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GAFA, Geom. funct. anal.DOI 10.1007/s00039-008-0675-6ONLINE FIRST

c! Birkhauser Verlag, Basel 2008

GAFA Geometric And Functional Analysis

LINES OF MINIMA AND TEICHMULLER GEODESICS

Young-Eun Choi, Kasra Rafi and Caroline Series

Abstract. For two measured laminations !+ and !! that fill up a hy-perbolizable surface S and for t ! ("#,#), let Lt be the unique hy-perbolic surface that minimizes the length function etl(!+) + e!tl(!!) onTeichmuller space. We characterize the curves that are short in Lt andestimate their lengths. We find that the short curves coincide with thecurves that are short in the surface Gt on the Teichmuller geodesic whosehorizontal and vertical foliations are respectively, et!+ and e!t!!. By de-riving additional information about the twists of !+ and !! around theshort curves, we estimate the Teichmuller distance between Lt and Gt. Wededuce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is boundedindependently of t.

1 Introduction

Suppose that !+ and !" are measured laminations which fill up a hyper-bolizable surface S. The object of this paper is to compare two paths inthe Teichmuller space T (S) of S determined by !+ and !". The first isthe Teichmuller geodesic G = G(!+, !"), whose time t Riemann surfaceGt ! G supports a quadratic di!erential qt whose horizontal and verticalfoliations are !+

t = et!+ and !"t = e"t!", respectively [GM]. The sec-

ond is the Kerckho! line of minima L = L(!+, !") [Ke3]. At time t,Lt ! L is the unique hyperbolic surface that minimizes the length functionl(!+

t ) + l(!"t ) = etl(!+) + e"tl(!") on T (S). (Recall that G can be char-

acterized as the locus in T (S) where the product of the extremal lengthsExt(!+) Ext(!") is minimized [GM].) Lines of minima have many proper-ties in common with Teichmuller geodesics, see [Ke3], and have been shownto be closely linked to deforming Fuchsian into quasi-Fuchsian groups bybending, see [S2].

Keywords and phrases: Teichmuller geodesics, lines of minima, hyperbolic metric2000 Mathematics Subject Classification: 30F45, 30F60, 32G15, 57M50

2 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

We are interested in comparing the two trajectories G and L, in particu-lar, to see whether or not they remain a bounded distance apart. If both Gt

and Lt are contained in the thick part of T (S), it is relatively easy to showthat the Teichmuller distance between them is uniformly bounded indepen-dently of t, see Theorem 3.8. A more surprising fact is that, in general,the sets of short curves on G and L are the same. Writing lGt("), lLt(") forthe geodesic lengths of a simple closed curve " in the hyperbolic metricson Gt,Lt respectively, we proveTheorem A (Proposition 7.1 and Corollary 7.9). The set of short curveson Gt and Lt coincide. More precisely, there exist universal constants#1, . . . , #4 > 0 such that, for each t, lGt(") < #1 implies lLt(") < #2 andlLt(") < #3 implies lGt(") < #4.

Finding combinatorial estimates for the lengths of these short curvesoccupies the main part of the paper and leads to a coarse estimate of thedistance between Gt and Lt. It turns out that along both G and L there aretwo distinct reasons why a curve " can become short: either the relativetwisting of !+ and !" about " is large, or !+ and !" have large relativecomplexity in S \" (the completion of the surface S minus "), in the sensethat every essential arc or closed curve in S \" must have large intersectionwith !+ or !". Our results give su"cient control to construct exampleswhich show that Gt and Lt may or may not remain a bounded distanceapart.

The estimates for curves which become short along G are based on Rafi[R1]. For convenience we say a curve is ‘extremely short’ on a given surfaceif its hyperbolic length is less than some fixed constant #0 > 0 defined interms of the Margulis constant, see section 2.1. Rafi’s results implyTheorem B (Theorem 5.10). Suppose that " is extremely short on Gt.Then 1

lGt(")$ max

!Dt("), log Kt(")

".

The terms Dt(") and Kt(") correspond respectively to the relativetwisting and large relative complexity mentioned above. More precisely,

Dt(") = e"2|t"t!|d!(!+, !") ,

where t! is the balance time at which i(", !+t ) = i(", !"

t ) and d!(!+, !") isthe relative twisting, that is, the di!erence between the twisting of !+ and!" around ", see section 4.3. The term Kt(") depends on the (possiblycoincident) thick components Y1, Y2 that are adjacent to " in the thick-thindecomposition of Gt. Let qt be the area 1 quadratic di!erential on Gt whose

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 3

horizontal and vertical foliations are respectively !+t and !"

t . Associated toqt is a singular Euclidean metric; we denote the geodesic length of a curve$ in this metric by lqt($), see section 2.6. By definition

Kt(") = max#

%Y1

lqt("),

%Y2

lqt(")

$,

where %Yi is the length of the shortest non-trivial, non-peripheral simpleclosed curve on Yi with respect to the qt-metric, see section 5.3.

One of the main results of this paper is a similar characterization ofcurves which become short along L(!+, !"). We prove that the hyperboliclength lLt(") of a short curve in Lt is estimated as follows:Theorem C (Theorem 7.13). Suppose that " is extremely short on Lt.Then 1

lLt(")$ max

!Dt("),

%Kt(")

".

The main tool in the proof is the well-known derivative formula of Kerck-ho! [Ke2] and Wolpert [W1] for the variation of length with respect toFenchel–Nielsen twist, together with the extension proved by Series [S1] forvariation with respect to the lengths of pants curves.

To estimate the Teichmuller distance between two surfaces that have thesame set of short curves, one uses Minsky’s product region theorem [Mi3].To apply this, in addition to Theorems B and C, we need to estimate the Te-ichmuller distance between the hyperbolic thick components of Gt and Lt,and also the di!erence between the Fenchel–Nielsen twist coordinates cor-responding to the short curves in the two surfaces. In Theorem 7.10 andCorollary 7.11, we show that the Teichmuller distance between the corre-sponding thick components is bounded. In Theorem 6.2, we estimate thetwist of !+ and !" around " at Lt. Combined with the analogous estimatefor Gt proved in [R2], we are able to show that the contribution to theTeichmuller distance between Gt and Lt from the twisting is dominated bythat from the lengths, leading toTheorem D (Theorem 7.15). The Teichmuller distance between Gt andLt is given by

dT (S)(Gt,Lt) =12

log max#

lGt(")lLt(")

$± O(1) ,

where the maximum is taken over all curves " that are short in Gt.

Theorems B, C, and D enable us to construct the various examplesalluded to above. Because Kt(") can become arbitrarily large while Dt(")remains bounded, it follows that Gt and Lt do not always remain a bounded

4 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

distance apart. However, in the case in which S is a once-punctured torusor four-times-punctured sphere, it turns out that the quantity Kt(") isalways bounded and therefore that the two paths are always within boundeddistance of each other. These ideas are taken further in [CRS], where weshow that Lt is a Teichmuller quasi-geodesic.

The greater part of the work of this paper is contained in the proof ofTheorem C. It is carried out in several steps. First, using the derivativeformulae mentioned above, we show in Theorem 6.1 that the length may beestimated by a formula identical to that in Theorem C, except that Kt(")is replaced by another geometric quantity

Ht(") = sup"#B

lqt(&)lqt(")

.

Here B are those pants curves in a short pants decomposition of Lt (seesection 3.1) which are boundaries of pants adjacent to ", while as abovelqt denotes length in the singular Euclidean metric associated to qt. This isthe content of section 6.

We now need to compare Ht(") and Kt("). From the definition it isquite easy to show (Proposition 7.1) that Ht(") % Kt("). In particular,it follows that a curve that is short in Gt is at least as short in Lt. Next,we show in Proposition 7.8 that on a subsurface whose injectivity radiusis bounded below in Gt, the injectivity radius with respect to Lt is alsobounded below, perhaps by a smaller constant. The main point in the proofis Proposition 7.4, which shows that the hyperbolic metric on Lt not onlyminimizes the sum of lengths l(!+

t ) + l(!"t ), but also, up to multiplicative

error, it minimizes the contribution of l(!+t )+l(!"

t ) to each thick componentof the thick-thin decomposition of Gt. This proves Theorem A.

Having set up a one-to-one correspondence between the thick compo-nents of Gt and Lt, we show in Theorem 7.10 and Corollary 7.11 that theTeichmuller distance between corresponding thick components is bounded.Finally we are able to prove in Proposition 7.12 that Ht(") & Kt("), com-pleting the proof of Theorem C.

Prior to this paper, the only results related to the relative behavior of Gand L were some partial results about their behavior at infinity. Results ofMasur [Ma] (for Teichmuller geodesics) and of Dıaz and Series [DS] (for linesof minima) show that if either !± are supported on closed curves, or if !±

are uniquely ergodic, then G and L limit on the same points in the Thurstonboundary of T (S). In general, the question of the behavior at infinityremains unresolved, but see also [L] which shows that there are Teichmuller

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 5

geodesics G which do not converge in Thurston’s compactification of T (S).It is not hard to apply the results of this paper to show the same is trueof lines of minima in Lenzhen’s example; we hope to explore this in moredetail elsewhere.

The motivation for our approach stems in part from a central ingredientof the proof of the ending lamination theorem [BrCM]. Suppose that N isa hyperbolic 3-manifold homeomorphic to S ' R. The ending laminationtheorem states that N is completely determined by the asymptotic invari-ants of its two ends. A key step is to show that if these end invariants areinduced by the laminations !+ and !", then the curves on S which haveshort geodesic representatives in N can be characterized in terms of theircombinatorial relationship to !+ and !". (The relationship is expressedusing the complex of curves of S, details of which are not needed in whatfollows. Roughly speaking, a curve is short in N if and only if the distancebetween the projections of !+ and !" to some subsurface Y ( S is largein the curve complex of Y .) In [R1], Rafi found a similar combinatorialcharacterization which shows that the curves which are short in N are al-most, but not quite, the same as those curves which become short alongG(!+, !"). Our definition of Kt is closely related to Rafi’s study [R3] of therelationship between the thick-thin decomposition of a hyperbolic surfaceS and a quadratic di!erential metric on the same surface. The relationshipbetween these two metrics plays a key role throughout the paper.

The paper is organized as follows. In section 2, we recall some back-ground facts about lines of minima and Teichmuller geodesics. In section 3,we prove Theorem 3.8 mentioned above, which states that if both Gt andLt are contained in the thick part of T (S), then the Teichmuller distancebetween them is bounded. We hope that treating this special case sepa-rately early on will give some intuition about what needs to be done ingeneral. In section 4, we review twists and Fenchel–Nielsen coordinatesand in section 5, after reviewing some fundamental facts about quadraticdi!erential metrics, we derive Theorem B and state the estimates for twistsabout short curves proved in [R2]. In section 6, we prove Theorem 6.1 andderive estimates for twists about the short curves. Finally, in section 7, weprove Theorems A, C and D. Throughout the paper, we make use of severalbasic length estimates on hyperbolic surfaces. The proofs, being somewhatlong but relatively straightforward, are relegated to the Appendix.

Acknowledgments. We thank the referee for carefully reading the manu-script and leading us to clarify the exposition.

6 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

2 Preliminaries

Throughout, S is an orientable hyperbolizable surface of finite type, possi-bly with punctures but with no other boundary.

2.1 Thick-thin decomposition. Let S denote the set of free homo-topy classes of non-peripheral, non-trivial simple closed curves on S. If(S,') is a surface with hyperbolic metric ' and " ! S, we write l#(")for the hyperbolic length of the unique geodesic representative of " withrespect to '. The Margulis lemma provides a universal constant #M > 0such that all components of the #M-thin part of (S,') (i.e. the subset of Swhere the injectivity radius is less than #M) are horocyclic neighborhoodsof cusps or annular collars about short geodesics. The #M-thick part of thesurface is the complement of the thin part.

For our purposes, it is necessary to choose a constant #0 > 0 su"cientlysmaller than #M, in order that the #0-thick-thin decomposition of a surfacesatisfies certain geometric conditions. These conditions will be mentionedwhen the context arises, but we assume that #0 has been chosen once andfor all so that these conditions are met. If l#(") < #0, we shall say that "is extremely short in '.

2.2 Notation. Since we will be dealing mainly with coarse estimates,we want to avoid keeping track of constants which are universal, in thatthey do not depend on any specific metric or curve under discussion. Forfunctions f, g we write f $ g and f

!$ g to mean respectively, that thereare constants c > 1, C > 0, depending only on the topology of S and thefixed constant #0, such that

1cg(x) " C ) f(x) ) cg(x) + C and 1

cg(x) ) f(x) ) cg(x) .

The symbols &,!&, %,

!% are defined similarly. For a positive quantity X ,we often write X = O(1) instead of X & 1 to indicate X is boundedabove by a constant depending only on the topology of S and #0, and moregenerally we write X = O(Y ) to mean that X/Y = O(1) for a positivefunction Y .

2.3 Measured laminations. We denote the space of measured lami-nations on S by ML(S). Given any hyperbolic metric ' on S, a measuredlamination ( ! ML(S) can be realized as a geodesic measured lamina-tion with respect to '. The hyperbolic length function extends by lin-earity and continuity to ML(S); we write l#(() for the hyperbolic lengthof a lamination ( ! ML(S). The geometric intersection number i(",&)of curves ",& ! S also extends continuously to ML(S). Laminations

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 7

µ, ! ! ML(S) are said to fill up S if i(µ, ()+ i(!, () > 0 for all ( ! ML(S).For ( ! ML(S), we denote the underlying leaves by |(|.

2.4 Teichmuller space. The Teichmuller space T (S) of S is the spaceof all conformal structures on S up to isotopy. The Teichmuller distancedT (S)(#,#$) between two marked Riemann surfaces #,#$ ! T (S) is[log K]/2, where K is the smallest quasiconformal constant of a homeo-morphism from # to #$ which is isotopic to the identity.

Each conformal structure # ! T (S) is uniformized by a unique hy-perbolic structure ', and conversely, each hyperbolic structure ' has anunderlying conformal structure #. Thus, we also consider T (S) to be thespace of all hyperbolic metrics on S up to isotopy. The thick part Tthick(S)of T (S) will be defined as the subset of all hyperbolic metrics such thatevery closed geodesic has length bounded below by the constant #0.

2.5 Kerckho! lines of minima. Suppose that !+, !" ! ML(S) fillup S. Kerckho! [Ke3] showed that the length function

' *+ l#(!+) + l#(!")

has a global minimum on T (S) at a unique surface L0. Moreover, as tvaries in ("#,#), the surface Lt ! T (S) that realizes the global minimumof l(!+

t ) + l(!"t ) for the weighted laminations !+

t = et!+ and !"t = e"t!"

varies continuously with t and traces out a path t *+ Lt called the line ofminima L(!+, !") of !±.

2.6 Quadratic di!erentials. We give a brief summary of facts aboutquadratic di!erentials that we use and refer the reader to [G], [St] for adetailed and comprehensive background. Let # be a Riemann surface andq a quadratic di!erential on # which is holomorphic, except possibly atpunctures, where q may have a pole of order one. This ensures that the areaof # with respect to the area element |q(z)dz2| is finite, and we normalize sothat the area is 1. Let Q(#) be the space of all such meromorphic quadraticdi!erentials on #.

The zeros and poles of q are called critical points. Away from the criticalpoints, we have two mutually orthogonal line fields defined respectively bythe conditions that Im

&%q(z)dz

'is zero and Re

&%q(z)dz

'is zero. This

defines a pair of measured singular foliations on # with singularities at thecritical points of q, respectively called the horizontal foliation Hq and thevertical foliation Vq. The measures on these foliations are determined byintegrating the line element

((%q(z)dz((. More precisely, for a curve ), its

8 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

horizontal and vertical measures are given respectively by

hq()) =)

$

(( Re&%

q(z)dz'(( , vq()) =

)

$

(( Im&%

q(z)dz'(( .

We call hq()) and vq()) respectively, the horizontal length and the verticallength of ).

Every essential simple closed curve $ in (S, q) has a unique q-geodesicrepresentative, unless it is in a family of closed Euclidean geodesics foli-ating an annulus whose interior contains no singularities. We denote theq-geodesic length of $ by lq($). It satisfies the following inequalities:

&hq($) + vq($)

'/,

2 ) lq($) ) hq($) + vq($) . (1)

By definition of intersection numbers for measured foliations, we havevq($) = i(Hq, $) and hq($) = i(Vq, $), so equation (1) implies

lq($) !$ i(Vq, $) + i(Hq, $) . (2)

This approximation will be used repeatedly.

2.7 Teichmuller geodesics. Suppose that #,#$ ! T (S) are markedRiemann surfaces with dT (S)(#,#$) = d. Then there is a unique quadraticdi!erential q on # such that the conformal structure on #$ is obtained fromthat of # by expanding in the horizontal direction of q by a factor ed andcontracting in the vertical direction by e"d. The homeomorphism whichrealizes this is called the Teichmuller map from # to #$, and has quasicon-formal distortion e2d. The 1-parameter family of quadratic di!erentials qt

whose horizontal and vertical foliations are respectively, etHq and e"tVq,for 0 ) t ) d, define the geodesic path from # to #$ with respect to theTeichmuller metric.

Gardiner and Masur [GM] showed that, for any pair of measured lami-nations !+, !" ! ML(S) which fill up S and such that i(!+, !") = 1, thereis a unique Riemann surface # ! T (S) and a unique quadratic di!erentialq ! Q(#) whose horizontal and vertical foliations are !+, !" respectively.(This uses the one-to-one correspondence between the space of measuredlaminations and the space of measured foliations.) For t ! R, set

!+t = et!+, !"

t = e"t!",

and let Gt and qt be the corresponding Riemann surface and quadraticdi!erential. The path t *+ Gt defines a Teichmuller geodesic which wedenote G = G(!+, !"). We abuse notation and use Gt to also denote thehyperbolic metric that uniformizes the Riemann surface Gt.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 9

2.8 The balance time. Let " ! S. We say " is vertical along G(!+, !")if its intersection i(", !") with the vertical foliation !" vanishes. In thiscase, " can be realized as a union of leaves of the vertical foliation. Simi-larly, " is horizontal if i(", !+) = 0. Mostly we shall be dealing with curves" which are neither horizontal nor vertical. In this case, there is always aunique time t! at which i(", !+

t!) = i(", !"t!). We call t! the balance time

of ". The length of " with respect to Gt is approximately convex along Gand is close to its minimum at t!, see [R2, Th. 3.1]. Our estimation ofthe hyperbolic lengths of short curves will mainly be made relative to theirbalance time.

3 Comparison on the Thick Part of Teichmuller Space

In this section we prove Theorem 3.8, which states that if Gt and Lt arein the #0-thick part Tthick(S) of Teichmuller space, then the Teichmullerdistance between them is uniformly bounded by a constant that dependsonly on the topology of S and #0. The idea is to first approximate thelength of a curve * for any ' ! Tthick(S) by its intersection with what wecall a short marking for ', and then to compare the short markings for Gt

and Lt. This method will be extended in section 7 when we consider theTeichmuller distance between Gt and Lt in general. We begin with somedefinitions.

3.1 Short markings. We call a maximal collection of pairwise disjoint,homotopically distinct, non-peripheral, non-trivial simple closed curveson S, a pants curve system on S. The terminology is due to the fact thatthe complementary components are pairs of pants, i.e three holed spheres(in which some boundary components may be punctures). Our notion of amarking is motivated by [MaM2]:Definition 3.1. A marking M on a surface S is a system of pants curves"1, . . . ,"k and simple closed curves +!1 , . . . , +!k such that

*+,

+-

i("i, +!j ) = 0 if i -= j ,

i("i, +!i) = 2 if two distinct pairs of pants are adjacent along "i ,

i("i, +!i) = 1 if "i is adjacent to only a single pair of pants .

We call +!i the dual curve of "i.

In the second case, "i . +!i fill a four-holed sphere (that is, a regularneighborhood of "i . +!i is homeomorphic to a four-holed sphere) and inthe third case "i . +!i fill a one-holed torus. It is easy to see that any

10 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

two markings which have the same pants system P have dual curves whichdi!er only by twists and half-twists around the curves in P.

The following well-known lemma states that for any hyperbolic metric,one can always choose a pants system whose length is universally bounded:Lemma 3.2 (Bers [Be2]). There exists a constant L > 0 such that forevery ' ! T (S) there is a pants curve system P with the property thatl#(") < L for every " ! P.

If the boundary curves of a pair of pants have bounded length as inBers’ lemma, the geometry of a pair of pants satisfies the following (for aproof, see Appendix):Lemma 3.3. Let P be a totally geodesic pair of pants with boundarycurves "1,"2,"3 of lengths l("i) < L for i = 1, 2, 3. Then the commonperpendicular of "i,"j (where possibly i = j) has length

log1

l("i)+ log

1l("j)

± O(1) ,

where the bound on the error depends only on L.

We will say that a pants curve system as in Lemma 3.2 is short in(S,'). A short marking M# for ' is a short pants system together with adual system chosen so that each dual curve +!i is the shortest among allpossible dual curves. For a given pants curve, notice that there may bemore than one shortest dual curve, in which case any choice will su"ce.Also notice that not all curves in a short marking are necessarily short ; ifa pants curve is very short, then the corresponding dual curve will be verylong. More precisely, we have the following easy consequence of Lemma 3.3:Corollary 3.4. Let M# be a short marking for ' and let ", +! ! M# bea pants curve and its dual. Then

l#(+!) = i(+!,") · 2 log1

l#(")± O(1) .

Observe that if ' ! Tthick(S), since the length of every curve in M# isuniformly bounded below, it follows from Lemma 3.2 and Corollary 3.4 thatthe length of every curve in M# is also uniformly bounded above. Thus, if' ! Tthick(S), we have l#(M#) !$ 1.

For surfaces in the thick part of T (S), short markings coarsely determinethe geometry. We express this in the following proposition whose proofcan be found in the proof of Lemma 4.7 in [Mi2] (see also the proof ofProposition 7.7 below). If M is a marking and ( ! ML(S), we write

i(M, () =.

%#M

i($, () .

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 11

Proposition 3.5. Let M# be a short marking for ' ! Tthick(S). Thenfor any * ! S,

l#(*) !$ i(M# , *) .

Since both length and intersection number scale linearly with weightsof simple closed curves, it follows thatProposition 3.6. Let M# be a short marking for ' ! Tthick(S). Thenfor any ( ! ML(S),

l#(() !$ i(M# , () .

3.2 Comparison on the thick part. We use the estimate in Propo-sition 3.6 to compare Gt and Lt in the thick part of T (S). The followingwell-known lemma is proved in greater generality in [R3] (see Theorem5.5(ii) below). Recall that # denotes the conformal structure associated tothe metric '.Lemma 3.7. Suppose that ' ! Tthick(S) and q ! Q(#). Then for every* ! S,

l#(*) !$ lq(*) .

Theorem 3.8. If Gt,Lt ! Tthick(S) then dT (S)(Gt,Lt) = O(1).

Proof. By Lemma 3.7, equation (2), and Proposition 3.6, we havelGt(MGt)

!$ lqt(MGt)!$ i(MGt , !

+t ) + i(MGt , !

"t )

!$ lGt(!+t ) + lGt(!

"t ) . (3)

Now, since Lt minimizes l#(!+t ) + l#(!"

t ) over all ' ! T (S), we havelGt(!

+t ) + lGt(!

"t ) / lLt(!

+t ) + lLt(!

"t ) .

Reversing the sequence of estimates in equation (3), we getlLt(!

+t ) + lLt(!

"t ) !$ i(MLt , !

+t ) + i(MLt , !

"t ) !$ lqt(MLt)

!$ lGt(MLt) .

Putting together the preceding three equations, we havelGt(MGt)

!% lGt(MLt) .

Since Gt ! Tthick(S), it follows from the observation following Corollary 3.4that

lGt(MLt)!& 1 . (4)

Notice also that lLt(MLt)!& 1. Lemma 4.7 of [Mi2] implies that for any

given B > 0, the diameter of the set {' ! T (S) : l#(MLt) < B}, withrespect to the Teichmuller distance, is bounded above by a constant thatdepends only on B. Thus it follows that dT (S)(Gt,Lt) = O(1). !

12 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

4 Twists and Fenchel–Nielsen Coordinates

In order to compare surfaces in the thin part of Teichmuller space, our maintool will be Minsky’s product region theorem [Mi3]. This uses Fenchel–Nielsen coordinates to give a nice coarse expression for Teichmuller dis-tance between surfaces which have common thin parts. To state the resultsprecisely, we first discuss twists and Fenchel–Nielsen coordinates.

4.1 Twists in hyperbolic metrics. There are various ways to definethe twist of one curve around another, all of which di!er by factors unim-portant to us here. We shall follow Minsky [Mi3]. Let ' ! T (S) be ahyperbolic metric and let " be an oriented simple closed geodesic on (S,').Let * be a simple geodesic that intersects " transversely and let p be a pointof intersection. In the universal cover H2, a lift * of * intersects a lift " of" at a lift p of p, and has endpoints *R, *L on ,%H2 to the right and leftof ", respectively (see Figure 1). Let pR, pL be the orthogonal projections

Figure 1: Defining the twist of * around ".

of *R, *L to " respectively. Then the twist of * around " at p is defined as

tw#(*,", p) = ±dH2(pR, pL)l#(")

,

where the sign is (+) if the direction from pL to pR coincides with theorientation of " and (") if it is opposite. For any other point q ! * 0 ",the twist satisfies ([Mi3, Lem. 3.1])((tw#(*,", q) " tw#(*,", p)

(( ) 1 .

To obtain a number that is independent of the point of intersection, Minskydefines

tw#(*,") = minp#&&!

tw#(*,", p) .

For convenience, we write Tw#(*,") for |tw#(*,")|.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 13

Note that the definition of twist is valid even if the simple geodesic * isnot closed, because the inequality

((tw#(*,", q) " tw#(*,", p)(( ) 1

depends only on the fact that di!erent lifts of * are disjoint. Thus if ! is ameasured geodesic lamination that intersects " transversely, we can definethe twist of ! around " by taking the infimum of twists over all leavesof ! that intersect ". We remark that although we will be working withmeasured geodesic laminations, when defining the twist, the measure is ir-relevant, in other words the twist tw#(!,") depends only on the underlyinglamination |!|.

4.2 Fenchel–Nielsen coordinates. We define the Fenchel–Nielsen co-ordinates /

l#("i), s!i(')0ki=1

associated to a pants curves system "1, . . . ,"k in the following standardway, see for example [Mi3]. Suppose that P is a pair of pants that is theclosure of a component of S \ {"1, . . . ,"k}. By a seam of P , we mean acommon perpendicular between two distinct boundary components of P .(Notice that the definition of seam refers to the internal geometry of Palone; two distinct boundary curves of P may project to the same curveon S.) Each boundary curve of P is bisected by the two points at whichit meets the two seams intersecting it. We first construct a base surface'0 = '0(l01, . . . , l0k) in which the pants curve "i has some specific choice oflength l0i . Each "i is adjacent to two (possibly coincident) pairs of pants; weglue these two pants together in such a way that seams incident on "i fromthe two sides match up. Since the seams meet the pants curves orthogonally,they glue up to form a collection of closed geodesics $j . Any other structure' ! T (S) comes endowed with an associated homeomorphism h : '0 + '.The length coordinates of ' are defined by l#("i). Let '0(l1, . . . , lk) denotethe surface in which "i (more precisely h("i)) has length li = l#("i), whilethe curves formed by gluing the new seams are exactly the images h($j).Now define -!i(') to be the signed distance that one has to twist around"i to obtain ' starting from '0(l1, . . . , lk), where the sign is determinedrelative to a fixed orientation on '0 and hence on '. Finally we define thetwist coordinates of ' by

s!i(') =-!i(')l#("i)

! R .

Lemma 4.1 (Minsky [Mi3, Lem. 3.5]). For any lamination ! ! ML(S)

14 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

that intersects " = "i and any two metrics ','$ ! T (S),(((tw#(!,") " tw#"(!,")) " (s!(') " s!('$))(( ) 4 .

In [Mi3], the statement is only given for closed curves. However the ar-gument extends without change to laminations. This is because the proof in[Mi3] is based on the observation that for any two simple closed curves *1, *2

intersecting ", the di!erence tw#(*1,") " tw#(*2,") is a topological quan-tity, independent of ', up to a bounded error of 1. More precisely, it followsfrom the proof in [Mi3] that if S is the annular cover of S correspondingto ", and if *1 and *2 are respectively, lifts of *1 and *2 intersecting the core" of S, then the di!erence tw#(*1,")" tw#(*2,") is the signed intersectionof *1 and *2 in the annulus S, up to a bounded error. This topologicalcharacterization holds even when *1 and *2 are simple geodesics which arenot necessarily closed.

4.3 Relative twist in an annulus. The above topological observationallows us to define the following:Definition 4.2. For any two laminations !1, !2 that intersect a curve ",define their algebraic intersection around " to be

i!(!1, !2) = inf#

&tw#(!1,") " tw#(!2,")

',

where the infimum is taken over all possible surfaces ' ! T (S).Often we need only the absolute value:

Definition 4.3. For any two laminations !1, !2 that intersect a curve ",define the relative twisting of !1, !2 around " to be

d!(!1, !2) =((i!(!1, !2)

(( .

Thus for any ' ! T (S), we have((tw#(!1,") " tw#(!2,")

(( = d!(!1, !2) + O(1) .

Notice that i!(!1, !2) and d!(!1, !2) are independent of the measures on!1, !2, depending only on the underlying laminations |!1| and |!2|.

Using Definition 4.2, one sees easily thati!(!1, !2) = i!(!1, () " i!(!2, () + O(1) , (5)

for any curve ( transverse to ". It is also easily seen that d!(!1, !2) agrees upto O(1) with the definition of subsurface distance between the projectionsof |!1| and |!2| to the annular cover of S with core ", as defined in [MaM2,§2.4] and used throughout [R1,2].

Another essentially equivalent way of measuring twist is to look at theintersection with the shortest curve transverse to ":

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 15

Lemma 4.4. Let " be a pants curve and let +! be a shortest dual curve of" in some marking for '. Then for any simple closed curve * intersecting ",we have |tw#(*,") " i!(*, +!)| = O(1).

Proof. Since i!(*, +!) = tw#(*,")" tw#(+!,"), up to a bounded error of 1,it is su"cient to show that |tw#(+!,")| = O(1).

Let +! be a lift of +! in the universal cover H2 and let ", "$ be thetwo lifts of " containing the endpoints of +! (see Figure 2). Let ) bethe perpendicular between " and "$ and let p, p$ be the endpoints of ) on

Figure 2: Bound on the twist of +! around ".

", "$, respectively. Since +! is the shortest dual curve, the endpoints of +!

must be within distance l#(") from p, p$. Let q, q$ be points on ", "$ atdistance l#(") from p, p$, respectively, on opposite sides of ). It is easy tosee that |tw#(+!,")| ) |tw#(&,")|, where & is the geodesic through q, q$.Let r be the foot of the perpendicular as shown. As in the first part ofthe proof of Lemma 3.5 in [Mi3], we note that since the images of "$ underthe translation along " are disjoint, the projection of "$ on " has length atmost l#("). Thus l(pr) < l(pq) = l("). Hence,

((tw#(&,")(( = 2

l(pq) + l(pr)l(")

< 4 . !

4.4 The product region theorem. Let A ( S be a collection ofdisjoint, homotopically distinct, simple closed curves on S and letTthin(A, #0) ( T (S) be the subset in which all curves " ! A have hyperboliclength at most #0. Extend A to a pants decomposition and define Fenchel–Nielsen coordinates as above. Let SA denote the surface obtained fromS by removing all the curves in A and replacing the resulting boundarycomponents by punctures.

16 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Following [Mi3], we now define a projection

$ : T (S) + T (SA) ' H!1 ' . . . ' H!r ,

where A = {"1 . . . ,"r} and H!i is the upper half-plane. The first compo-nent $0 which maps to T (SA) is defined by forgetting the coordinates ofthe curves in A and keeping the same Fenchel–Nielsen coordinates for theremaining surface. For " ! A define $! : T (S) + H! by

$!(') = s!(') + i/l#(") ! H! .

Let dH! be half the usual hyperbolic metric on H!. Minsky’s product regiontheorem states that, up to bounded additive error, Teichmuller distance onTthin(A, #0) is equal to the sup metric on

T (SA) ' H!1 ' . . . ' H!r .

Theorem 4.5 (Minsky [Mi3]). Let ', - ! Tthin(A, #0). Then

dT (S)(', -) = max!#A

!dT (SA)($0('),$0(-)), dH!($!('),$!(-))

"± O(1) .

We remark that Minsky makes several assumptions on the size of #0 inorder to prove the above theorem. We may assume that our initial choice of#0 satisfies these assumptions. Recall from section 2.1 that a curve " ! Sis said to be extremely short if l#(") < #0. The distance between theprojections to H! can be approximated as follows:

Lemma 4.6. Suppose " ! S is extremely short in both ', - . Then

exp 2dH!

/$!('),$!(-)

0 !$ max!l#l' |s!(') " s!(-)|2, l'/l# , l#/l'

",

where l# = l#(") and l' = l' (").

Proof. This is a simple calculation using the formula

cosh 2dH(z1, z2) = 1 +|z1 " z2|2

2 Im z1 Im z2,

where dH is as above half the usual hyperbolic distance in H. !

The lemma reveals a useful fact from hyperbolic geometry: unless thedi!erence |x " x$| is extremely large, the distance between two pointsx + iy, x$ + iy$ ! H is dominated by |log[y/y$]|. In our situation this meansthat unless the di!erence between the twist coordinates is extremely largein comparison to the lengths, their contribution to hyperbolic distance canbe neglected. We quantify this in the following useful corollary which showsthat as long as each twist coordinate is bounded by O(1/l), it can be ne-glected when estimating the contribution to Teichmuller distance comingfrom the same short curve in two surfaces.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 17

Corollary 4.7. Suppose that '1,'2 ! Tthin(", #0) and that, for i = 1, 2,! ! ML(S) satisfies

Tw#i(!,") l#i(") = O(1) .

Then

dH!

/$!('1),$!('2)

0=

12

((((logl#1(")l#2(")

(((( ± O(1) .

Proof. This follows easily from Lemma 4.6, using Lemma 4.1 to approx-imate s!('1) " s!('2) by tw#1(!,") " tw#2(!,"). (Cross terms such ass!('1)l#2(") may be rearranged as [s!('1)l#1(")][l#2(")/l#1(")].) Note thatthe multiplicative error in Lemma 4.6 translates to an additive error in thedistance. !

5 Short Curves along Teichmuller Geodesics

In this section we prove Theorem B, stated more precisely as Theorem 5.10.This gives a combinatorial estimate for the hyperbolic length of an ex-tremely short curve along the Teichmuller geodesic G(!+, !"). We alsorecall the estimate for the twist of !± around such curves (Theorem 5.11)proved in [R2].

Deriving the length estimate is largely a matter of putting together re-sults proved in [Mi1], [R1,3]. Both estimates are made by a careful studyof the relationship between the hyperbolic metric Gt and the quadratic dif-ferential metric qt. As indicated in the Introduction, there are two distinctreasons why a curve " may become extremely short: one is that the relativetwisting of !+ and !" around " may be very large, the other that !+ and!" may have large relative complexity in S \ ". We express the latter interms of a scale factor which controls the relationship between the quadraticdi!erential and hyperbolic metrics on the components of the thick part ofGt adjacent to ", as made precise in Rafi’s thick-thin decomposition forquadratic di!erentials, Theorem 5.5.

We begin in sections 5.1 and 5.2 with some essential facts about thegeometry of annuli with respect to quadratic di!erential metrics.

5.1 Annuli in quadratic di!erential metrics. We review the no-tions of flat and expanding annuli from [Mi3]. These concepts provide theframework with which to analyze short curves.

Let # ! T (S) and let q be a quadratic di!erential on #. Let $ be apiecewise smooth curve in (S, q). At a smooth point p, the curvature .(p)is well defined, up to a choice of sign. If $ is the boundary component of

18 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

a subsurface Y , we choose the sign to be positive if the acceleration vectorat p points into Y . At a singular point P , although the curvature is notdefined, we shall say $ is non-negatively curved at P if the interior angle/(P ) is at most 0 and say it is non-positively curved at P if /(P ) is atleast 0. By interior angle, we mean the angle that is on the same side of$ as Y . We say $ is monotonically curved with respect to Y either if thecurvature is non-negative at every point, or non-positive at every point.The total curvature of $ is given by

.Y ($) =)

%.(p) +

. &0 " /(P )

',

where the sum is taken over all singular points P on $. The Gauss–Bonnettheorem gives .

.Y ($) " 0.

ord P = 201(Y ) , (6)

where ord P is the order of the zero at P , the first sum is over all boundarycomponents $ of Y , and the second sum is over the zeros P of q in theinterior of Y .

Let A be an annulus in (S, q) with piecewise smooth boundary. Thefollowing definitions are due to Minsky [Mi1]. We say A is regular if bothboundary components ,0, ,1 are monotonically curved with respect to Aand if ,0, ,1 are q-equidistant from each other. Suppose that A is a reg-ular annulus such that .A(,0) ) 0. We say A is an expanding annulus if.A(,0) < 0 and we call ,0 the inner boundary and ,1 the outer boundary.Expanding annuli are exemplified by an annulus bounded by a pair of con-centric circles in R2. The inner boundary is the circle of smaller radius andhas total curvature "20, while the outer boundary has total curvature 20.

A regular annulus is primitive with respect to q if it contains no singu-larities of q in its interior. By equation (6), its boundaries satisfy .A(,0) =".A(,1). A regular annulus is flat if .A(,0) = .A(,1) = 0. By (6), aflat annulus is necessarily primitive, and is foliated by Euclidean geodesicshomotopic to the boundaries. Thus a flat annulus is isometric to a cylin-der obtained as the quotient of a Euclidean rectangle in R2. Note that aprimitive annulus must either be flat or expanding.

One reason for introducing flat and expanding annuli is that their mod-uli are easy to estimate. The following result can be deduced from Theo-rem 4.5 of [Mi1] and is proved in [R1]:Theorem 5.1. Let A ( S be an annulus that is primitive with respectto q and with inner and outer boundaries ,0 and ,1, respectively. Let d bethe q-distance between ,0 and ,1. Then either

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 19

(i) A is flat and Mod A = d/lq(,0); or(ii) A is expanding and Mod A $ log[d/lq(,0)].

5.2 Modulus of annulus and length of short curve. The link be-tween the hyperbolic and quadratic di!erential metrics on a surface is madeusing annuli of large modulus. Let ' be the hyperbolic metric that uni-formizes #. If " is short in ', Maskit [M] showed that the extremal lengthExt!(") and hyperbolic length l#(") are comparable, up to multiplicativeconstants. Moreover, there is an embedded collar C(") around " in (S,')whose modulus is comparable to 1/l#(") (see [Mi3] for an explicit calcu-lation), and therefore also to 1/Ext!("). By the (geometric) definition ofextremal length, this implies that the maximal annulus around " in # hasmodulus comparable to 1/l#("). The following theorem of Minsky allowsus to replace any annulus of su"ciently large modulus with one that isprimitive:

Theorem 5.2 [Mi1, Th. 4.6]. Let A ( # be any homotopically non-trivial annulus whose modulus is su!ciently large and let q ! Q(#). ThenA contains an annulus B that is primitive with respect to q and such thatMod A $ Mod B.

(The statement of Theorem 4.6 in [Mi1] should read ModA / m0 notMod A ) m0.) Thus, we have

Theorem 5.3. If " is a simple closed curve which is su!ciently short in(S,'), then for any q ! Q(#), there is an annulus A that is primitive withrespect to q with core homotopic to " such that

1l#(")

$ Mod(A) .

We may assume that #0 was chosen so that if l#(") < #0, then l#(") issmall enough that this theorem is valid.

We can now apply Theorem 5.1 in the following way. It follows fromequation (6) that every simple closed curve $ on (S, q) either has a uniqueq-geodesic representative, or is contained in a family of closed Euclideangeodesics which foliate a flat annulus [St]. Denote by F ($) the maximalflat annulus, which necessarily contains all q-geodesic representatives of $.If the geodesic representative of $ is unique, then F ($) is taken to be thedegenerate annulus containing this geodesic alone. Denote the (possibly co-incident) boundary curves of F ($) by ,0, ,1 and consider the q-equidistantcurves from ,i outside F ($). Let ,i denote the first such curve which isnot embedded. If ,i -= ,i, then the pair ,i, ,i bounds a region Ei($) whose

20 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

interior is an annulus with core homotopic to $, and which is by its con-struction regular and expanding. Combining the preceding two theoremswith Theorem 5.1 we haveCorollary 5.4. If " is an extremely short curve on (S,'), then

1l#(")

$ max!Mod F ("),Mod E0("),Mod E1(")

".

Proof. Since " is extremely short, we have 1/Ext!(") $ 1/l#(") and henceit follows from the (geometric) definition of extremal length that

1l#(")

$ 1Ext!(")

/ max!ModF ("),Mod E0("),Mod E1(")

".

By Theorem 5.3, there is a primitive annulus A whose core is homotopicto " such that 1/l#(") $ Mod(A). We will show that

Mod(A) & max!Mod F ("),Mod E0("),Mod E1(")

".

If A is flat, then A must be contained in the maximal flat annulus F (").In this case, Mod(A) ) Mod F ("). If A is expanding, then although itmay not be contained in either E0(") or E1(") it must be disjoint fromthe interior of F ("). Without loss of generality, let us assume that A lieson the same side of F (") as E0("). Let ,0 and ,0 be respectively, theinner and outer boundaries of E0("). Let C0 and C1 be respectively, theinner and outer boundaries of A. Since lq(,0) is equal to the q-length ofthe geodesic representative of ", the q-length of the inner boundary of Asatisfies lq(C0) / lq(,0). Let 2 be a q-shortest arc in E0(") from ,0 toitself; its length is 2dq(,0, ,0). The intersection 2 0 A is a union of twoarcs, each of which goes from one boundary component of A to another.Since lq(20A) ) lq(2), it follows that dq(C0, C1) ) lq(20A)/2 ) dq(,0, ,0).Thus, it follows from Theorem 5.1 that

Mod(A) $ logdq(C0, C1)

lq(C0)) log

dq(,0, ,0)lq(,0)

$ Mod/E0(")

0. !

The idea of our basic length estimates for extremely short curves " inTheorem 5.10, is to combine this corollary with the estimates for the moduliof F (") and Ei(") in Theorem 5.1.

5.3 Thick-thin decomposition and the q-metric. The thick-thindecomposition for quadratic di!erentials developed in [R3] describes therelationship between the q-metric on the surface # and the uniformizinghyperbolic metric ' in the thick components of the thick-thin decompositionof '. It states that on the hyperbolic thick parts of (S,') the two metrics arecomparable, up to a factor which depends on the moduli of the expanding

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 21

annuli around the short curves in the boundary of the thick component.This factor will be crucial in our estimates below.

To make a precise statement, for a subsurface Y of S, define the q-geodesic representative of Y to be the unique subsurface Y of (S, q) withq-geodesic boundary in the homotopy class of Y that is disjoint from theinterior of F ($i) for all components $i of ,Y . Notice that Y is q-geodesicallyconvex, so that if a closed curve * is contained in Y , it has a q-geodesicrepresentative contained in Y . (It is possible for Y to be degenerate. See[R3] for an example where the area of Y is zero.)

If Y is not a pair of pants, define %Y to be the length of the q-shortestnon-peripheral simple closed curve contained in Y . If Y is a pair of pants,define %Y to be max{lq($1), lq($2), lq($3)} where $1, $2, $3 are the threeboundary curves of Y . The thick-thin decomposition for quadratic di!er-entials is the following:

Theorem 5.5 (Rafi [R3]). Let ' be the hyperbolic metric that uni-formizes # and let Y be a thick component of the hyperbolic thick-thindecomposition of (S,'). Then

(i) diamq Y!$ %Y ;

(ii) For any non-peripheral simple closed curve * in Y , we have

lq(*) !$ %Y l#(*) .

5.4 Twist in the q-metric. In order to compare two surfaces, weneed to estimate not only the lengths but also the twist parameters of shortcurves. To do this we use a signed version of Rafi’s definition [R2, §4] of thetwist of a simple curve * about another curve " in a quadratic di!erentialmetric q on S.

Let A ( S be a regular annulus with core curve ", let S be the annularcover of S corresponding to ", and let " be a q-geodesic representative of thecore of S. Suppose that * is a simple q-geodesic (i.e. geodesic with respectto the q-metric) that intersects ", and let * be a lift which intersects ".Let & be a bi-infinite q-geodesic arc in S that is orthogonal to ". Wewould like to define the twist twq(*,") to be the sum aS(*, &) of the signedintersection numbers over all intersections between * and &. The followinglemma shows that aS(*, &) is, up to a bounded additive error, independentof the choices of & and * .

Lemma 5.6. Let ", ", and S be as above and suppose that * is a simpleq-geodesic transverse to ". Suppose that *, * $ are di"erent lifts of * thatintersect " and that &, &$ are di"erent bi-infinite q-geodesic arcs orthogonal

22 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

to ". ThenaS(*, &) = aS(* $, &$) ± O(1) .

Proof. Let F (") be the lift to S of the maximal flat annulus F (") around" and let aF (*, &) denote the sum of signed intersection numbers overintersection points within F ("). A simple Euclidean argument shows that,for any two disjoint arcs *, * $, we have

aF (*, &) = aF (* $, &$) ± O(1) .

We claim that outside F ("), any two q-geodesics can intersect at mosttwice. Outside F ("), S is made up of two regular expanding annuli E1, E2,one attached to each boundary of F ("). These annuli extend out to in-finity in S (which can be compactified using the hyperbolic metric on S,see [MaM2, §2.4]). The key point is that in any expanding annulus E,two geodesic arcs can intersect at most once. For if they intersected twice,we would get a piecewise geodesic loop $ homotopic to the inner bound-ary, made up of two geodesic arcs that go from one intersection point tothe other. Along each arc, the geodesic curvature vanishes. The Gauss–Bonnet theorem in equation (6) applied to the annulus bounded by $ andthe inner boundary shows this is impossible. (Notice that if E is not prim-itive, then the singularities of q in E only improve the desired inequality inequation (6).) !

We define twq(*,") to be the minimum of the numbers aS(*, &) overall choices of *, &. Notice that the argument requires only that the lifts of* to S be disjoint, so that we can similarly define twq(!,") for a geodesiclamination ! where as usual, twq(!,") depends only the underlying support|!| of !.

The following key result allows us directly to compare the twists in thehyperbolic and quadratic di!erential metrics.Proposition 5.7 [R2, Th. 4.3]. Suppose that ' is a hyperbolic metricuniformizing a surface # ! T (S) and that q ! Q(#), and let ! be a geodesiclamination intersecting ". Then((tw#(!,") " twq(!,")

(( = O/1/l#(")

0.

The statement in [R2] has i!(!, +!) in place of tw#(!,"), however byLemma 4.4, this distinction is unimportant. The result in [R2] is stated forclosed curves but extends immediately to the case of geodesic laminationsas explained above.

5.5 Length and twist along G(!+, !!). As explained at the endof section 5.2, we can use Theorem 5.1 and Corollary 5.4 to estimate the

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 23

length of an extremely short curve " in Gt. We call a flat or expandingannulus which achieves the maximum modulus in Corollary 5.4 a dominantannulus for ". There may be more than one dominant annulus, but thiswill not a!ect our reasoning and we will refer to ‘the dominant annulus’.The estimates depend on whether the dominant annulus is flat or expand-ing, corresponding to the two terms Dt(") and Kt(") in the main resultTheorem 5.10 of this section.

Suppose first that the maximal flat annulus Ft(") is dominant. Providedthat " is neither vertical nor horizontal (see section 2.8), the followingproposition expresses Mod Ft(") in terms of the relative twisting d!(!+, !")of !+, !" around " defined in section 4.3. The case in which " is eitherhorizontal or vertical, so that either !+ or !" has empty intersection with ",is easier and is dealt with in section 5.6.Proposition 5.8. Let " be a curve in (S, q) that is neither vertical norhorizontal and suppose that the maximal flat annulus Ft(") is dominant.Then

Mod Ft(") $ e"2|t"t!|d!(!+, !") .

Proof. Since a flat annulus is Euclidean, its geometry is very simple. Let) be a qt-geodesic arc in Ft(") joining the two boundaries of Ft(") that isorthogonal to the geodesic representatives of ". For a simple geodesic *transverse to ", define twFt(*,") to be the signed intersection number of* with ) in Ft("). It is independent of the choice of ) up to a boundederror of 1. Assuming that " is neither vertical nor horizontal, then atthe balance time t! (see section 2.8) the horizontal and vertical foliationsboth make an angle of 0/4 with the qt!-geodesic representatives of ". Inthis case, a leaf of !+

t! or !"t! intersects ) approximately (up to an error

of 1) lqt!())/lqt!

(") times, so the modulus of Ft!(") is approximated bytwFt!

(!+,") = twFt!(!","). More generally, the horizontal leaves make an

angle 3t with ", where |tan 3t| = e2(t"t!). From this it is a straightforwardexercise in Euclidean geometry, see section 4.1 of [R2], to prove((twFt(!

±,") " e'2(t"t!) Mod Ft(")(( ) 1 . (7)

We will show that((twFt(!+,") " twFt(!

",")(( = d!(!+, !") ± O(1) , (8)

from which the proposition follows. From the proof of Lemma 5.6 we havetwqt(*,") = twFt(*,")±O(1). Now it was observed in section 4.2 (see alsothe subsequent discussion in section 4.3) that although tw#(!,") dependson the metric ' in which it is measured, the di!erence in twist of !+ and !"

equals (up to a bounded error) the number of times a leaf of !+ intersects

24 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

a leaf of !" in the annular cover of S corresponding to ". Since this alsoholds for a quadratic di!erential metric, we get

((twqt(!+,") " twqt(!

",")(( =

((tw#(!+,") " tw#(!",")(( ± O(1)

= d!(!+, !") ± O(1) .

Equation (8) follows. !Suppose now that one or other of the expanding annuli around " is

dominant. The estimate of modulus in this case is given byProposition 5.9. Let q ! Q(#). Suppose that " is extremely short in 'and let Y be a thick component of the hyperbolic thick-thin decompositionof (S,'), one of whose boundary components is ". Let " be the q-geodesicrepresentative of " on the boundary of Y and let E(") be a maximalexpanding annulus on the same side of " as Y . If E(") is dominant, then

Mod E(") $ log%Y

lq(").

Proof. Let dq denote the q-metric. By Theorems 5.2 and 5.1(ii), it issu"cient to show that dq(,0, ,1) $ %Y .

Note that although E(") is not necessarily contained in Y , the outerboundary ,1 must intersect Y . Hence, dq(,0, ,1) ) diamq(Y ) and so, byTheorem 5.5(i), we have dq(,0, ,1) & %Y .

Now we prove the inequality in the other direction. Observe that sinceE(") is maximal, the outer boundary ,1 intersects itself and so there isa non-trivial arc 2 with endpoints on " whose length is 2dq(,0, ,1). Firstsuppose that 2 is contained in Y . A regular neighborhood of ".2 is a pairof pants whose boundary curves are homotopic to " and two additionalcurves *1, *2. Note that for i = 1, 2,

lq(2) + lq(") / lq(*i) .

Thus, if either *1 or *2, say *1 is non-peripheral in Y , thendq(,0, ,1)

lq(")% lq(*1)

lq(")/ %Y

lq(").

If both *1, *2 are peripheral, then Y is a pair of pants, and we havedq(,0, ,1)

lq(")% lq(*1) + lq(*2)

lq(")$ %Y

lq(").

If 2 exits Y , we replace it with a new arc 2$ as follows. Let p be thefirst exit point and let $ be the boundary component of Y that contains p.Let 2$ be the arc that first goes along 2 to p, then makes one turn around$ from p to itself, then comes back to " along the first path. Because $

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 25

is in the boundary of Y , its hyperbolic length is extremely short and thus,by Theorem 5.3, the original arc 2 must pass through an annulus of largemodulus with core curve $. Therefore, lq($) & lq(2) and so we have

lq(2$) ) 2lq(2) + lq($) & lq(2) .

Now we can run the same argument as above with 2$ in place of 2 to deducethe desired inequality. !

We are now able to write down the desired length estimate. Supposethat the curve " is extremely short in some surface Gt along the Teichmullergeodesic G(!+, !"). Let Y1, Y2 be the thick components of the thick-thindecomposition of (S,Gt) that are adjacent to " (where Y1 may equal Y2).Define

Kt(") = max#

%Y1

lqt("),

%Y2

lqt(")

$(9)

and

Dt(") = e"2|t"t!|d!(!+, !") . (10)

Then, combining Corollary 5.4 and Propositions 5.8 and 5.9, we obtain ourfirst main result which is essentially Theorem B of the Introduction:

Theorem 5.10. Let " be a curve on S that is neither vertical norhorizontal. If " is extremely short in Gt, then

1lGt(")

$ max!Dt("), log Kt(")

".

It was shown in [R1] that Kt(") can be estimated combinatorially asthe subsurface intersection of !+, !" in the corresponding component of thethick part of Gt adjacent ". Since this is not necessary for our development,we shall not go into this here.

We need to estimate not only the lengths but also the twist param-eters about short curves. Combining Proposition 5.7 with equation (7),Lemma 5.6, Proposition 5.8, and Theorem 5.10, we get

Theorem 5.11 [R2, Th. 1.3]. With the same hypotheses as Theorem 5.10,

TwGt(!+,") = O

11

lGt(")

2if t / t! ,

TwGt(!",") = O

11

lGt(")

2if t ) t! .

26 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

5.6 Estimates for horizontal and vertical short curves on G. Wealso need the analogue of Theorems 5.10 and 5.11 for extremely short curves" which are either horizontal or vertical.

For definiteness, assume " is vertical so that i(", !") = 0. The definitionof balance time no longer makes sense. Instead, we work relative to timet = 0. Let d be the height (i.e. distance between the two boundaries) ofF0("). At an arbitrary time t,

lqt(") = i(", !+t ) = etlq0(") ,

while the height of Ft(") is e"td. HenceMod Ft(") = e"2t ModF0(") . (11)

This is the analogue of Proposition 5.8.If " is vertical, the discussion in Proposition 5.9 about expanding annuli

is unchanged. Thus we obtainTheorem 5.12. Let " be a vertical curve on S. If " is extremely shortin Gt, then

1lGt(")

$ max!e"2t Mod F0("), log Kt(")

".

If " is horizontal, the estimate is the same except that the first term isreplaced by e2t ModF0(").

We also want the analogue of Theorem 5.11. If " is vertical, thenTwGt(!",") is undefined. However, we haveTheorem 5.13. If " is extremely short in Gt, and if " is vertical then

TwGt(!+,") = O

11

lGt(")

2,

while if " is horizontal then

TwGt(!",") = O

11

lGt(")

2.

Proof. If " is vertical, the q-twist twq(!+,") in Ft(") vanishes, while if "is horizontal, then twq(!",") = 0. The result follows from Lemma 5.6 andProposition 5.7. !

6 Short Curves along Lines of Minima

In this section we prove Theorem B, stated more precisely as Theorem 6.1,which gives our combinatorial estimate for the length of a curve whichbecomes extremely short at some point along the line of minima L(!+, !").

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 27

We also estimate the twist of !± around " in Theorem 6.2. This will formthe basis for our comparison of the metrics Lt and Gt. It turns out that,in a close parallel to the case of the Teichmuller geodesic, there are tworeasons why a curve can be extremely short: either the relative twisting of!+, !" about " is large, or one or other of the pants curves in a pair ofpants adjacent to " in a short marking of Lt has large intersection witheither !+ or !".

More precisely, suppose that " is an extremely short curve in Lt and letPLt be a short pants system in Lt, which necessarily contains ". Define

Ht(") = sup"#B

lqt(&)lqt(")

, (12)

where B is the set of pants curves in PLt which are boundaries of pantsadjacent to ", and qt is the quadratic di!erential metric of area 1 (onthe corresponding surface Gt) whose horizontal and vertical foliations arerespectively, !+

t and !"t .

Let Dt(") be as in equation (10). Our main estimates are

Theorem 6.1. Let " be a curve on S which is neither vertical norhorizontal. If " is extremely short in Lt, then

1lLt(")

$max!Dt("),

%Ht(")

".

Theorem 6.2. With the same hypotheses as Theorem 6.1, the twistsatisfies

TwLt(!+,") = O

11

lLt(")

2if t / t! ,

TwLt(!",") = O

11

lLt(")

2if t ) t! .

To prove Theorems 6.1, 6.2, we note that since the surface Lt is on theline of minima, we have at the point Lt,

dl(!+t ) + dl(!"

t ) = 0 . (13)

The pants curves in PLt (together with seams) define a set of coordinates(l#("), -!(')) on T (S) as explained in section 4.2, which in turn defineinfinitesimal twist ,/,-! and length ,/,l(") deformations for " ! PLt .Theorems 6.1 and 6.2 will follow from the relations we get by applyingequation (13) to ,/,-! and ,/,l("). For ,/,-!, we use the well-knownformula of Kerckho! [Ke2] and Wolpert [W1], while for ,/,l(") we use theanalogous formula for the length deformation derived in [S1].

28 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

6.1 Di!erentiation with respect to twist. Suppose as above that "is an extremely short curve in Lt. If we apply equation (13) to ,/,-!, thederivative formula in [Ke2] and [W1] gives

0 =,l(!+

t ),-!

+,l(!"

t ),-!

=)

!cos /+d!+

t +)

!cos /"d!"

t ,

where /± is the function measuring the angle from each arc of |!±t | to ".

Assume that " is neither vertical nor horizontal, so that neither i(!+,")nor i(!",") is zero. Then we may define the average angle %±

t by

cos %±t =

1i(!±

t ,")

)

!cos /±d!±

t .

Setting T = t " t!, the preceding two equations give

eT cos %+t + e"T cos %"

t = 0 . (14)

If a particular leaf L of a lamination |!| cuts " at an angle / at a point p,then from the definition of the twist (see section 4.1) and simple hyperbolicgeometry we have

cos / = tanhtwLt(L,", p)lLt(")

2.

Since the twists twLt(L,", p) for di!erent leaves L di!er by at most 1, if "is su"ciently short we obtain the estimate

|cos / " cos %±t | = O

/lLt(")

0, (15)

from which we deduce that either cos %±t and twLt(!±,") have the same

sign, or that |cos %±t | = O(lLt(")) so that |twLt(!±,")| = O(1).

Note also that equation (14) implies that !+, !" twist around " inopposite directions and that the lamination whose weight on " is smallerdoes more of the twisting.

6.2 Di!erentiation with respect to length. For the length defor-mation, we shall apply the extension of the Wolpert formula derived in [S1],which gives a general expression for dl(*), for * ! S, with reference to apants curves system P. Let "1, . . . , "n be the lifts of the pants curves in Psuccessively met by *, where the segment of the lift * of * between "1 and"n projects to one complete period of *. Let dj be the length of the com-mon perpendicular 0j between "j , "j+1 and let Sj be the signed distancebetween 0j"1 and 0j along "j , where the sign is positive if the directionfrom 0j"1 to 0j coincides with the orientation of "j. (Note that if "j and"j+1 project to the same curve " and if " is adjacent to two distinct pairsof pants, then 0j projects to an arc perpendicular to " that is not a seam.)

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 29

Then equation (3) of [S1] states that

dl(*) =n.

j=1

cosh uj ddj +n.

j=1

cos /j dSj , (16)

where /j is the angle from * to "j measured counter-clockwise and uj isthe complex distance from * to the complete bi-infinite geodesic whichcontains 0j. Replacing sums by integrals, we see that this formula, derivedin [S1] for closed curves, pertains equally to a measured lamination.

In our case, we take P to be PLt and apply this formula to ,/,l(") for" ! PLt. The non-zero contributions will be from terms dSj correspondingto lifts of ", and from two types of terms ddj: those corresponding toperpendiculars with endpoints on lifts of ", and those corresponding toperpendiculars which do not intersect any lift of ", but whose projectionsare contained in a common pair of pants with ".

We first estimate the contribution from the terms ddj . Suppose as abovethat PLt is a short pants decomposition for Lt. Let P be a pair of pantsin S \ PLt that has " as a boundary component. The geometry of P iscompletely determined by the lengths of the three boundary curves ",&, $.A common perpendicular joining two (not necessarily distinct) boundarycomponents of P may or may not have one of its endpoints on a boundarycurve which projects to " on S. We say that the common perpendicularsof the first kind are adjacent to ", while those of the second type are not.The terms ddj are estimated by the following lemma which is proved in theAppendix:Lemma 6.3. Suppose that " is extremely short in Lt and let P be a pairof pants in S \ PLt that has " as a boundary component. Let v denotethe length of a common perpendicular adjacent to ", and let w denote thelength of a common perpendicular not adjacent to ". Then

,v

,l(")!$ " 1

l(")and

,w

,l(")!$ l(") ,

where the partial derivatives are taken with respect to the coordinates(l("), -!)!#PLt

.

We remark that the first of these estimates coincides with the heuris-tic computation that since the collar around " has length comparable tolog[1/l(")], the derivative should be approximately "1/l(").

We also need to bound the coe"cient cosh uj of ddj in equation (16).Notice that the bound applies to all the pants curves (not just the extremelyshort ones) in a short marking.

30 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Lemma 6.4. If the pants curves system P is short, then for all j,

|cosh uj |!$ 1 .

Proof. Since P is short, by definition all curves "j have length boundedabove, and hence the length dj of the common perpendicular 0j to "j , "j+1

is bounded below.First suppose that * intersects the infinite geodesic 0j that contains 0j .

In this case, uj = i4j , where 4j is the angle between * and 0j at their inter-section point o. Consider the case when o is contained in the segment 0j .Let xj be the distance between o and the endpoint oj of 0j that lies on "j .Since * intersects both "j and "j+1, the angle of parallelism formula gives

|tan 4j | < 1/ sinh xj and |tan 4j| < 1/ sinh(dj " xj) .

Since at least one of xj and dj " xj is bounded below, this gives a uniformupper bound on |tan 4j|. Thus 4j is uniformly bounded away from 0/2 and|cosh uj| = |cos 4j| is bounded below by a universal positive number.

Now consider the case when o lies outside of 0j. Let o$j, o$j+1 denote

respectively, the points of intersection between *j and "j , "j+1 and let oj ,oj+1 denote respectively, the points of intersection between 0j and "j, "j+1.If d(o, oj) / d(o, oj+1), then replace * with the geodesic * $ that passesthrough o$j and oj+1 and if d(o, oj) ) d(o, oj+1), then replace * with thegeodesic * $ that passes through o$j+1 and oj. The angle 4$

j of intersectionbetween * $ and 0j satisfies |cos 4j| > |cos 4$

j |. We now run the precedingargument with * $ in place of * to conclude |cos 4$

j| is bounded below.Now, suppose that * does not intersect 0j . Then uj is the hyperbolic dis-

tance from * to 0j. Denote by p the point where the common perpendicularfrom * to 0j meets 0j; this point may lie outside the segment 0j between"j , "j+1. Let yj, y$j denote the (unsigned) distances from p to "j , "j+1 re-spectively. The quadrilateral formula gives sinh yj sinhuj = |cos /j| andsinh y$j sinhuj = |cos /j+1|, where /j, /j+1 are the angles between * and"j , "j+1 respectively. Whether or not p ! 0j , at least one of yj and y$j isbounded below by dj/2. Since there is a uniform lower bound on dj , itfollows that |sinh uj| and hence |cosh uj | is uniformly bounded above. Theresult follows. !

We now consider the second sum in equation (16).

Lemma 6.5. Let "j be a lift of the curve " along which the curve * hasshift coordinate Sj = Sj(*). Then

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 31

,Sj

,l(")= twLt(*,") " s!(Lt) ± O(1) ,

where s!(Lt) is the Fenchel–Nielsen twist along " at Lt as defined insection 4.2.

Proof. Homotope the lift * to the piecewise geodesic path * that runs alongthe successive lifts "i and common perpendiculars 0i. The projection of* to S is homotopic to *. Then Sj(*) equals the signed distance that *travels along "j. We need to express this shift in a usable form.

Recall the definition of Fenchel–Nielsen twist coordinates from sec-tion 4.2. As above, we denote the pants curves in a short marking forLt by "1, . . . ,"k. The curve "j forms the boundary of the lifts of two,possibly coincident, pairs of pants Pj and Pj+1. The projection "j of "j toPj is bisected by the endpoints of the two seams of Pj which join "j to eachof the other two boundaries of Pj (before identification in the surface S).Likewise the projection "$

j of "j to Pj+1 is bisected by the endpoints ofexactly two seams of Pj+1.

The zero twist surface '0 = '0(l!1 , . . . , l!k) is formed by gluing Pj toPj+1 along "j and "$

j in such a way as to match these two pairs of points.Thus on '0, the distance along "j between incoming and outgoing per-pendiculars 0j and 0j+1 may be expressed in the form nj(*)l(")/2 + ej(*),where nj(*) ! Z is the (signed) number of seams * intersects along "j andej(*) is an error term which allows for the possibility that 0j ,0j+1 may notbe seams of Pj and Pj+1, but rather common perpendiculars from "j or "$

jto itself. In all cases however, |ej(*)| < l#("j) and ej(*) depends only onthe geometry of Pj and Pj+1, see [S1].

Now at Lt, the incoming and outgoing perpendiculars 0j and 0j+1 arefurther o!set by -!j(Lt) = lLt(")s!j (Lt) giving the formula

Sj(*) = 12nj(*)lLt(") + ej(*) + -!j (Lt) ,

see also section 4.2 of [S1].We can now proceed to estimate ,Sj/,l("). Since the partial deriva-

tives are taken with respect to the coordinates (l#("), -!(')), the term,-!j/,l(") vanishes. To avoid an unpleasant calculation, we get rid of theterm ej(*) as follows. Modify * to a path which still runs along the lifts ofthe pants curves and their common perpendiculars, but which never goesalong a perpendicular from a lift of " to itself. Specifically, let 0j be sucha common perpendicular which projects to a pair of pants P one of whoseboundary components is ". Let & be one of the other boundary compo-nents and let ) be the perpendicular from " to &. The projection of 0j to

32 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

P is homotopic, with fixed endpoints, to an arc which runs along ", thenalong ), then along &, back along ), finally back to the final point on ",see Figure 3. Modify * by replacing 0j by the lift of this alternate path.

Figure 3: Homotoping the perpendicular.

Doing this in each instance gives a replacement for *, with respect to whichone can define all quantities occurring in (16) as before. The derivation ofequation (16) in [S1] will still work for this new path. Denoting the newlydefined shift also by Sj, we thus have ,Sj/,l(") = nj/2 ± 1.

We claim that nj/2 = tw#0(*,")±O(1). By definition, * traverses liftsof the pants curves to H in the same order as *. Thus the segment of *running along the lift " = "j to H is the interval between the footpointsQj, Qj+1 of the perpendiculars 0j and 0j+1 from "j"1 and "j+1 (the liftsof pants curves adjacent to "j) to "j . Now Qj lies within the interval on"j bounded by the footpoints of the perpendiculars from the two endpointsof "j"1 on ,H to "j ; and similarly for Qj+1. Thus our claim follows as in[Mi3, Lem. 3.1], see the discussion in section 4.1. The proof of the presentlemma can now be completed by applying Lemma 4.1. !

We can now put the above results together to obtain an estimate of,l(!)/,l("). Let {0j}j#J be the subset of perpendiculars whose projectionsare contained in a common pair of pants with " but are disjoint from ".Then by Lemmas 6.3(ii) and 6.4, we have

.

j#J

cosh uj,dj

,l(")!$

.

j#J

,dj

,l(")!$

.

j#J

l(") .

In the case that we have a measured lamination ! instead of a curve *, weobtain by the same reasoning

.

j#J

cosh uj,dj

,l(")!$

.

j#J

l(") · w((0j) ,

where w((0j) is the !-weight of the leaves of ! that go from "j to "j+1.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 33

Let us denote&((") =

.

j#J

w((0j) . (17)

By applying equation (15) and Lemmas 6.3 – 6.5 to equation (16), we obtainthe following:Lemma 6.6. Let " be a curve in a short pants decomposition PLt

of Lt and let ! be a measured lamination transverse to ", with aver-age intersection angle %. If " is extremely short and neither horizontalnor vertical, then using coordinates (l("i), -!i) relative to PLt, we have,l(!)/,l(") = "A + B + C, where

A!$ i(!,")

1l(")

, B!$ &((")l(") and

C!$ i(!,")

/(twLt(!,") " s!(Lt)) cos % + O(1)

0.

6.3 Proof of the main estimates. We are ready to prove our mainresults Theorems 6.1 and 6.2. If we apply equation (13) to ,/,l("), thenby Lemma 6.6, we obtain

0 =,l(!+

t ),l(")

+,l(!"

t ),l(")

= "(A+ + A") + B+ + B" + C+ + C", (18)

where

A± !$ i(!±t ,")

l("), B± !$ &(±

t(")l(") and

C± !$ i(!±t ,")

/(twLt(!

±,") " s!(Lt)) cos %±t + O(1)

0.

Since A+ + A" = B+ + B" + C+ + C", we get1

l(")!$

&(+t(") + &(#

t(")

i(!+t ,") + i(!"

t ,")l(") +

C+ + C"

i(!+t ,") + i(!"

t ,"). (19)

Notice that the term C+ + C" simplifies: definingD± = C± + i(!±

t ,")s!(Lt) cos %±t ,

it follows immediately from equation (14) that C+ + C" = D+ + D".Lemma 6.7. Let Ht(") be defined as in equation (12). Then we have

Ht(") $&(+

t(") + &(#

t(")

i(!+t ,") + i(!"

t ,").

Proof. The strands of !±t which intersect pants adjacent to " but which are

disjoint from ", must intersect one of the curves in B. Hence, by definitionof &(±

t("), we have

&(+t(") + &(#

t(")

i(!+t ,") + i(!"

t ,")&

.

"#B

lqt(&)lqt(")

!$ Ht(") .

34 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

To prove the inequality in the other direction, let & ! B be the curvethat realizes the maximum in the definition of Ht("). For ! = !±

t , let!"!, !"" , !"% be the collections of strands of ! that run between & and ",from & to itself, and between & and $, respectively. (As usual, there aredi!erent possible configurations of strands in each pants, in particular !""

may be empty. The inequalities which follow are however valid in all cases.)Denote the !-weight of these by w(!"!), w(!""), and w(!"%), respectively.Then

Ht(") =lqt(&)lqt(")

!$.

I=+,"

w(!I"!) + w(!I

"") + w(!I"%)

lqt(")

&.

I=+,"

w(!I"") + w(!I

"%)lqt(")

&&(+

t(") + &(#

t(")

i(!+t ,") + i(!"

t ,"). !

Proof of Theorem 6.1. Lemma 6.7, equation (19), and the remark followinggives 1/lLt(") $ Gt + HtlLt("), where Ht = Ht(") and

Gt = Gt(") =D+ + D"

i(!+t ,") + i(!"

t ,").

Hence, we must have either1

lLt(")$ Gt or

1lLt(")

$ HtlLt(") ,

from which we obtain1

lLt(")$ max

!Gt,

%Ht

".

We simplify the expression for Gt as follows. By the discussion fol-lowing equation (15) we see that either twLt(!+,") cos %+

t is positive, or|twLt(!+,") cos %+

t | = O(1), and likewise for !". Also note that twLt(!+,")and twLt(!",") are either O(1) or have opposite signs, so that

d!(!+, !") = TwLt(!+,") + TwLt(!

",") ± O(1) .

As before, let T = t " t!. Then by applying equation (14), we get

Gt(") =eT |cos %+

t |TwLt(!+,") + e"T |cos %"t |TwLt(!",")

eT + e"T+ O(1)

$ eT |cos %+t |d!(!+, !")

eT + e"T=

e"T |cos %"t |d!(!+, !")

eT + e"T. (20)

This almost completes the proof, except it remains to be shown that if1/lLt(") $ Gt, then

Gt $ e"2|t"t!|d!(!+, !") .

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 35

By equation (20), it is su"cient to show that there is some constant c > 0,independent of ", such that |cos %"

t | > c whenever T > 0 and |cos %+t | > c

whenever T < 0.Our assumption that 1/lLt(") $ Gt and the fact that lLt(") is su"-

ciently small, together with equation (20) imply that1

lLt(")!& TwLt(!

",") + TwLt(!+,") .

Let Xt = TwLt(!",")lLt(") and Yt = TwLt(!+,")lLt("). The above in-equality states that

Xt + Yt!% 1 . (21)

If T > 0, then by equation (14), |cos %"t | > |cos %+

t | so Xt > Yt"O(lLt("))by equation (15). Thus, reducing the value of the upper bound #0 on lLt(")if necessary, it follows from equation (21) that Xt is bounded below by somepositive constant, and thus the same is true of |cos %"

t |. The analogousstatement holds for | cos %+

t | when T < 0. !

Proof of Theorem 6.2. From equation (14), |cos %±t | ) e"2|T |. It follows

from equation (15) that if T 1 0 thenTwLt(!

+,") lLt(") & e"2T .

The argument for T 2 0 is similar. Now suppose that |T | = O(1). SinceTwLt(!

±,") lLt(") & d!(!+, !") lLt(")and since by Theorem 6.1,

d!(!+, !") lLt(") & e2|T |,

the result follows. !

6.4 Estimates for horizontal and vertical short curves on L. Asin section 5.6, we need the analogue of Theorems 6.1 and 6.2 for extremelyshort curves " which are either horizontal or vertical. As in that section,assume " is vertical so that i(", !") = 0.

As before, we shall obtain the estimates by applying equation (13) to,/,-!, ,/,l("). Since " is vertical,

0 =,l(!+

t ),-!

+,l(!"

t ),-!

=,l(!+

t ),-!

=)

!cos /+d!+

t .

Hence equation (14) is replaced by cos %+t = 0. Furthermore, equation (15)

gives |twLt(!+,")| = O(1).Let m"(") be the weight on " of !" = !"

0 , in other words, !" =m"(")" + ), where ) has support disjoint from ". Then following the line

36 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

of discussion in section 6.2, it is easy to check that,l(!"

t ),l(")

= e"tm"(") + &(#t(")l(") .

Hence, in place of equation (18), we obtain

0 =,l(!+

t ),l(")

+,l(!"

t ),l(")

= "A+ + B+ + C+ + e"tm"(") + B",

where A+, B±, C+ are defined as before. SinceC+ !$ i(!+

t ,")&(twLt(!,") " s!(Lt)) cos %+

t + O(1)' !$ i(!+

t ,") ,

we get1

lLt(")$ Ht(")lLt(") +

e"tm"(")i(!+

t ,")= Ht(")lLt(") + e"2t m"(")

i(!+,").

Thus we obtainTheorem 6.8. Let " be a curve which is vertical on S. If " is extremelyshort in Lt, then

1lLt(")

$max#

e"2t m"(")i(!+,")

,%

Ht(")$

.

If " is horizontal, the estimate is the same except that the first term isreplaced by e2tm+(")/i(!","), where now m+(") is the weight on " of !+.

Theorem 6.9. If " is extremely short in Lt, then the twist satisfiesTwLt(!+,") = O(1) if " is vertical and TwLt(!",") = O(1) if " is hori-zontal.

7 Comparing Lt and Gt

In this section we prove our final results. We compare the geometry of Lt

and Gt by looking at their respective thick-thin decompositions. Specifi-cally, we prove that

Ht(") $ Kt(") . (22)Combined with Theorems 5.10 and 6.1, this completes the proof of Theo-rems C and A. We show further in Theorem 7.10 that on correspondingthick components, the two metrics Lt and Gt almost coincide. Combiningthis with the information about twisting given in Theorems 5.11 and 6.2,we can then estimate the Teichmuller distance between Lt and Gt, thuscompleting the proof of Theorem D (Theorem 7.15).

As explained in the Introduction, the logical flow in the proof of equation(22) is not straightforward. We first show relatively easily in Proposition 7.1

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 37

that Ht(") % Kt("). The key point in proving the other half of equation(22) is Proposition 7.4, which shows that the metric Lt not only minimizesl#(!+

t ) + l#(!"t ), but that it also in a suitable coarse sense minimizes the

contribution to the sum made by the parts of !±t which lie in the thick part

of Lt. This is proved in section 7.2. In section 7.3 we use Proposition 7.4 todeduce that a curve that is extremely short in Gt is also extremely short in Lt

(Proposition 7.8). This is used in proving Theorem 7.10 mentioned above,from which in section 7.4 we are finally able to show that Ht(") & Kt(")(Proposition 7.12).

7.1 Curves are shorter in L(!+, !!).

Proposition 7.1. If " is extremely short in Gt, then Ht(") % Kt(").Therefore,

1lLt(")

% 1lGt(")

.

Proof. Once we show that Ht(") % Kt("), the second statement followsfrom Theorems 5.10 and 6.1.

The only case of interest is when Kt(") is large. Let Et(") be one ofthe expanding annuli Ei(") around " of larger modulus, defined as in thediscussion preceding Corollary 5.4. Denote the inner and outer boundarycurves of Et(") by ,0 and ,1. Let 2 be an essential arc from " to itselfsuch that lqt(2) = 2dqt(,0, ,1), where as usual dqt denotes distance in theqt-metric. The annulus Et(") intersects the qt-representative Y of a thickcomponent Y of (S,Gt) adjacent to ". Let us first suppose that 2 is con-tained in Y . A small regular neighborhood of ".2 has boundary consistingof " and two curves, *1, *2, which together with " bound a pair of pants.Therefore, either both *1 and *2 are contained in B or one of these twocurves must intersect a curve in B transversely. (As in equation (12), Bis the set of pants curves in a short pants decomposition PLt which areboundaries of pants adjacent to ".)

First consider the case when *1, *2 ! B. If either *1 or *2 is non-peripheral in Y , then by definition of Ht(") and %Y ,

Ht(") = max"#B

#lqt(&)lqt(")

$/ max

#lqt(*1)lqt(")

,lqt(*2)lqt(")

$/ %Y

lqt(").

If both *1, *2 are peripheral in Y , then

Ht(") % lqt(*1) + lqt(*2) + lqt(")lqt(")

$ %Y

lqt("). (23)

38 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Now consider the case when either *1 or *2 intersects a curve & ! Btransversely. Note that, for i = 1, 2,

lqt(*i) ) lqt(2) + lqt(") ) 4 diamqt(Y ) !$ %Y .

Suppose that & intersects *i and that *i is not peripheral in Y . Then itfollows from the above inequality and the definition of %Y that lqt(*i)

!$ %Y .Since by Theorem 5.5 we have lqt(*i)

!$ %Y lGt(*i), we see that lGt(*i)!$ 1.

Then, by the collar lemma for quadratic di!erentials [R1], we havelqt(&)

!% lqt(*i)!$ %Y .

The only remaining possibility to consider is when both *1, *2 are peripheralso that Y is a pair of pants. If & intersects *i, then since & must pass throughan annulus around *i which has large modulus, we conclude lqt(&) % lqt(*i).If & does not intersect *i, every arc ) of & 0 Y has both endpoints on theother curve *j, j -= i. The endpoints of ) divide *j into two arcs, oneof which together with ) forms a curve homotopic to *i. Since & passesthrough an annulus of large modulus around *j , this implies

lqt(*i) ) lqt()) + lqt(*j)!& lqt(&) + lqt(&) .

Either way, we have lqt(&)!% lqt(*i) and thus lqt(&)

!% lqt(*1) + lqt(*2).Finally, if the original arc 2 was not contained in Y , we can replace it

with an arc that is contained in Y of comparable length as in the proof ofTheorem 5.9, and run the same argument. !

7.2 Length estimates on subsurfaces. The object of this section isto prove Proposition 7.4. We begin with estimates that are necessary toanalyze the contribution to the length of a lamination associated to thethick part of the surface S. Thus if (S,') is a hyperbolic surface andY ( S is a subsurface of the thick part, we want to find an approximationto l#(!± 0 Y ). To consider the problem in general, we consider l#(!± 0 Q)for a subsurface Q with geodesic boundary. Suppose that * is a geodesicthat intersects Q but is not entirely contained in Q. The essential idea isthat we can approximate * 0 Q by piecewise geodesic arcs homotopic to* 0 Q, which alternately run along arcs perpendicular to ,Q and parallelto ,Q. The length of the parallel portion is determined by the twistingof * about the curves in ,Q, while the portion *Q perpendicular to ,Q isdefined and estimated as explained below.

Let " be a collection of disjoint simple closed geodesics on (S,') andlet Q be a totally geodesic surface which is the metric completion of acomponent of S \ ". (It is possible for two distinct boundary componentsof Q to be identified in S to a single curve " ! ", so strictly speaking,

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 39

Q is not a subsurface of S.) If ) is an essential geodesic arc with endpointson ,Q, let )Q be the shortest arc in Q that is freely homotopic to ),relative to ,Q. In this case, clearly )Q is orthogonal to ,Q. If 5 is ameasured geodesic lamination whose support is entirely contained in Q, let5Q = 5. For convenience we allow the possibility that the support of 5contains components of ,Q, remarking that this is not quite the same asthe definition in [Mi3].

Suppose ( is a measured geodesic lamination on S. Then the intersection( 0 Q is a union of components of ( that are entirely contained in Q andarcs with both endpoints on ,Q. If ) is an arc of ( 0 Q, let n()Q) denotethe transverse measure of arcs in the homotopy class [)Q]. The orthogonalprojection of ( into Q is (Q =

3n()Q))Q +

35Q, where the first sum

is taken over a representative )Q from each class of arcs in ( 0 Q and thesecond sum is taken over all components 5 of ( that are entirely containedin Q. Define

l#((Q) =.

n()Q)l#()Q) +.

l#(5Q) .

If all curves in ,Q are of uniformly bounded length, then we have thefollowing estimate of l#(( 0 Q) in terms of (Q and Tw#((,"):Lemma 7.2. Suppose l#("j) < 6 for every component "j of ,Q. Thenthere exists a constant K = K(6) such that, for any measured lamination( on S,((((l#(( 0 Q) "

4l#((Q) +

.l#("j)

Tw#((,"j)2

i((,"j)5(((( ) Ki((, ,Q) ,

where the sum is taken over all "j that intersect ( transversely.

For a proof, see the Appendix. The next lemma can be proved similarly,applying the same property of hyperbolic triangles. We omit the proof.Lemma 7.3. Suppose l#(") < #0. Let A be an embedded annulus in(S,') such that one component of ,A is the geodesic " and the othera hyperbolically equidistant curve of length #0. Then there is a uniformconstant K such that, for any measured lamination ( on S that intersects" transversely,((((l#(( 0 A) "

4log

1l#(")

+ l#(")Tw#((,")

2

5i((,")

(((( ) Ki((,") .

Here, log[1/l#(")] approximates the width of A, up to a bounded addi-tive error.

We will now apply Lemmas 7.2 and 7.3 to prove Proposition 7.4 below,which in turn is the key step to proving Theorem 7.10.

40 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

For 7 > 0, and a hyperbolic metric ' ! T (S), defineS)(') =

!" ! S : l#(") < 7

".

Proposition 7.1 implies that if " is extremely short in Gt, then we canchoose a constant # < #0, depending only on #0, such that if lGt(") < #,then lLt(") < #0. In other words, we can choose # < #0 so that

S*(Gt) ( S*0(Lt) . (24)Now, let Q = Qt be a component of S \S*(Gt). The metric Lt naturally

endows Q with the structure of hyperbolic surface with geodesic bound-ary, which by the above, satisfies lLt(") < #0 for all components " of ,Q.Henceforth, fix a constant c that satisfies #0 < c < #M. For ' = Gt,Lt,or in general, any metric that makes Q a hyperbolic surface with geodesicboundary components that are extremely short, define C(",') to be thecollar of " in (Q,') such that one component of ,C(") is (the geodesicrepresentative of) ", and the other, the equidistant curve of length c. Be-cause c < #M, the collars are all disjoint from one another. Let (QT ,') bethe metric subsurface of Q defined by

(QT ,') = (Q,')6 7

!#+Q

C(",') .

In particular, every component of ,QT has length c.Since Lt is on the line of minima, we have

lLt(!+t ) + lLt(!

"t ) ) lGt(!

+t ) + lGt(!

"t ) .

The contribution to this inequality from QT is given as follows:Proposition 7.4. If Q is a component of S \ S*(Gt), then

lLt(!+ 0 QT ) + lLt(!

" 0 QT )!& lGt(!

+ 0 QT ) + lGt(!" 0 QT ) .

To prove Proposition 7.4, we need the following two lemmas.Lemma 7.5. Suppose (Q,') is a hyperbolic surface with geodesic bound-ary such that l#(") < #0 for all " ! ,Q. Then for any measured geodesiclamination ( on (S,'), we have

l#(( 0 QT ) !$ l#((Q 0 QT ) .

Proof. We consider l#(( 0 QT ) = l#(( 0 Q) " l#(( 08

C(")) and applyLemmas 7.2, 7.3 to the right-hand side. The proof is completed by observingthat

l#((Q 0 QT ) = l#((Q) ".

!#+Q

log1

l#(")i((,") + O

/i((, ,Q)

0

and that l#(( 0 QT )!% i((, ,QT ) = i((, ,Q), due to the fact that every

component of ,QT has an annular neighborhood of definite width. !

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 41

Lemma 7.6. Suppose that (Q,') and (Q,'$) are two hyperbolic sur-faces with geodesic boundary whose boundary components are all extremelyshort. Suppose that there is a short pants decomposition of (Q,') with re-spect to which the Fenchel–Nielsen coordinates for (Q,') and (Q,'$) agree,except possibly for the lengths and twists corresponding to componentsof ,Q. Then, for any simple closed curve or arc ) with endpoints on ,Q,

l#()Q 0 QT ) !$ l#"()Q 0 QT ) .

Proof. This is essentially the same as a discussion in Minsky [Mi3, p. 283].The idea is that there is a K-bilipschitz homeomorphism (QT ,') + (QT ,'$)with constant K depending only on #0. To see this, cut Q along the pantscurves into pairs of pants and further cut each pair of pants into hexagons.Corresponding hexagons in the two surfaces have the same side lengths,except those whose edges form part of ,Q. Now truncate those hexagonswhich have an edge on ,Q by cutting o! the collar round ,Q in such a waythat the boundary of the truncated hexagon is the corresponding compo-nent of ,QT . By our construction, the non-geodesic edges of the hexagonsin the two surfaces are both equidistant curves of the same length c/2.

We define the required map piecewise from each possibly truncatedhexagon in (QT ,') to the corresponding one in (QT ,'$). Since all theFenchel–Nielsen coordinates agree in the interior of QT , we only have to seethat there is a bilipschitz map between the truncated parts of two hexagonsH and H $ with alternate sidelengths l1, l2, l3 and l$1, l2, l3 coming from thepants curves, where l1, l$1 < c/2. Since l2, l3 are uniformly bounded aboveand below, the distance between the corresponding sides in both hexagonsis also uniformly bounded above and below, see the proof of Lemma 3.3.The distance between the side of length l1 and the equidistant curve oflength c/2 is equal to log[c/2l1], up to a bounded additive error. Henceby Lemma 3.3, the distances between the sides of lengths l2, l3 and theequidistant curve of length c/2 are bounded above, while they are boundedbelow by choice of c.

It is now easy to define a bilipschitz homeomorphism between the trun-cations of H and H $. For example, fix a point O whose distance from allsides of H is uniformly bounded above and below and divide H into sixtriangles by joining O to the vertices of H. Note that if we are given twohyperbolic triangles whose side lengths are uniformly bounded above andbelow, we can map the three sides linearly to each other and then extendto a uniformly bilipschitz map on the interiors. We can do the same evenwhen one side is an equidistant curve rather than a geodesic. Now define

42 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

the required map from H to H $ triangle by triangle making it agree on theedges joining O to the vertices of H. It is clear that the resulting boundon K depends only on the initial upper bound on the lengths of the pantscurves. Note also that K + 1 as #0 + 0. !

We may assume that in the definition of the truncated surfaces QT ,the constants #0 and c are chosen small enough that any non-peripheralsimple geodesic loop contained in (Q,') is completely contained in QT .In particular, if P# is a short-pants system for ' and if & ! P# \ ,Q iscontained in QT , then so is its dual +". Let M# be the short marking of' associated to P#. We call the subset of M# thus defined, the restrictionM#|Q of M# to Q [MaM2]. Equivalently, M#|Q is the set of curves in M#

that are completely contained in Q and are non-peripheral in Q. If Q is apair of pants then M#|Q is empty.Proof of Proposition 7.4. Where convenient, we drop the subscript t.Since Q is a component of S \ S*(Gt), the curves in ,Q are included inthe set of pants curves in both ML and MG . Define a new metric - = -t

on S interpolating Gt and Lt as follows. Let X be the metric completionof S \ Q. First we choose a new pants system P' for S. The system P'

contains all the curves in ,Q, in the interior of Q it consists of the pantscurves in MG |Q, while in the interior of X it consists of the pants curvesin ML|X . We define -t by specifying its Fenchel–Nielsen coordinates withrespect to P' . The metric -t will have the same Fenchel–Nielsen coordinatesassociated to the pants curves in MG |Q as Gt and the same Fenchel–Nielsencoordinates associated to the curves in ML|X . ,Q as Lt.

Since L is on the line of minima we havelL(!+

t ) + lL(!"t ) ) l' (!+

t ) + l' (!"t ) . (25)

Let us estimate both sides of this inequality. Applying Lemma 7.2 to !=!±t ,

we obtainl' (! 0 X) = l' (!X) + 1

2

.

!#+Q

l' (")i(", !)Tw' (!,") + O/i(!, ,Q)

0,

lL(! 0 X) = lL(!X) + 12

.

!#+Q

lL(")i(", !)TwL(!,") + O/i(!, ,Q)

0.

By construction, l' (!X) = lL(!X).((l' (! 0 X " lL(! 0 X)

((

) 12

.

!#+Q

l' (")i(!,")((Tw' (!,") " TwL(!,")

(( + O/i(!, ,Q)

0.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 43

By construction, the Fenchel–Nielsen twist coordinates for - and L onany component " of ,Q coincide. Therefore, by Lemma 4.1, we have|Tw' (!,") " TwL(!,")| ) 4. Thus((l' (! 0 X) " lL(! 0 X)

(( ) 2.

!#+Q

l' (")i(!,") + O/i(!, ,Q)

0.

Substituting this into equation (25) and noting that we are working underthe assumption that all components " of ,Q are extremely short in - , weobtain

lL(!+t 0 Q) + lL(!"

t 0 Q) < l' (!+t 0 Q) + l' (!"

t 0 Q) + O/lqt(,Q)

0.

Since every component of ,Q is extremely short in - , the collar lemmaimplies that i(!, ,Q)

!& l' (! 0Q) so we may replace the last approximationby

lL(!+t 0 Q) + lL(!"

t 0 Q)!& l' (!+

t 0 Q) + l' (!"t 0 Q) .

Since for ' = L, - and ! = !±t , we have by Lemma 7.5

l#(! 0 Q) = l#(! 0 QT ) + l#/! 0 (Q \ QT )

0

!$ l#(!Q 0 QT ) + l#/! 0 (Q \ QT )

0,

we can apply Lemma 7.3 to subtract the contribution of the collars formingQ \ QT from both sides to obtain

lL(!+Q 0 QT ) + lL(!"

Q 0 QT )!& l' (!+

Q 0 QT ) + l' (!"Q 0 QT ) .

To complete the proof, we apply Lemmas 7.5 and 7.6:l' (!+

Q 0 QT ) + l' (!"Q 0 QT ) !$ lG(!+

Q 0 QT ) + lG(!"Q 0 QT )

!$ lG(!+t 0 QT ) + lG(!"

t 0 QT ) . !

7.3 Correspondence between thick components. This section con-tains the meat of our comparison between the geometries of Lt and Gt. Weshow that (Corollary 7.9) the sets of short curves on Lt and Gt coincide.Generalizing Theorem 3.8, we prove (Theorem 7.10 and Corollary 7.11)that the geometries of the thick parts of Lt and Gt are close. As in thatproof, our strategy is to use short markings to estimate lengths. The mainpoint is to use Proposition 7.4 as a substitute for the length minimizationproperty of Lt.

We need the following result which generalizes Proposition 3.6 to thickcomponents.Proposition 7.7. Assume that l#(") < #0 for every component " of,Q. Let 7 > 0 and suppose that l#(*) / 7 for every non-peripheral simpleclosed curve * in Q. Then for any simple closed geodesic $ on (S,'),

l#($ 0 QT ) !$ i(M# |Q, $) , (26)where the multiplicative constants depend only on 7.

44 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Proof. By Lemma 7.5 it is su"cient to prove thatl#($Q 0 QT ) !$ i(M# |Q, $) .

We modify the argument in [Mi2, Lem. 4.7].Notice that since QT is 7-thick, it follows from Corollary 3.4 that the

lengths of all the curves in M#|Q are bounded above. Cutting QT along thecurves in M#|Q, we obtain a collection of convex polygons {Di}, togetherwith annuli {Aj}, where one boundary component ,0Aj is a component of,QT , while the other component ,1Aj is made up of arcs in M#|Q.

Since the total length of curves in M#|Q is uniformly bounded above,the length of ,Di is uniformly bounded above, and therefore, Di has uni-formly bounded diameter. We claim that the annuli Aj also have uniformlybounded diameter. Since the length of ,0Aj is bounded below by #0, anarea argument shows that the distance between ,1Aj and ,0Aj is uniformlybounded above. Since, furthermore, the lengths of ,0Aj and ,1Aj are uni-formly bounded above, it follows that Aj has uniformly bounded diameter,as claimed. Setting

D = max{diam Di,diam Aj}gives the upper bound

l#($Q 0 QT ) ) i($,M# |Q) · D .

Since the lengths of all the curves in M#|Q are bounded above, by thecollar lemma, there is an embedded collar of definite radius d around everycurve in M#|Q. Therefore, if $ crosses & ! M#|Q, then l#($) > d · i($,&).Let k be the number of pants curves in Q. Since there must be some& ! M#|Q such that i($,&) / i($,M# |Q)/(2k), we have

l#($Q 0 QT ) > d2k · i($,M# |Q) ,

giving the desired lower bound. !Applying Proposition 7.7, we can now deduce from Proposition 7.4 that

a non-peripheral curve in Q cannot be too short in Lt:Proposition 7.8. Let Q be a component of S\S*(Gt) where # is chosen asin equation (24). Then for any non-peripheral simple closed curve * in Q,

we have lLt(*)!% 1.

Proof. First, we claim thatlGt(!

+t 0 QT ) + lGt(!

"t 0 QT ) !$ %Q . (27)

To see this, let MGt be a short marking for Gt and let MGt |Q denote itsrestriction to Q. By Proposition 7.7, we havelGt(!

+t 0 QT ) + lGt(!

"t 0 QT ) !$ i(MGt |Q, !+

t ) + i(MGt |Q, !"t ) !$ lqt(MGt |Q) .

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 45

On the other hand, lGt(MGt |Q) !$ 1. Hence by Theorem 5.5

lqt(MGt |Q) !$ %Q ,

and the claim is proved.Now if * is a non-peripheral simple closed curve in Q with lLt(*) < #0,

then consideration of the collar about * gives the estimate

lLt(! 0 QT )!% i(!, *) log

1lLt(*)

,

for any ! ! ML(S), so in particular,

lLt(!+t 0 QT ) + lLt(!

"t 0 QT )

!% lqt(*) log1

lLt(*).

Proposition 7.4 and equation (27) give

%Q!% lq(*) log

1lLt(*)

.

From the definition of %Q we have %Q/lq(*) ) 1, so that lLt(*)!% 1. !

Proposition 7.8 and Proposition 7.1 together prove Theorem A of theIntroduction, that the sets of extremely short curves on Lt and Gt coincide.More precisely, we can reformulate Proposition 7.8 as

Corollary 7.9. Let # be as in equation (24). Then there exists #$ > 0such that S*"(Lt) ( S*(Gt).

It is also now easy to complete the proof of our main comparison betweenthe thick parts of Lt and Gt:

Theorem 7.10. Let Q be a component of S \ S*(Gt) which is not a pairof pants, and let MLt be a short marking for Lt. Then

lGt(MLt |Q) !$ 1 .

Proof. By Theorem 5.5, we have

lGt(MLt |Q) !$ 1,Q

lqt(MLt |Q)!$ 1

,Q

&i(MLt |Q, !+) + i(MLt |Q, !")

'.

By Proposition 7.8, there is a constant 7 = 7(#0) depending only on #0 suchthat lLt(*) > 7(#0) for every non-peripheral curve * in Q. Therefore, wecan apply Proposition 7.7 to get

i(MLt |Q, !+t ) + i(MLt |Q, !"

t ) !$ lLt(!+t 0 QT ) + lLt(!

"t 0 QT ) .

Since the lower bound lGt(MLt |Q)!% 1 is trivial, the result now follows from

Proposition 7.4 and equation (27). !

46 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Equivalently, we can formulate Theorem 7.10 in terms of the surface Q0

obtained from Q by replacing every boundary component with a puncture.Let Lt|Q0 and Gt|Q0 be respectively, the surface Q0 equipped with the met-rics obtained from Lt and Gt by pinching the curves in ,Q but otherwiseleaving the metric unchanged. In other words, in the notation of the prod-uct region theorem in section 4.4, the collection of curves in ,Q is A andthe metrics on Q0 are defined by $0(Lt) and $0(Gt), respectively, restrictedto the component Q0 of SA. Then we haveCorollary 7.11. Let Q be a component of S \ S*(Gt). Then

dT (Q0)(Lt|Q0 ,Gt|Q0) = O(1) .

Proof. The boundary components of both (Q,Lt) and (Q,Gt) are extremelyshort. In this case, it was shown in [Mi3] (see the proof of Lemma 7.6) that(Q,Lt) and (Q,Gt) can be embedded into (Q0,Lt|Q0) and (Q0,Gt|Q0) re-spectively, by a K-quasi-conformal map, where K depends only on #0. Sincesimple curves do not penetrate the thin part of Q0, the restriction MLt |Q isa short marking for (Q0,Lt|Q0). By Theorem 7.10, we have lGt(MLt |Q) !$ 1.The result now follows as in the proof of Theorem 3.8. !

7.4 Comparison of lengths of short curves. Theorem 7.10 allowsus to complete the proof of equation (22):Proposition 7.12. Let " be an extremely short curve on Lt. Then

Ht(") & Kt(") .

Proof. With B as in equation (12), let & ! B be the curve that has thelargest qt-length, so that

Ht(") !$ lqt(&)lqt(")

.

Since S*(Gt) ( S*0(Lt), it follows that the curves in PLt are disjoint fromS*(Gt). Thus, " and & are contained in the closure of a common componentQ of S \ S*(Gt).

Suppose first that & is not peripheral in Q. Then & ! MLt|Q, so thatby Theorem 5.5 and 7.10,

lqt(&) !$ %QlGt(&) !$ %Q .

If in addition, " is not peripheral, then lqt(") !$ %Q so that Ht(") !$ 1 andthe desired inequality holds trivially. If " is peripheral, then

Ht(") !$ lqt(&)lqt(")

!$%Q

lqt(")) Kt(") .

Now suppose that & is peripheral in Q. If Q is a pair of pants, thenthe desired inequality follows from the definition of Ht(") and Kt("). If

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 47

Q is not a pair of pants, then since the component of Q \ MGt |Q contain-ing & is an annulus whose one boundary component is & and the other afinite (at most 4) union of arcs coming from curves .$i in MGt |Q, again byTheorem 5.5 we obtain

lqt(&) ).

lqt($i)!$

.%QlGt($i)

!$ %Q ,

from which the result follows as before. !Theorem C now follows immediately from Theorem 6.1, Proposition 7.1,

and Proposition 7.12, completing our comparison between short curves onGt and Lt.Theorem 7.13. Let " be any curve on S which is neither vertical norhorizontal. If " is extremely short in Lt, then

1lLt(")

$ max!Dt("),

%Kt(")

".

In case " is vertical or horizontal, we haveTheorem 7.14. If " is vertical, then

1lLt(")

$ max!e"2t Mod F0("),

%Kt(")

".

If " is horizontal, then the estimate is the same except that the first termis replaced by e2t Mod F0(").

Proof. There are multiplicative constants depending only on the fixed lam-inations !± such that

m'(")i(!±,")

!$ Mod F0(")

(see Theorems 6.8 and 5.12) holds independently of " in a tautological way,due to the fact that the total number of vertical (or horizontal) curves isfinite; it is bounded above by "1(S). The proofs of Proposition 7.1 and 7.12go through in this case, so that Ht(") $ Kt("). Hence the estimate followsfrom Theorem 6.8. !

7.5 Teichmuller distance. With the preceding collection of results atour disposal, Theorem D of the Introduction becomes an easy applicationof Minsky’s product region theorem 4.5.Theorem 7.15. The Teichmuller distance between Gt and Lt is given by

dT (S)(Gt,Lt) = max!#S"(Gt)

12

((((loglGt(")lLt(")

(((( ± O(1) .

Proof. As noted before, lLt(") ) #0 for every " ! S*(Gt). To simplifynotation, let Et = S*(Gt). By Theorem 4.5, we havedT (S)(Gt,Lt) = max

!#Et

!dT (SEt )

($0(Gt),$0(Lt)), dH!($!(Gt),$!(Lt))"±O(1) ,

48 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

where SEt is the surface obtained from S by removing Et and replacing theresulting boundary components by punctures and $0,$! are defined as insection 4.4. From Corollary 7.11 we deduce that

dT (SE )

/$0(Gt),$0(Lt)

0= O(1) .

If t " t! > 0, then by Theorem 5.11 or 5.13 we haveTwGt(!

+,")lGt(") = O(1) ,

and by Theorem 6.2 or 6.9 we haveTwLt(!

+,")lLt(") = O(1) .

Applying Corollary 4.7 to Gt and Lt and the lamination !+, we findexp 2dH!

/$!(Gt),$!(Lt)

0 !$ lGt(")/lLt(") .

If t " t! < 0, the same result holds by applying a similar argumentwith !". !

7.6 Examples. The combinatorial nature of our length estimates al-lows us to use Theorem 7.15 to construct examples in which L and G havea variety of di!erent relative behaviors. As a special case, if S is a once-punctured torus or four-times-punctured sphere, every thick componentmust be a pair of pants. In this case, Kt(") is bounded and therefore, acurve gets short only if d!(!+, !") is large:Corollary 7.16. If S is a once-punctured torus or a four-times punc-tured sphere, then for any measured laminations !+, !", the associatedTeichmuller geodesic and line of minima satisfies

dT (S)(Gt,Lt) = O(1) .

On surfaces of higher genus, it is possible to have !+, !" and " such thatd!(!+, !") is bounded while Kt(") is arbitrarily large. We can constructa simple example as follows. Take two Euclidean squares each of area 1/2and cut open a slit of length 8 at each of their centers. Although it isnot necessary, for concreteness we can assume that in both squares, theslit is parallel to a pair of sides. Foliate each square by the two mutually

Figure 4: Glue two slit tori along slit.

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 49

orthogonal foliations that both make angle 0/4 with the slit and, for each,take the transverse measure induced by the Euclidean metric. Identify pairsof sides in each square to obtain two one-holed tori T1, T2 and glue T1, T2

along their boundaries, as shown in the figure, to obtain a genus two surfaceS with waist curve ". The two foliations match along " and specifying oneto be the vertical foliation defines a quadratic di!erential q = q0 on S. LetG and L be respectively, the Teichmuller geodesic and the line of minimadefined by the vertical and horizontal foliations !" and !+ of q0. In thisexample, the foliations are rational, but it is easy to see that varying theinitial angle of the slit gives more general foliations.

Let qt be the associated family of quadratic di!erentials. Note that attime t = 0 the curve " is balanced. The qt–geodesic representative of "is unique and the flat annulus corresponding to " is degenerate. Thus, byProposition 5.8, we have d!(!+, !") = O(1). But since %T1 = %T2 = 1/

,2

at t = 0, we have K0(") $ 1/8 (the shaded region indicates a maximalexpanding annulus). Assuming 8 is very small, Theorems 5.10 and 6.1 give

1lG0(")

$ log18

and1

lL0(")$ 1,

8.

Thus, by Theorem 7.15,

dT (S)(G0,L0) %12

loglG0(")lL0(")

$ log1,

8 log[1/8].

In fact, because for any two hyperbolic metrics ', - we have [W2]

dT (S)(', -) / 12

log sup&#S

l#(*)l' (*)

,

and because the length of " along G is (coarsely) shortest at the balancetime t! = 0 [R1], the following stronger inequality holds:

inft#R

dT (S)(Gt,L0) / inft#R

12

loglGt(")lL0(")

% 12

loglG0(")lL0(")

.

Taking 8 small enough we can ensure that L0 is as far as we like fromany point on G. This example can be easily extended to any surface oflarge complexity, by which we mean a surface whose genus g and numberof punctures p satisfies 3g " 4 + p / 1. In summary,Corollary 7.17. If S is a surface of large complexity, then given anyn > 0, there are measured laminations !+(n), !"(n) on S which dependon n, such that for the associated Teichmuller geodesic G(n) and line ofminima L(n),

inft#R

dT (S)

/Gt(n),L0(n)

0> n .

50 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

It is also possible to construct examples for any such surface wherethe two measured laminations are fixed and the associated Teichmullergeodesic and line of minima satisfy dT (S)(Gtn ,Ltn) > n for a sequence oftimes tn + # as n + #. This however, is beyond the scope of this paper.

8 Appendix

We give proofs of the length estimates that were deferred in previous sec-tions.

Proof of Lemma 3.3. Let H be one of the two right-angled hexagonsobtained by cutting P along its three seams. Let li = l("i)/2 and di be thelength of the seams, labeled as shown in Figure 5. By the cosine formula

Figure 5: Half a pair of pants.

for right-angled hexagons, we have

cosh d3 =cosh l3 + cosh l1 cosh l2

sinh l1 sinh l2.

By hypothesis, li < L/2, so sinh li!$ li and cosh li

!$ 1, where the multi-plicative constants involved depend only on L. Therefore,

cosh d3!$ 1

l1l2.

It follows that d3 = log[1/l1] + log[1/l2] ± O(1), where the bound on theadditive error depends only on L.

Now we estimate the length of the perpendicular from "i to itself. Letx be the length of the perpendicular as in Figure 5. By the formula forright-angled pentagons, we have

cosh x = sinh l2 sinhd3 .

Since sinh l2!$ l2 and by the above, sinh d3

!$ 1/[l1l2], it follows thatcosh x

!$ 1/l1. Hence, x = log[1/l1]±O(1). Since P is made of two isometriccopies of H, we obtain the desired estimate. !

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 51

To prove Lemma 6.3, in addition to the standard hexagon and pentagonformulae (see for example [B] or [S1]), we need the following expression forderivatives of side-lengths derived in [S1, Prop. 2.3]:Lemma 8.1. Let H be a planar right-angled hexagon with sides labeledi = 1, . . . , 6 in cyclic order about ,H. Let li denote the length of side i andfor n mod 6, let pn,n+3 be the perpendicular distance from side n to siden + 3. Letting $ denote derivative with respect to some variable x, we have

(cosh pn,n+3)l"n = l

"n+3 " (cosh ln"2)l

"n"1 " (cosh ln+2)l

"n+1 . (28)

It is convenient to subdivide Lemma 6.3 into two parts, Lemma 8.2and Lemma 8.3, depending on whether or not the common perpendicularin question is adjacent to ". We begin with a somewhat more detaileddiscussion of the possible configurations.

Let P be a pair of pants in S\PLt that has " as a boundary component.For clarity, we distinguish between the three boundary curves ",&, $ of Pand their projections 0("),0(&),0($) to S. We may always assume that0($) -= 0("). There are then two possible configurations depending onwhether or not 0(&) = 0("). Figure 6(a) represents the case in which0(&) = 0(") and Figure 6(b), the case in which 0(&) -= 0("). In (b), we donot rule out the possibility that 0(&) = 0($). This leads to a dichotomyin the formulae used in the proofs, but not in the final estimates. Let d bethe length of the perpendicular between " and $ and let h! be the lengthof the shortest perpendicular from " to itself.

Figure 6: (a) 0(") = 0(&); (b) 0(") -= 0(&).

Lemma 8.2. Suppose that " is extremely short in Lt. Then the derivativesof the perpendiculars adjacent to " are as follows:

(i) d$ =,d

,l(")!$ " 1

l("), (ii) h$

! =,h!

,l(")!$ " 1

l(").

52 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Proof. Let x, y, z be the lengths of the perpendiculars as shown in Figure 6.(i) In case (a), the formula (28) and the pentagon formula ([S1] lemma

2.1) give, respectively,

d$ cosh x = "cosh d

2, cosh x = sinh d sinh

l(")2

.

If l(") is small, then sinh l(") !$ l(") and by Lemma 3.3, d % 1/ log l(") sothat coth d = O(1). Hence

d$ = " coth d

sinh[l(")/2]!$ " 1

l(").

In case (b), using the same formulae as above, we get

d$ cosh x =1 " cosh d

2, cosh x = sinh d sinh

l(")2

.

Now by Lemma 3.3, d = 2 log[1/l(")]±O(1) and therefore sinh d!$ 1/l(")2 !$

cosh d. Hence,

d$ =1 " cosh d

2 sinh d sinh[l(")/2]!$ l(")[1 " cosh d] !$ " 1

l(").

(ii) In case (a), h! = 2y and cosh y = sinh d sinh[l($)/2]. Di!erentiatingboth sides, we get

y$ =sinh[l($)/2] cosh d

sinh yd$.

By Lemma 3.3, cosh d!$ 1/[l(")l($)] and sinh y

!$ 1/l("), so we obtainy$

!$ "1/l("), as desired.In case (b), h! = d and by [S1] equation(6),

d$ cosh z = " cosh d .

Substituting cosh z!$ 1/l($) and cosh d

!$ 1/[l(")l($)] gives the desiredestimate. !

Now consider perpendiculars in P that are disjoint from ". Let h%

be the length of the perpendicular from $ to itself, as shown in Figure 7.Further, if 0(&) -= 0("), let D be the length of the perpendicular between&, $. We have h% = D when 0(&) = 0($) (see Figure 7(b)).Lemma 8.3. Suppose that " is extremely short in Lt. Then the derivativesof the perpendiculars not adjacent to " are as follows:

(i) h$% =

,h%

,l(")!$ l(") , (ii) D$ =

,D

,l(")!$ l(") .

Proof. (i) Assume that 0(&) -= 0($) so that we are in the configurationof Figure 7 (a) or (c). Consider the ‘front’ hexagon in P and denote the

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 53

Figure 7: (a) 0("), 0(&), 0($) all distinct; (b) 0(&) = 0($); (c) 0(") = 0(&)

lengths of the sides as shown in Figure 8, so that l1 = l(")/2, l2 = l($)/2,and z = h%/2. By the pentagon formula, cosh z = sinh d3 sinh l1. Takingthe derivative with respect to l1, we get

z$ sinh z = d$3 cosh d3 sinh l1 + sinh d3 cosh l1 .

In the proof of Lemma 8.2, we had d$3 = " coth d2/ sinh l1, and by the

Figure 8: Front right-angled hexagon.

cosine formula for right-angled hexagons, we have

cosh l1 =cosh d1 + cosh d2 cosh d3

sinhd2 sinh d3.

Substituting these, we get

z$ =1

sinh z· cosh d1

sinhd2.

Now by the sine formula for right-angled hexagons,1

sinhd2=

sinh l1sinh d1 sinh l2

.

With this, we have

z$ = sinh l1 ·cosh d1

sinhd1· 1sinh l2 sinh z

!$ sinh l1!$ l1 ,

since coth d1!$ 1 and sinh z

!$ 1/l2 when l1, l2 are respectively, bounded.

54 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

In the case 0(&) = 0($), we have h$% = D$, which is computed below.

(ii) Let y be the length of the perpendicular between " and the com-mon perpendicular of &, $, as in Figure 6. Then by equation (28) and thepentagon formula, we have

D$ cosh y = 1 , cosh y = sinh d sinhl($)2

.

By Lemma 3.3,d = log

&1/l(")

'+ log

&1/l($)

'± O(1) ,

and therefore sinh d!$ 1/[l(")l($)]. Since the pants system is short,

sinh[l($)/2] !$ l($). Thus D$ !$ l("), as claimed. !Lemmas 8.2 and 8.3 together prove Lemma 6.3.

Proof of Lemma 7.2. If ( has a component 5 whose support is containedin Q, then 50Q = 5 = 5Q, which has no e!ect on the inequality. Thus, weassume that no component of ( has support entirely contained in Q. Then,for simplicity, let us further assume that ( is a simple closed curve. Sinceboth sides of the inequality scale linearly with weights, it is su"cient toprove the lemma under this assumption. The basic idea is to approximatean arc ) of (0Q with the union of )Q and segments pp, qq which run along,P between the endpoints p, q of )Q and the corresponding endpoints p, qof ). Let "p,"q denote the components of ,P that contain p, q, respectively.It is possible that "p = "q.

We will show that there are uniform constants C,C $ such that((l#(pp) " l#("p) · Tw#((,"p)/2(( < C (29)((l#(qq) " l#("q) · Tw#((,"q)/2(( < C((l#()) " [l#(pp) + l#()Q) + l#(qq)]

(( < C $. (30)It is convenient to consider the picture in the universal cover H2, as shownin Figure 9. Let ) be a lift of ) and let "p, "q be the lifts of "p,"q thatcontain the endpoints of ). Since )Q is homotopic to ) relative to ,Qand is perpendicular to ,Q, its lift )Q is the unique perpendicular between"p, "q, drawn as the segment pq in the figure. There are two possible cases,depending on whether or not ) intersects )Q.

Let p$, p$$ be the feet of the perpendiculars as shown. That is, p$ is thefoot of the projection from the geodesic ray extending ) to "p and p$$ is thefoot of the projection from "q to "p. In either case, by definition of twist,

Tw#((,"p, p) = 2l#(pp$)/l#("p) ,

and furthermore, ((l#(pp) " l#(pp$)(( < l#(pp$$) . (31)

GAFA LINES OF MINIMA AND TEICHMULLER GEODESICS 55

Figure 9: Approximating the length of ).

On the other hand, by trigonometry in H2, we havecosh l#(pp$$) = 1/ tanh l#(pq) = 1/ tanh l#()Q) .

Since )Q goes from "p to "q and since by hypothesis l#("p), l#("q) < 6,it follows from the collar lemma that l#()Q) > c(6) for some constant c(6)depending only on 6. This implies that there is a constant C = C(6)depending only on 6 such that l#(pp$$) < C(6). Therefore, equation (31)gives equation (29), as desired. Of course, the same argument appliesto "q, so ((l#(qq) " l#("q) · Tw#((,"q, q)/2

(( < C . (32)To show equation (30), we use the well-known fact that, for any /0 > 0,

there exists a constant k(/0) such that, for any hyperbolic triangle withsidelengths a, b, c and angle / opposite to c with / / /0, we have a+ b" c <k(/0). In the case where ) intersects )Q, as in the figure on the left, weapply this to the triangles 3opp and 3oqq and get

l#(pp) + l#(pq) + l#(qq) " l#(pq) < k(0/2) .

In the case on the right, we apply this to triangles 3pqp and 3pqq. Tosee that !qqp is bounded below by some /0, observe that

!qqp = 0/2 " !pqp ,

and thatsin(!pqp) <

1cosh l#(pq)

.

Since l#(pq) = l#()Q) > c(6), it follows that !pqp is bounded away from0/2 and so !qqp is bounded below by some constant /0 = /0(6), as desired.Thus in this case,

l#(pp) + l#(pq) + l#(qq) " l#(pq) < k(0/2) + k//0(6)

0,

completing the proof of equation (30).

56 Y.-E. CHOI, K. RAFI AND C. SERIES GAFA

Combining equations (29),(30), and (32) we obtain((((l#()) "

4l#()Q) + l#("q)

Tw#((,"q)2

+ l#("p)Tw#((,"p)

2

5(((( < K(6) .

Summing over all arcs ) in ( 0 Q, we obtain((((l#(( 0 Q) "

4l#((Q) +

.

j

l#("j)Tw#((,"j)

2i((,"j)

5(((( < K(6)i((, ,Q) . !

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Young-Eun Choi, Education Program for Gifted Youth, Ventura Hall, StanfordUniversity, Stanford, CA 94305-4101, USA [email protected]

Kasra Rafi, Department of Mathematics, University of Chicago, 5734 S. Uni-versity Avenue, Chicago, IL 60637, USA [email protected]

Caroline Series, Mathematics Institute, University of Warwick, Coventry CV47AL, UK [email protected]

Received: May 2006Revision: November 2006Accepted: February 2007