{-1}(U)-K)$

$U$

in

$S_{Q}$

and a.

If is the topology on $c1_{GM}(M)^{Q}$ generated by the base consisting of all open sets
in $M$ and all the sets
, thenis called the Q-compactification of$M$ . By definition, $M$ is open
in $c1_{GM}(M)^{Q}$ since contains the topology of $M$ .
$\mathfrak{T}$$U_{K}$$(c1_{GM}(M)^{Q}, \mathfrak{T})$$\mathfrak{T}$Theorem 5.2 (Gardiner-Masur compactification revisited). Let be a dense subset ofandan unbounded set in $(-\infty, 0]$ . Set. Then,the identity mapping
extends to a homeomorphism from the Qcompactification to the Gardiner-Masur compactification.
$\Sigma$$\mathcal{P}M\mathcal{F}$$T_{0}$$Q=\mathcal{L}_{0}(\Sigma,T_{0})$$\mathcal{T}(X)arrow \mathcal{T}(X)$

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–)$|$

DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, OSAKA UNIVERSITY, MACHIKANEYAMA
1-1, TOYONAKA, OSAKA 560-0043, JAPAN