Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory

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by  Athanase Papadopoulos

SOME LIPSCHITZ MAPS BETWEEN HYPERBOLIC SURFACES
WITH APPLICATIONS TO TEICHMU¨LLER THEORY
ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
Abstract. IntheTeichmu¨llerspaceofahyperbolicsurfaceoffinitetype,we
constructgeodesiclinesforThurston’sasymmetricmetrichavingtheproperty
0
that when they are traversed in the reverse direction, they are also geodesic
1
lines(uptoreparametrization). Thelinesweconstructarespecialstretchlines
0
inthesenseofThurston. Theyaredirectedbycompletegeodesiclaminations
2
that are not chain-recurrent, and they have a nice description in terms of

n Fenchel-Nielsencoordinates. Atthebasisoftheconstructionarecertainmaps
a withcontrolledLipschitzconstants betweenright-angledhyperbolichexagons
J havingthreenon-consecutiveedgesofthesamesize. Usingthesemaps,weob-

3 tain Lipschitz-minimizing maps between hyperbolic particular pairs of pants
1 and, more generally, between some hyperbolic sufaces of finite type with ar-
bitrarygenusandarbitrarynumberofboundarycomponents. TheLipschitz-

] minimizingmapsthatwecontructaredistinctfromThurston’sstretchmaps.
T
G AMSMathematics SubjectClassification: 32G15;30F30;30F60.
Keywords: Teichmu¨llerspace, surfacewithboundary, Thurston’s asymmetric
.
h metric, stretch line, stretch map, geodesic lamination, maximal maximally
t stretchedlamination,Lipschitzmetric.
a
m
[

1 1. Introduction
v
8 In this paper, we prove some results on Thurston’s asymmetric metric on Te-
8 ichm/”uller space. This metric was introduced by Thurston in his paper ??.
0 we start by constructing Lipschitz homeomorphisms with controlled Lipschitz
2 constantbetweensymmetric right-angled hyperbolic hexagons,thatis,convexright-
.
1 angledhyperbolichexagonshavingthreenon-adjacentedgesofequallength. Using
0 these Lipschitz homeomorphisms, we obtain, by doubling the hexagons, Lipschitz
0
homeomorphisms between symmetric hyperbolic pairs of pants, that is, hyperbolic
1
pairs of pants which have three geodesic boundary components of equal lengths.
:
v These Lipschitz homeomorphisms between symmetric pairs of pants are extremal
i
X in the sense that their Lipschitz constant is minimal among all Lipschitz constants
of homeomorphismsin the same isotopyclass. But these Lipschitz extremalhome-
r
a omorphisms between pairs of pants are not stretch maps in the sense of Thurston.
ByvaryingtheLipschitzconstantsofthehomeomorphismsweconstruct,weobtain
apathintheTeichmu¨llerspaceofthepairofpantswhichactuallycoincideswitha
stretch line in the sense of Thurston, and we exploit the properties of such stretch
lines.
We recallthatstretchlines aregeodesicswithrespectto Thurston’sasymmetric
metric, defined by minimizing the Lipschitz constant between marked hyperbolic
surfaces.
Bygluingpairsofpantsalongtheirboundarycomponents,andbycombiningthe
mapsweconstructbetweenpairsofpants,weobtainstretchlinesintheTeichmu¨ller
spaceofhyperbolicsurfacesoffinite type,ofarbitrarygenusandofarbitrarynum-
berofboundarycomponents,whicharealsogeodesics(uptoreparametrization),for
Date:January13,2010.
1
2 ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
Thurston’s asymmetric metric, when they are traversed in the opposite direction.
These are the first examples we know of such geodesics for this metric.
We also recall that by a result of Thurston, given any two points g and h in
Teichmu¨ller space, there is a unique maximally stretched chain-recurrent geodesic
laminationµ(g,h)fromgtohwhichismaximal(withrespecttoinclusion),andthat
if g and h lie in that order on a stretch line directed by a complete chain-recurrent
geodesic lamination µ, then µ(g,h) = µ. We obtain the following results that are
variations on this theme: We show that if two elements g and h in Teichmu¨ller
space lie (in that order) on a stretch line we construct, the lamination µ(g,h) is
strictlysmallerthanthelaminationthatdirectsthatline,andthatthereareseveral
(nonchain-recurrent)maximalmaximallystretchedgeodesiclaminationsfromg to
h. In other words,the stretch lines we constructare directed by complete geodesic
laminationsthatarenotchain-recurrent,andunlikethe chain-recurrentcase,these
laminations are not uniquely defined.
2. Thurston’s stretch maps between hyperbolic ideal triangles and
between pairs of pants
Inthissection,werecallthedefinitionofastretchmapbetweenhyperbolicideal
triangles and between pairs of pants. This construction is due to Thurston (see
[9]).
We start with a stretch map from a hyperbolic ideal triangle to itself.
Considera hyperbolic idealtriangle equipped with the partialfoliationby horo-
cyclic segments that are perpendicular to the boundary. Up to isometry, there is
a unique such object. There is a non-foliated region at the center of the triangle,
bounded by three pieces of horocycles (see Figure 1). This horocyclic foliation is
equippedwithanaturaltransversemeasure,whichischaracterizedbythefactthat
thetransversemeasureassignedtoanyarccontainedinanedgeoftheidealtriangle
coincides with the Lebesgue measure induced by the hyperbolic metric.
The non-foliated region of a hyperbolic triangle intersects each edge of the tri-
angle at a point called the center of that edge.
horocycles
perpendicular
to theboundary
non-foliated
horocyclic arc
region
of length one
Figure 1. The horocyclic foliation of an ideal triangle.
Let T be the hyperbolic ideal triangle equipped with its horocyclic measured
foliation, and consider a real number k 1. The stretch map of magnitude k of T
≥
is a homeomorphism f :T T satisfying the following properties:
k
→
(1) The restriction of f to the non-foliated region of T is the identity map of
k
that region.
(2) OneachedgeofT,f sendsanypointatdistancexfromthecenterofthat
k
edge to a point at distance kx.
LIPSCHITZ MAPS 3
(3) The map f preserves the horocyclic foliation of T; that is, it sends leaves
k
to leaves.
(4) On each leaf of the horocyclic foliation, f contracts linearly the length of
k
that leaf.
By gluing stretch maps between ideal triangles we construct stretch maps be-
tween hyperbolic pairs of pants.
A hyperbolic pair of pants is a sphere with three open disks removed, equipped
with a hyperbolic metric in which the three boundary components are closed
geodesics(the liftofsuchacurveto the hyperbolicuniversalcoverseenasasubset
of the hyperbolic plane H2 is a geodesic in H2).
LetP beahyperbolicpairofpants. Wechooseacompletegeodesiclaminationλ
in P. Such a complete geodesic lamination necessarily consists of three disjoint bi-
infinite geodesics that spiral around the boundary components of P, decomposing
thatsurfaceintotwohyperbolicidealtriangles. The horocyclicmeasuredfoliations
of the two ideal triangles fit together smoothly since they are both perpendicular
to the edges of the ideal triangles, and therefore they form a Lipschitz line field on
the surface. Foreachk 1,considerastretchmapofmagnitudek definedoneach
≥
oftheidealtrianglescomposingP. WeobtainanewhyperbolicpairofpantsP by
k
gluingtheidealtrianglestogetheralongtheirboundariesaccordingtoidentifications
thatare compatible withthe stretchmaps. This defines a homeomorphismfromP
toanotherhyperbolicpairofpantsP ,whichis calledastretch map(ofmagnitude
k
k) from P to P .
k
The above construction can be repeated on several copies of hyperbolic pairs of
pants. Bygluingtogetherthesepairsofpantsaccordingtotheidentificationsgiven
by the stretch maps, we obtain a stretch map of magnitude k from a hyperbolic
surface S to another S . Note that the complete geodesic laminations giving the
k
decompositions into ideal triangles of the pairs of pants in S give, together with
the pants decomposition of S, a complete geodesic lamination on the surface S.
Remark 2.1. The reader should be aware that stretch maps are actually defined
in a much wider generality than the one presented here. The underlying complete
geodesic lamination giving the decomposition of the surface into ideal triangles
can be chosen arbitrarily among the complete geodesic laminations and it is not
necessarily the completion of a geodesic pants decomposition as above. However,
in this paper, we shall only need the special case of stretch maps described above.
3. Extremal Lipschitz maps between symmetric right-angled
hexagons
Given two metric spaces (X,d ) and (Y,d ) and a map f : X Y between
X Y
→
them, the Lipschitz constant Lip(f) of f is defined as
d f(x),f(y)
Lip(f)= sup Y R .
(cid:0)d x,y (cid:1) ∈ ∪{∞}
x=y X X
6 ∈
(cid:0) (cid:1)
We shall say that the map f is Lipschitz if its Lipschitz constant is finite.
Thestretchmapsf betweenhyperbolicidealtrianglesthatweconsideredinthe
k
last section are examples of Lipschitz homeomorphisms, with Lipschitz constant
equalto k. Note that the fact that this Lipschitz constantis at leastk can be seen
fromtheactionofthese mapsonthe boundaryoftheidealtriangles. Thefactthat
theLipschitzconstantisexactlykisimplicitinThurston’spaper[9]. Italsofollows
from the computations below (see Remark 3.4). By using these maps as building
blocks, we recalled in 2 how one obtains Lipschitz homeomorphismsof hyperbolic
§
4 ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
pairsofpantsand,moregenerally,ofhyperbolicsurfaces. Thesestretchmapshave
Lipschitz constants k.
Inthissection,weshalldefineLipschitzmapsbetweensomeparticularhyperbolic
right-angled hexagons, which will also have controlled Lipschitz constants, and
whichcanbe usedto define Lipschitz homeomorphismsbetweenspecialhyperbolic
pairs of pants, by gluing hyperbolic right-angledhexagons and taking the union of
Lipschitz maps between them. By gluing together these special pairs of pants in
anappropriatemanner,thiswilleventuallyyieldhomeomorphismsbetweenspecial
hyperbolic surfaces of arbitrary finite type, with controlled Lipschitz constants.
A symmetric right-angled hexagon is a geodesic hexagon H in the hyperbolic
plane H2 with three pairwise non-consecutiveedges having the same length. (Note
that this implies that the remaining three edges also have the same length.)
We considera symmetric right-angledhexagonH,and we choosethree pairwise
non-consective edges of H, which we call the long edges. We denote their common
lengthby2L. Theotherthreenon-consecutiveedgesarecalledshort,andwedenote
their common length by 2l. An easy computation using well-known formulae for
right-angled hexagons gives
(1) 2sinh(l)sinh(L)=1.
For each real number k 1, we let H be the symmetric right-angled hexagon
k
≥
obtained by multiplying the lengths of the long edges of H by the factor k. We
note that this property determines the isometry type of H in a unique way. We
k
call the edges of H that are the images of the long edges of H by this dilatation
k
map the long edges of H and we denote their common length by 2L . We let 2l
k k k
denote the length of the other edges of H , which we call the short ones.
k
In this section, all the maps between symmetric right-angled hexagons that we
shall consider will be homeomorphisms sending the long (respectively short) edges
to the long (respectively short) edges, and in general we shall not repeat this con-
dition.
The three lengths of any three non-consecutive edges of H (respectively of H )
k
satisfy the triangle inequality. Therefore, we can equip H (respectively H ) with
k
a partial measured foliations F (respectively F ) whose leaves are loci of equidis-
k
tant points from the short edges. In the hyperbolic plane, equidistant points from
geodesics are classicaly called hypercycles, and we shall use this terminology. The
foliations of H (respectively H ) by hypercycles are shown in Figure 2, and such
k
foliations have already been considered by Thurston in his compactification the-
ory of Teichmu¨ller space (see [2, expos´e 6]). There is a non-foliated region of F
(respectively F ) at the center of H (respectively H ).
k k
TheintersectionnumberofF (respectively,F )withanedgeofH (respectively,
k
H )iseither2Lor0(respectively,2kLor0)dependingonwhethertheedgeislong
k
or short.
WealsoequipH (respectivelyH )withthepartialfoliationG(respectivelyG )
k k
whose leaves are geodesic arcs perpendicular to the leaves of F (respectively F ).
k
In Theorem 3.3, we shall construct a map, h : H H which (leafwise) sends
k k
→
F to F , and G to G and whose Lipschitz constant is k. Such a map is Lipschitz-
k k
extremalinitshomotopyclassrelativetotheboundary,sincetheLipschitzconstant
of any map f : H H which sends long (respectively short) edges of H to long
k
→
(respectively short) edges of H is bounded below by k. The Lipschitz-extremal
k
mapsweshallconstructare“canonical”inthesensethattheypreserveapairofhy-
percyclic/geodesic foliations, and they are reminiscent of Thurston’s stretch maps
between ideal triangles. In some precise sense that we specify below, Thurston’s
stretch maps between ideal triangles are limits of the Lipschitz-extremal maps be-
tween symmetric hexagons.
LIPSCHITZ MAPS 5
Beforedefining the maph ,wemakeageometricalremark. Considerthe family
k
of all symmetric right-angled hexagons H as k varies from 1 to infinity. Each of
k
these hexagons has a center which is the center of the rotation that permutes each
triple ofnon-consecutiveedges. Foreachsuchhexagon,considerthe three geodesic
raysemanating from its center and meeting the shortedges perpendicularly. Place
all the hexagons H in the hyperbolic plane so that all their centers coincide and
k
such that all the above geodesic rays coincide as well. Now for each such hexagon
H , consider the associated extended hexagon H defined as the region of infinite
k k
areaenclosedbythethreegeodesicsinH2extendingthelongedgesofH . Itfollows
b k
fromEquation(1)thatasL decreases,l increases,andconversely. Fromthis,we
k k
deduce that for any 1 k k′, we have Hk′ Hk.
≤ ≤ ⊂
We alsonote thatas k tends to infinity, the extendedhexagonH as wellas the
b b k
hexagon H itself converge, in the Hausdorff topology associated to the Euclidean
k b
metric (using as in Figure 3 the disk-model of the hyperbolic plane) to an ideal
triangle. Likewise,ask ,themeasuredfoliationF convergestothehorocyclic
k
→∞
foliation of the ideal triangle (represented in Figure 1) and the non-foliated region
of F converges to the non-foliated region of that horocyclic foliation.
k
The following two lemmas will be used in the proof of Theorem 3.3 below.
Lemma 3.1. For k′ >k 1, the non-foliated region of Fk′ is strictly contained in
≥
the non-foliated region of F .
k
Proof. We work in the disk model of the hyperbolic plane. The statement will
follow from the construction of the symmetric hexagons, represented in Figure 3.
In the upper part of that figure, the hexagon H (also with its edges extended) is
k
drawn in bold lines, and the hexagon Hk′ (with its edges extended) is drawn in
dashed lines. We have chosen the hexagons to be symmetric with respect to the
Euclidean center O of the unit disk. In the upper figure, the point p (respectively
q)is the Euclideancenter ofthe hypercyclethatis onthe boundaryofnon-foliated
regionofHk (respectivelyHk′). The pointa (respectivelyb)is avertexofthe non-
foliated region of Fk (respectively Fk′). A more detailed view of a region drawn
in the the upper part of Figure 3 is represented in the lower part. The point a
′
(respectively b) is the center of a boundary hypercycle of the non-foliated region
′
of Fk (respectively Fk′). The Euclidean triangles Opa and Oqb are homothetic
by a Euclidean homothety of center O and factor < 1. This homothety sends
the Euclidean circle arc aa to the Euclidean circle arc bb. Thus, there exists a
′ ′
Euclidean homothety of center O that sends the non-foliated regionof Hk′ strictly
into the non-foliated region of H , which proves the lemma. (cid:3)
k
Figure 2. The foliation by curves equidistant to the short edges of a
symmetricright-angledhexagon. Thecentralregionisnotfoliated, and
it is boundedby three hypercycleswhich meet each other tangentially.
6 ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
Lemma 3.2. In the upper half-plane model of the hyperbolic plane, consider the
geodesic represented by the imaginary axis iR+ = ir,r > 0 , and a hypercycle
{ }
making an angle π θ with this geodesic, with 0 < θ < π/2. Let ℓ be the length
2 − 1 1
of a geodesic arc α joining perpendiculary the vertical geodesic and the hypercycle.
Then, we have
cosθ =tanhℓ.
1
Proof. We refer to Figure 4. We parametrize the geodesic arc α by the map
α:[θ ,π/2] H2
1
→
θ (cosθ,sinθ).
7→
Usingtheformulafortheinfinitesimallengthelementintheupperhalf-planemodel,
we can write
π/2 α(θ) π/2 dθ
′
ℓ= k k dθ = .
Z Im(α(θ)) Z sinθ
θ1 θ1
Computing the integral, we find
e ℓ =tan(θ /2)
− 1
b
′
a
′
q
p
b
a
O b′ a′ q p
b
a
Figure 3. The upper figure represents, in bold lines, a symmetric
right-angledhexagonHk,andindashedlines,asymmetricright-angled
hexagon Hk′ with k′ > k, together with their extensions Hˆk and Hˆk′.
The fact that the non-foliated region of the symmetric hexagon H′ is
k
included in the non-foliated region of the symmetric hexagon Hk, for
k′ >k,asitisrepresentedintheupperfigure,canbededucedfromthe
Euclideanconstructioninthelowerfigure,inwhichthearcsaa′ andbb′
areontheboundariesofthenon-foliated regionsofHk andHk′ respec-
tively.
LIPSCHITZ MAPS 7
and after transformation we obtain
cosθ =tanhℓ.
1
(cid:3)
We now construct the map h :H H .
k k
→
From the inclusion of the non-foliatied regionof H into the non-foliated region
k
ofH forallk 1(Lemma3.1),itwillfollowthatthemaph weshallconstructcan
k
≥
be chosen to be contracting from the non-foliated region of H to the non-foliated
region of H .
k
To define the map h , it suffices to do it in a component of the foliated region
k
of H. Consider such a component. It is isometric to the region C in the upper
half-plane model of the hyperbolic plane defined in polar coordinates by
C = z =Reiθ : 1 R e2l, θ θ π/2 ,
1
{ ≤ ≤ ≤ ≤ }
where θ is chosen so that the geodesic parameterized by θ Reiθ, θ θ π/2,
1 1
7→ ≤ ≤
has length L.
From Lemma 3.2, we have
cosθ =tanhL.
1
Likewise, the image by h of the component C of the complement in H of the
k
non-foliated region is isometric to the region C in the upper half-plane model of
k
H2 given by
C = z =Reiθ : 1 R e2lk, θ θ π/2 ,
k k
{ ≤ ≤ ≤ ≤ }
where
cos(θ )=tanh(kL).
k
In these descriptions,the foliations F andF , aregivenby the hypercyclesdefined
k
by θ = cst, while the foliations G and G , are given by the geodesics defined by
k
R = cst. The short sides of C and C correspond to θ = π/2. Our map h maps
k k
a point A C which is at distance d from the short side of C to a point which is
∈
at distance kd from the short side of C . If the point A lies on the leaf of G which
k
cuts the short side of C at distance h, then the image of A by h belongs to the
k
leaf that cuts the short side of C at distance hl /l.
k k
We need to have an explicit formula for h in order to compute the norm of its
k
derivative.
LetAbeapointinCgiveninpolarcoordinatesby(R,θ). Denotethecoordinates
of the point h (A) C by (R,θ ). We also describe the points A and h (A) by
k k ′ ′ k
∈
their distances from the short sides, namely d and kd, and by their distances from
the lowest geodesic boundary of C and C , as above.
k
i ℓ
α
θ
1
Figure 4. ℓ is the length of a segment α joining perpendicularly the
vertical geodesic and the hypercycle making an angle θ1 with the hori-
zontal. Wehavecosθ1 =tanhℓ.
8 ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
Let us first compute R. The logarithm of R and of R are the distances of the
′ ′
points A and h (A) from the lowest geodesic boundary of C and C , respectively.
k k
By what has been previously said, we have
l
k
logR = logR.
′
l
Therefore,
R′ =Rlk/l.
Let us now compute θ . The same computation as for the formula giving θ
′ 1
establishes
1
sinθ = , or cosθ =tanhd.
coshd
Therefore,
1
d=argcosh .
sinθ
(cid:16) (cid:17)
Now,
θ =arccos(tanh(kd)).
′
Thus we get the following formula for h , viewed as a map from C to C ,
k k
1
h (R,θ)= Rlk/l,arccos(tanh kargcosh ) .
k
(cid:18) (cid:0) (cid:16)sinθ(cid:17)(cid:1) (cid:19)
Nowthatthehomeomorphismh isdefined,weproceedtoshowthatitsLipschitz
k
constant equals k. For this, we compute the norm of its derivative.
We easily have
∂R l ∂R ∂θ
′ = kR(lk/l)−1, ′ =0, ′ =0.
∂R l ∂θ ∂R
1
Since arccos(x)= , we get
′
−√1 x2
−
∂θ 1 ∂ 1
′
= tanh kargcosh
∂θ − 1 tanh2(kargcosh 1 )∂θ(cid:16) (cid:0) (cid:16)sinθ(cid:17)(cid:1)(cid:17)
r − (cid:16)sinθ(cid:17)
1 ∂ 1
= cosh(kargcosh ) tanh kargcosh .
− (cid:16)sinθ(cid:17) ∂θ(cid:16) (cid:0) (cid:16)sinθ(cid:17)(cid:1)(cid:17)
1
Now, since tanh′(x)= , we have
cosh2(x)
∂ 1 k ∂ 1
tanh kargcosh = argcosh .
∂θ(cid:16) (cid:0) (cid:16)sinθ(cid:17)(cid:1)(cid:17) cosh2(kargcosh 1 )∂θ (cid:16)sinθ(cid:17)
sinθ
(cid:16) (cid:17)
Hence, since argcosh′(x)= 1 ,
√x2 1
−
∂θ k ∂ 1
′
= − argcosh
∂θ cosh(kargcosh 1 )∂θ (cid:16)sinθ(cid:17)
sinθ
(cid:16) (cid:17)
k 1 ∂ 1
= −
cosh(kargcosh 1 ) 1 1∂θsinθ
(cid:16)sinθ(cid:17) qsin2θ −
ksinθ cosθ
= .
cosθcosh(kargcosh 1 )sin2θ
sinθ
(cid:16) (cid:17)
Finally, we have
∂θ k 1 1
′ = cosh(kargcosh ) − .
∂θ sinθ sinθ
h (cid:16) (cid:17) i
LIPSCHITZ MAPS 9
The last partial derivative can also be written as
∂θ coshd
′
=k .
∂θ cosh(kd)
We now proceed to compute the norm of the differential dh . Recall that the
k
square of the norm of a vector (dx,dy) in the tangent plane T (H2) of the upper
z
half-plane model of the hyperbolic plane is given by
dx2+dy2
,
y2
where z =x+iy. In polar coordinates, this is written as
dR2+R2dθ2
.
R2sin2θ
Let V = (V ,V ) be a non-zero tangent vector at the point (R,θ). We compute
R θ
the norm of the differential dh at the point (R,θ). We have
k
∂h ∂h
(dh ) V 2 = ( kdR+ kdθ) V 2
|| k (R,θ)· || || ∂R ∂θ · ||
1 ∂R ∂R ∂θ ∂θ
= ′V + ′V 2+R2 ′V + ′V 2
R2sin2θ(cid:16)(cid:0)∂R R ∂θ θ(cid:1) (cid:0)∂R R ∂θ θ(cid:1) (cid:17)
1 ∂R ∂θ
= ′V 2+R2 ′V 2 .
R2sin2θ(cid:16)(cid:0)∂R R(cid:1) (cid:0)∂θ θ(cid:1) (cid:17)
Note that
1
V 2 = (V2+R2V2).
|| || R2sin2θ R θ
Therefore, since ||(dhk)(R,θ)||=supV6=0 ||(dhk|)|(VR|,|θ)·V||, we get
∂R′V 2+R2 ∂θ′V 2
(dh ) 2 = sup ∂R R ∂θ θ
|| k (R,θ)|| V=0(cid:16)(cid:0) V(cid:1)R2+R2V(cid:0)θ2 (cid:1) (cid:17)
6
∂R′V 2+ ∂θ′RV 2
= sup ∂R R ∂θ θ
(cid:0) V2(cid:1)+(R(cid:0)V )2 (cid:1)
V=0(cid:16) R θ (cid:17)
6
∂R′ 2 ∂θ′ 2
= sup V + RV
R θ
∂R ∂θ
VR2+(RVθ)2=1(cid:16)(cid:0) (cid:1) (cid:0) (cid:1) (cid:17)
∂R 2 ∂θ 2
′ ′
= max , .
∂R ∂θ
n(cid:16) (cid:17) (cid:16) (cid:17) o
We have
1 R e2l.
≤ ≤
Since l /l 1, we get
k
≤
1 Rlk/l−1 e2(lk−l) >0,
≥ ≥
that is,
∂R
′
0 1.
≤ ∂R ≤
Now, since
∂θ cosh(d)
′
=k ,
∂θ cosh(kd)
we get, for all (R,θ),
∂θ
′
0 k
≤ ∂θ ≤
10 ATHANASEPAPADOPOULOSANDGUILLAUMETHE´RET
and the equality ∂θ′ = k is realized at the points d = 0, that is, on the short side
∂θ
of C. Therefore, we obtain
sup (dh ) =k.
k (R,θ)
|| ||
(R,θ) C
∈
The supremum of the norm of dh bounds from above the Lipschitz constant of
k
h : If x,y are two points of C and if γ is the geodesic path from x to y, we get
k
d(x,y)
d(h (x),h (y)) l(h (γ))= (dh ) γ (t) dt sup (dh ) d(x,y).
k k k k γ(t) ′ k z
≤ Z0 || · || ≤ z || ||
Therefore,ifL(h )denotestheLipschitzconstantofh ,wegetfromwhatprecedes,
k k
L(h ) k.
k
≤
Since the long edges are dilated by the factor k, we have L(h ) k. Finally,
k
≥
L(h )=k.
k
Putting all pieces together, the map we constructed from H to H has Lipschitz
k
constant k.
We summarize the preceding construction in the following:
Theorem 3.3. The map h :H H is k-Lipschitz. Furthermore for any k <k,
k k ′
→
there is no k -Lipschitz map from H to H .
′ k
Proof. The first part follows from the construction. Since, by definition, a map
h : H H sends the long edges of H to the long edges of H , we immediately
k k k
get Lip(→h ) k. This proves the second part of the theorem. (cid:3)
k
≥
Remark 3.4. We already observed that, reasoning in the disk model of the hy-
perbolic plane and using the notion of Hausdorff convergence on bounded closed
subsets of that disk with respect to the underlying Euclidean metric, we can make
a sequence of symmetric right-angled hexagons converge to an hyperbolic ideal
triangle, in such a way that the following three properties hold:
(1) The partial measured foliation of the hexagons by hypercycles converges to
the partial measured foliation of the hyperbolic ideal triangle by horocycles.
(2) The partial foliation of the hexagons by geodesics perpendicular to the fo-
liation by hypercycles converges to the partial foliation of the ideal triangle by
geodesics perpendicular to the horocycles.
(3) The non-foliated regionsof the hexagonsconvergeto the non-foliatedregion
of the ideal triangle.
Furthermore,forallk 1,wecanmaketheconvergenceofhexagonstotheideal
≥
triangle in such a way that k-Lipschitz maps f : H H converge uniformly on
k k
→
compact sets to the stretch maps f : T T between hyperbolic ideal triangles.
k
→
This shows in particular that the stretch maps f have Lipschitz constant k.
k
We note that Lipschitz maps between pairs of pants are also consideredby Otal
in his paper [4], in relation with the Weil-Peterssonmetric of Teichmu¨ller space.
4. Asymmetric metrics on Teichmu¨ller spaces of surfaces with or
without boundary
In this section, S is a surface of finite type (g,b), which may have empty or
nonempty boundary(g denotes the genusofS andb the number ofboundarycom-
ponents). WeassumethattheEulercharacteristicofS isnegative. Thehyperbolic
structuresweconstructonS aresuchthatallthe boundarycomponents areclosed
smooth geodesics. We denote by T(S) or by T the Teichmu¨ller space of S, that
g,b
is, the space of homotopy classes of hyperbolic metrics on that surface.