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T H E V E L L I N G - K I R I L L O V M E T R I C O N T H E U N I V E R S A L T E I C H M U L L E R C U R V E

By LEE-PENG TEO

A b s t r a c t . We extend Velling's approach and prove that the second variation of the spherical areas of a family of domains defines a Hermitian metric on the universal Teichmiiller curve, whose pull-back to Diff +(S 1)/S 1 coincides with the Kirillov metric. We call this Hermitian metric the Velling-Kirillov metric. We show that the vertical integration of the square of the symplectic form of the Velling-Kirillov metric on the universal Teichmiiller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichrliiller space. Restricted to a finite dimensional Teichmiiller space, the vertical integration of the corresponding form on the Teichmiiller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmiiller space.

1

I n t r o d u c t i o n

Let T(1) be the universal Teichmtiller space and T(1) be the corresponding universal Teichmiiller curve. T(1) and T(1) have the natural structure of infinite dimensional complex manifolds, and the natural projection p : 7"(1) --+ T(1) is a holomorphic fibration. In [Vel], J. Velling introduced a metric on T(1) by using spherical areas. Namely, consider the Bers embedding of T(1) into the Banach space

Aoo(A) = ~r holomorphic on A : sup [r -[z12)2[ < c~ ~,

k zEA )

For every Q E Am(A) and t small, the solution to the where A is the unit disc.

equation

(1.1) S ( f tQ) = tQ,

where S ( f ) is the Schwarzian derivative of the function f, defines a family of domains fit = ftQ(A). Here ftQ is normalized so that ftQ(O) = O, f~Q(o) = 1

271 JOURNAL D'ANALYSE MATH~MATIQUE, Vol. 93 (2004)

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2 7 2 L.P. T E O

and f t ~ ( 0 ) = 0. Veiling proved that the spherical area A s ( ~ t ) o f the domain f~t satisfies

~2t2As(f~t)lt= o >_ O.

This defines a Hermitian metric on the tangent space to T(1) at the origin, identified Our first result, T h e o r e m 3.4, is the following explicit formula for with Am(A). this metric: 1 d 2 oo 11 Q N2s - 27r "d-~ As(f~)[~=o = ~ nlant2' n = 2 c ~ 3

w h e r e Q(z) = ~ n = : ( n -- n)anz n-~. T h e series converges for all Q 9 A m ( A ) . However, since the spherical area o f the d o m a i n f t Q ( A ) is not independent o f the choice o f the function f Q that satisfies (1.1), II 9 tls does not naturally define a metric on T(1) by right group translations. 1 Nevertheless, Vclling's approach can be generalized to define a metric on the universal Teichmiiller curve T(1). This is achieved by a natural identification o f T ( 1 ) with the space Homeoqs (S 1)/S 1 - - the subgroup o f orientation preserving quasisymmetric h o m e o m o r p h i s m s o f the unit circle that fix the point 1, and with the space

79 = { f : A ---+ ~ a univalent function : f ( 0 ) = 0, i f ( 0 ) = 1, f has a quasiconformal extension to ~:},

which we prove in Section 2. This endows T ( 1 ) with a group structure. 2 Following Velling's approach to T(1), given a one-parameter family o f univalent functions

f t : A ~ C 9 ~ , ftlt=o = id, which defines a tangent vector v corresponding to

dft]t=O at the origin, we define a metric on the tangent space to T(1) at the origin by

IIv

II ~=

~~-~As(ft(A))tt=o

and extend it to every point o f T(1) by right translations. This metric is Hermitian and K~ihler. M o r e remarkably, its pull-back via the e m b e d d i n g Diff+(S1)/S 1 Homeoqs(S1)/S 1 ~ T ( 1 ) is precisely the metric

oo

II v 112= ~ nlc=l 2

r t = l

o n Diff+(S1)/S 1 introduced b y Kirillov [Kir87, KY87] via the coadjoint orbit method. Here v = ~ n CneinO0/00' C-n = ~nn is a vector field on S 1. We call this 1The metric on T(1) defined as a pull-back of the Hermitian metric on Aoo(A) given by II " IIs is not natural. It does not induce a metric on finite dimensional Teichrniiller spaces embedded in T(1) since these embeddings are base-point dependent.

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THE VELLING-KIRILLOV METRIC 273

K~fihler metric on T(1) the Velling-Kirillov metric and prove that it is the unique fight invariant K~Jhler metric on T(1).

Let n be the symplectic form of the Velling-Kirillov metric on T(1). We consider the (1, 1) form w on T(1), which is the vertical integration of the (2, 2) form a/X~ on T(1), i.e., integration of ~/x~ over the fibers of the fibrationp : T(1) -~ T(1). We show that this is equivalent to Velling's suggestion of averaging the Hermitian form ]l " Ils along the fibers of T(1) over T(1). Our second result, which we prove in Theorems 4.2 and 4.3, is that w is the symplectic form of the Weil-Petersson metric on T(1), defined only on tangent vectors which correspond to H 3/2 vector fields on S 1 .

When F is a cofinite Fuchsian group, the Teichmiiller space T(F) of F embeds holomorphically in T(1). The Bers fiber space/3)v(r) is the inverse image of T(F) under the projection map T(1) --+ T(1), and the Teichmtiller curve Y'(F) is a quotient space of BY'(F). The symplectic form n is well-defined when restricted to bt'(F). We prove in Theorems 4.9 and 4.10 that the vertical integration of t~/x via the map 5r(F) --r T(F) is the symplectic form that defines the Weil-Petersson metric on F.

In the Appendix, we consider an analogue of the Bers embedding for T(1). We prove that T(1) embeds into the Banach space

Aoo(A) = ( ~ holomorphic on A : sup ]r -[z[2)[ < o c t ,

l zEA )

and its image contains an open ball about the origin of Aoo (A). We also verify that ,400 (A) and Aoo (A) @ C induce the same complex structure on T(1). These results are not used in the main text.

The content of this paper is the following. In Section 2, we review different models for the universal Teichmfiller space and the universal Teichmiiller curve and study their relations with the homogeneous spaces of Homeoqs(S1). In Sec- tion 3, we review Velling's approach and define a metric on the universal Te- ichmtiller curve. We prove that its pull-back to D i f f + ( S 1 ) / S 1 coincides with the Kirillov metric. In Section 4, we prove that the vertical integration of the square of the symplectic form of the Velling-Kirillov metric is the symplectic form that defines the Weil-Petersson metric on Teichmtiller spaces. In the Appendix, we consider an embedding of T(1).

A c k n o w l e d g e m e n t s . This work is an extension of a part of my Ph.D. thesis. I am especially grateful to my advisor, Leon A. Takhtajan, for the stimulating discussions and useful suggestions. I would also like to thank him for bringing this subject to my attention. J. Velling kindly made his unpublished manuscript [Vel]

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274 L.P. TEO

available, which has been a great stimulation for the present work. The author has quoted or reproduced some of his results for the convenience of the reader.

2 U n i v e r s a l T e i c h m i i l l e r s p a c e a n d t h e u n i v e r s a l T e i c h m i i l l e r c u r v e

2.1 T e i e h m i i l l e r t h e o r y . Here we collect basic facts from Teichmtiller theory. For details, see [Nag88, Ah187, Leh87].

Let T(1) be the universal Teichmtiller space. There are two classical models of this space.

Let A be the open unit disc and A* = ~ \ N = {z E C U {oo} I

Izl

> 1} the exterior of the unit disc. Let L ~ (A*) (resp., L ~~ (A)) be the complex Banach space of bounded Beltrami differentials on A* (resp., A) and let L ~176 (A*)~ be the unit ball of L ~ (A*). For any # E L ~ (A*)1, we consider the following two constructions.

(I) Model A: w~, theory.

We extend # by reflection to A, i.e.,

(2.1) #(z) = # 2-~, z E A.

There is a unique quasiconformal map w~, fixing - 1 , - i and 1, which solves the Beltrami equation

It satisfies (2.2)

by the reflection symmetry A*.

(II) Model B: w i' theory.

(w~,)~ = ~ ( w . ) z .

1 ( 1 )

w.(z)

~ "

(2.1). As a result, w , fixes the unit circle S 1, A and

We extend # to be zero outside A*. There is a unique quasiconformal map w ~', holomorphic on the unit disc, which solves the Beltrami equation

~ = , ~ ,

and is normalized such that f = w~'lzx satisfies f(0) = 0, f'(0) = 1 and f"(0) = 0. The universal Teichm(iller space T(1) is defined as a set of equivalence classes of normalized quasiconformal maps

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THE VELLING-KIRILLOV METRIC 275

where lz " u if and only if wv = w~ on the unit circle, or equivalently, w ~' = w" on the unit disc.

Using model B, we can identify T(1) with the space

79 = { f : A --r C univalent : f(0) = 0, if(0) = 1, f"(0) = 0; f has a quasiconformal extension to C}.

Let

S(f)

be the Schwarzian derivative of the function f , which is given by s ( f ) = \ T ] , - -~ ~ , T ]

Let Aoo(A) be the Banach space

z2, 3

= A 2 \ T ] "

A~176 ={ e h~176176 ~ A : sup

<oo).

The Bets embedding T(1) ~ Aoo(A), which maps [#] - - the equivalence class o f # to S(w~'ih), endows T(1) with a unique structure of a complex Banach manifold such that the projection map

0 : L~176 -+ T(1)

is a holomorphic submersion. In particular, L ~ ( A * ) I and Aoo (A) induce the same complex structure on T(1).

The derivative of the map r at the origin

Do~I , : L~176 *) ~, ToT(1)

is a complex linear surjection, with kernel ~'(A*) - - the space o f infinitesimally trivial Beltrami differentials. Explicitly,

~(A*)= {'eL~176 ff vr VOeAI(A')}

where A I (A*) is the Banach space of L t (with respect to Lebesgue measure on A*) holomorphic functions on &*.

Define

~r holomorphic on A ' : sup ]r -[zl2)21 < c ~ Aoo(A*)

[ zEA* )

and its complex anti-linear isomorphic space

= ~#(z) = (1 - t z ] 2 ) 2 5 i z ) : , E Aoo(A*)}, n - l , l ( A *)

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2 7 6 L.P. TEO

the space of harmonic Beltrami differentials on A *. There is a canonical splitting L ~ ( A *) = N'(A*) @ a - l ' l ( ~ * ) ,

which identifies the tangent space at the origin of T(1) with f~-l,1 (A*). Moreover, the Bers embedding induces the isomorphism f~-l, 1 (A*) -7+ Aoo (A) given by

(2.3) tt ~ r = - ~ (~--_ z)4 .

A*

L~(A*)I has a group structure induced by the composition of quasiconformal maps, A * # = V , Explicitly, it is given by V --- where w u = w x o w u . /~ + (h o~,, "~ (~_z_~ ~*J (w~)~ 1 + g ( A o w ~ kv.eb. UJ (w~,),

This group structure descends to T(1). Moreover, the right group translation by [#], R M : T(1) --+ T(1), [A] ~ [A 9 #] is biholomorphic. However, the left group translation is not even a continuous map on T(1) (see, e.g., [Nag88, Leh87]).

R e m a r k 2.1. Conventionally, the model of the universal Teichmiiller space is the complex conjugate of the one we define above. Consider the natural complex anti-linear isomorphism

L~176 - + L ~ 1 7 6

# ~+ # = t z 2-- ~, z ~ A.

Setting/~ to be zero outside A, we obtain a unique solution of the Beltrami equation

w~ = ~w~,

which is holomorphic on A* and normalized such that g = w~lA. has Laurent expansion at c~ given by

[ a2 a 3 \

(2.4) g ( z ) = z~l + ~ + )-g + - . - ) . Thus T(1) is identified with the space

D* = {g : A* ~ ~ univalent : g has Laurent expansion at oo given by (2.4) and has quasiconformal extension to C}.

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THE V E L L I N G - K I R I L L O V METRIC 277 The universal Teichmiiller curve T(1) is a fiber space over T(1). The fiber over each point [#] is the quasidisc wU(A *) E C with the complex structure induced

from ~,

7-(1) = {([,],z) : [,] e T(1), z e

(2.5)

It is a Banach manifold modeled on A~,(A) @ (2. We have a real analytic isomor- phism between T(1) x A* and 7-(1) given by

([u],z) ([u],w"(z)).

2,2 H o m o g e n e o u s spaces of HomeOqs(S 1). Let HomeOqs(S 1) be the

group of orientation preserving quasisymmetric homeomorphisms of the unit circle S 1. It contains the subgroup of orientation preserving diffeomorphisms - - Diff+ (S 1). We denote by MSb(S 1) the subgroup of MSbius transformations and, abusing notation, denote by S 1 the subgroup of rotations.

Consider the model A of the universal Teichmiiller space T(1) given above. Clearly, the map T(1) 3 [#] ~

wulsl E

Homeoqs(S 1) is well-defined and one-to- one. The Ahlfors-Beurling extension theorem implies that its image consists of all normalized orientation preserving quasisymmetric homeomorphisms of the unit circle (see, e.g., [Ber72, Nag88, Leh87]); in other words,

T(1) ~ Homeoqs(SX)/MSb(S1).

Let # E f~-l'l(A*) be a tangent vector at the origin of T(1). It generates the one-parameter flow wtu; and the corresponding vector field is given by

~buO/Oz,

where

27ri (~ - z)(r + 1)(r + i)(~ - 1) de C

and/2 is the extension o f # by reflection to C. Restricted to S 1, we have ~bu(z) =

izu(z),

where u(eW)0/08 is the vector field on S 1.

It was proved by Reimann (see [Rei76, GS92, Nag93]) that the tangent space to Homeoqs (S 1) at the origin is the Zygmund space

A(S1)={u(ei~

and

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278 L. P. TEO

where

f

A(R) = t F : R ~ I1~ : (i)F is continuous, and

(ii) [F(z + t) + F ( z - t) - 2F(z)[ < B It[ for some B, Vz, t

6 ~}.

By imposing extra normalization conditions, we can characterize the tangent space at the origin o f Homeoqs(S' ) / S l and norneoqs(S t ) / M r b ( S 1 ) in a similar way.

R e m a r k 2.2. It is not known how to characterize the Z y g m u n d space A ( S ' ) using Fourier coefficients on S ~.

In [Kir87], Kirillov considered the Lie group Diff+(S 1) and proved that there is a natural bijection between the space/C o f smooth contours o f conformal radius 1 which contain 0 in their interior and the space D i f f + ( S X ) / S 1. We generalize this bijection in the following theorem.

T h e o r e m 2.3. There is a natural bijection between the space Homeoqs (S l ) / S l

a n d the space ICqe o f all quasicircles, i.e. images o f the unit circle under quasi- conformal maps o f conformal radius 1 which contain 0 in their interior. Moreover, f o r every 7 E Homeoqs(S1)/S 1, there exists two univalent functions f : A ~ C a n d

9 : A" ~ C determined by the following properties:

1. f and 9 admit quasiconformal extensions to quasiconformal mappings o f t ; 2. 7 = 9 - 1 ~ f l s ' m o d S 1 ;

3. f ( 0 ) : 0, f ' ( 0 ) = 1; 4. g ( ~ ) = ~ , g ' ( ~ ) > O.

P r o o f . By the A h l f o r s - B e u r l i n g extension theorem, an orientation preserving

quasisymmetric h o m e o m o r p h i s m 3' o f the unit circle can be extended to a quasi- conformal map w o f C satisfying the reflection property (2.2). Let/z be the Beltrami differential o f the map WIA-. Up to a linear fractional transformation, w agrees with w u as defined in Section 2.1, i.e., w = al o w u for some al E PSU(1,1). T h e corresponding map w u (Section 2.1) is h o l o m o r p h i c inside the unit disc A. Define g = a2 o wu o w -1, where a2 E PSL(2, C) is uniquely determined by the requirement that f = a2 o wU satisfy f(0) = 0, if(O) = 1 and g satisfy g(oo) = oo. T h e maps flA and glA" are holomorphic. T h e y do not depend on the extension o f 7, and we have 7 = g-1 o f l s , . The image o f S 1 under f , which is the same as

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THE VELLING-KIRILLOV METRIC 279

the image of S 1 under g, is by definition a quasicircle C with conformal radius 1. By post-composing w with a rotation, we can arrange for the map 9 also to satisfy g ' ( ~ ) > 0.

Conversely, by definition, a quasicircle C is the image o f S 1 under a quasicon- formal map h : C -+ C. Let #1 be the Beltrami differential of hl~, extended to A* by reflection. Let wul be a solution of the corresponding Beltrami equation. Then f = h o wu~l is a quasiconformal map which is holomorphic inside A. When 0 is in the interior of C, there is a unique way to normalize wul by post-composition with a PSU(1, 1) transformation such that f(0) = 0 and if(0) > 0. The image of S 1 under f is the quasicircle C. In fact, by the Riemann mapping theorem, f[ r, is uniquely determined by C and the normalization conditions f(0) = 0, if(0) > 0. That C has conformal radius 1 implies that if(0) = 1. Let # be the Beltrami differential of flA*, extended to A by reflection. Let w u be a solution of the corresponding Beltrami equation. Define g = f o w ; 1 o a, where a E PSU(1, 1) is uniquely de- termined so that 9(c~) = oe and g'(er > 0. The map ~/= 9 -1 o f l s l is then an orientation-preserving quasisymmetric homeomorphism of the unit circle. [] The decomposition "r = 9 -1 o f is known as conformal welding. Using the fact that the correspondence between f and the quasicircle C is one-to-one, we can identify Homeoqs(S1)/S 1 with the space of univalent functions

/b = { f : A > (2 a univalent function : f(0) = 0, if(O) = 1, f has a quasiconformal extension to C}.

7) is a complex subspace of the complex space of sequences {an} (Fourier coefficients of the holomorphic function f). This induces a complex structure on Homeoqs (S 1 ) / S 1.

R e m a r k 2.4. Observe that if "y = wt, lsl up to post-composition with a PSU(1, 1) transformation, then the corresponding f is equal to w ~ up to post- composition with a PSL(2, C) transformation.

We identify Homeoqs(SX)/S I as the subgroup of Homeoqs(S 1) consisting of quasisymmetric homeomorphisms that fix the point 1. Consider Homeoq~ (S 1 ) / M S b ( S 1) as the subspace of Homeoqs (S 1)/S 1 corresponding to the natural inclusion T(1) ~_ D ,-+ D ~_ Homeoqs(S1)/S 1. Analogous to the isomorphism T(1) "~ Homeoqs(S1)/MSb(S1), we have

T h e o r e m 2.5. There is an isomorphism between T(1) and Homeo q~ ( S1) / S 1 79. Moreover, the complex structure o f T(1) induced from A ~ ( A ) @ C coincides with the complex structure induced from 79.

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280 L.P. TEO

P r o o f . The fiber of Homeoqs( S1) / S 1 over7 E Homeoqs( S1) / MSb( S 1) consists of all quasisymmetric homeomorphisms of the form a o 7 mod S 1, where a E PSU(1, 1) mod S 1 are parametrized by w E A* ~ PSU(1, 1 ) / S 1, i.e.,

1 - z ~

o (z) -

Z - - W

f~,, gw) be the univalent functions corresponding to 7 (resp., 7~, = (2.6)

Let f, g (resp.

aw o 7), i.e.,

7 = 9 -1 o f , Using Remark 2.4, we have

aw O T = g~l o .fw. f~, = ,~w O f, for some Aw E PSL(2, C). imply that z (2.7) A w ( z ) - - - - ] , CwZ + The condition g~(cc) = c~ implies that

(2.8) cw = -

and hence gw = Aw o g o a~ 1

The normalization conditions on fw E 7) and f E D

1 .f"(o) where cw = - -

2

1

g(w)"

Let [#] be the equivalence class which corresponds to 7 under the isomorphism T(1) ~_ H o m e o q s ( S 1 ) / M t b ( S 1 ) . For w E A*, the point g(w) lies in f(A*) = w~'(A*), since f(A*) = g(A*). Hence the natural correspondence between Homeoqs (S 1 ) / S 1 (_~ 7)) and T(1), given by

a~, o 7 E Homeoqs(S1)/S 1 (fw = Aw o f E 7)),

(2.9) a~o o 7 (f~, = A~o o f ) ~ ([#], g(w)), is an isomorphism.

In the identification above, T(1) is the natural subspace {([#], oo) : [#] E T(1)} of T(1). The embedding ([/~], oo) ~ f of T(1) into 7) is the pre-Bers embedding. Hence the complex structure of T(1) ~ Aoo(A) agrees with the complex structure induced from 7). From (2.9), (2.7), (2.8), we see that if we fix [#] in ([#], z) E T(1), and change z holomorphically, the corresponding f E /3 associated to ([#], z) changes by post-composition with A = ( ~ 0 ) E PSL(2, C), where the coefficient c depends holomorphically on z. This implies that the complex structure of T(1) induced from the embedding T(1) ~ A m ( A ) ~ C agrees with the complex structure

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THE VELLING-KIRILLOV METRIC 281

We can identify each point in T(1) as an equivalence class of quasiconformal mappings as in the proof of the theorem above. This immediately implies that T(1) also has a group structure coming from composition of quasiconformal maps, which is an extension of the group structure on T(1). According to the definition and the identification given in the proof of Theorem 2.5, the group multiplication in terms of coordinates (2.5) is given by

(2.10) where

(2.11)

(N,z) .([u],z0) = (M,z'),

u = --" and z' = w v o w -1 o ( w ~ ) - l ( z ) . l + ~ ( A o w ) ~ ~ O z

Here w is the quasiconformal map corresponding to the point ([#], z0) E T(1) H o m e o q , ( S 1 ) / S 1. The right group translation by ([#],z0), R(M,zo) : T(1) ~ T(1) is biholomorphic (see [BerT3]). Thus we can identify the tangent space at ([#], z0) with the tangent space at (0, ~ ) - - the origin of T(1) - - via the inverse of the derivative of the map R([u],z0) at the origin, i.e., via the map (D(o,oo)R(M,zo))-1. Moreover, this identification and the group structure give rise to a splitting of the tangent space at each point of T(1) into horizontal and vertical directions. At the origin (0, co), the vertical direction is spanned by {0} @ C and the horizontal direction is spanned by ft - m (A*) (D {0}. A horizontal vector (u, 0), v E f~-1,1 (A*), at the origin (0, co) has a unique horizontal lift to each point (0, z) on the fiber at (0, co). Namely, let ([tu], z~), z6 = z be a curve that defines the horizontal lift of (v, 0) at the point (0, z). For t small, z~ is determined by the equation

([)~(t)], c~) * (O,z) = ([tu],z~), A(t) e L ~ 1 7 6

The point (0, z) corresponds to the map az defined by (2.6) (the subscript z does not indicate a derivative). Using the formulas (2.10), (2.11), taking the derivative with respect to t and setting t = 0 (which we denote by 9 ), we have

(2.12) ,~ = ( v a'z~ o a~ -t a ' ] and ~' = wU(z).

Hence the horizontal tangent vector (v, 0) at (0, c~) is lifted to the vector (v, w"(z)) at

(0,

z), and the latter is identified with the horizontal tangent vector (A, 0) at the

l

g

origin (0, oo) of T(1).

2.3 I d e n t i f i c a t i o n o f t a n g e n t s p a c e s . Here we want to identify the tan-

gent spaces of the different models of the universal Teichmtiller curve and universal Teichmtiller space. We need the following two lemmas.

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2 8 2 L, P, TEO

oo 3

L e m m a 2.6. Let Q(z) = ~ n = 2 ( n - n)anz n-2 E Aoo(A). Then the series ~ n ~ 2 n2"la,~l 2 is convergent f o r all real s < 1.

P r o o f . Since

Q E Aoo(A) = {r holomorphic on A : sup Ir -

1z12)21 < ~},

zEA

w e have for any o~ < 1,

dzdy

N )212

<

A

w h e r e z = x + iy. This integral is equal to

c o

7r - n) r i 4 + n - c01) la,~12"

r t : 2

Stirling's formula for the g a m m a function F implies that lim

r ( n -

1)(n z - n) 2 = 1. n ~ F ( 4 + n - a ) n l+a B y the comparison test, the series

o o

Z nl+~

2

r t : 2

is convergent for all a < 1, which implies the assertion. [] R e m a r k 2.7. We have used an idea o f Veiling [Vel] in the proof o f this theorem.

L e m m a 2.8 g Z y g 8 8 1 ) . I f the function f ( z ) = ao + alz + . . . + a,~z ~ + , . . is holomorphic on A and continuous on A t J S 1, and the series ~ n n[an[ 2 is convergent, then the series

~0 q" r ia + " ' " q- a n elnO -k . . " converges uniformly to f ( e ~~ on 0 < 0 < 2~r.

First, we look at the isomorphism between the universal Teichmtiller curve W : H o m e o q s ( S 1 ) / S 1 ~

~,

" r ~ f .

It establishes the relation between the real analytic (through Homeoqs(S1)/SX)) and complex analytic (through Z)) descriptions o f T(1). Infinitesimally, it takes the following explicit form.

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T H E V E L L I N G - K I R I L L O V METRIC 283 T h e o r e m 2.9. The derivative o f W at the origin is the linear mapping D o W : To Homeoqs(S1)/S 1 ~ ToT) given by

oo

~ n + l

ECneinO

r--). i E c n ~ .

n:r n = l

P r o o f . Consider the smooth one parameter flow 7 t = (gt)-i o ft]s~ , 7tit=0 _~ id. It is known (see, e.g., [Leh87]) that 7 t, f t and gt can be extended to quasi- conformal mappings of C, real analytic on C \ S 1 . The corresponding vector fields

d t d f t and d t

-di "Y ' dt -~g

are continuous on C, real analytic on C \ S 1. We write the perturbative expansion

f t ( z ) = z + tu + O(t 2) = z + t z ( a l z + a2z 2 + . . . ) + O(t2), for z E A, and

gt(z) = z + tv + O(t 2) = z + tz(bo + blz -1 + b2z -2 + . . . ) + O(t2), f o r z E A*.

We denote

d t d t

3 ' = ~ ' / ' t=0' ] = d f t t = o and g = ~ g l t = 0 ' so that .fl~ = u and glzx- = v.

Under the Bers embedding, S(ft]~x) belongs to a bounded subspace of A ~ ( A ) ; and the corresponding tangent vector to T(1) at the origin is

d e Aoo(A).

U z ~ = S ( f l A ) t=0

Since u = ~n~__l a,,z n+l is holomorphic on A and continuous on C, Lemma 2.6 (with s = 89 and Lemma 2.8 imply that the series

oo

E anei(n+l)

n---1

converges uniformly to the continuous function ulsl (e i~ on the unit circle S 1. Similar arguments imply that the series

oo

E bnei(1-n)

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2 8 4 L.P. TEO

converges uniformly to the continuous function

vl s~ (ei~

on S 1.

Taking the derivative with respect to t of the relation 7 t =

(gt)-i oft

and setting t = 0, we have

(2.13) '~ = - g + ].

This shows that the series

o o (X)

Z anei(n+l)O -- Z bnei(1-n)O

r ~ i r~,..~-O

converges uniformly to the function all sl. In particular, it is the Fourier series of a/Is1. Let

u(ei~

be the corresponding vector field, so that x / =

izu(z)

on S 1. We have proved that the Fourier series of u(e

io)

inO

cue , e-n = Cn

nEZ

converges uniformly to

u(e i~ .

Moreover,

cx)

i ~

Cne i(n+l)O ..~. Z anei(n+l)O -- Z

bnei(l-n)~

nEZ n = l n=O

Comparing coefficients, we have

an=iCn,

b n = - i c - n ,

n > l .

Moreover, we have the relation

an = bn.

[3

By imposing extra normalization conditions, we can pass from the models for T(1) to the models for T(1).

R e m a r k 2.10. In [Nag93], Nag proved a result similar to Theorem 2.9 for T(1) by using explicit formulas for "~ and ] from the theory of quasiconformal mappings. Here we use a slightly different approach.

For the second isomorphism between the universal Teichmiiller space, we combine the Ahlfors-Beurling extension theorem and the Bers embedding and get the map

B: Homeoqs(SX)/MSb(S

1) --~

(L~(A*),/,.o) ~

noo(A),

[t,l

s(w"lA),

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THE VELLING-KIRILLOV METRIC 285

T h e o r e m 2.11.

The derivative of the map 13 at the origin is the linear mapping

Do13: To (Homeoqs( S1) / MSb( S1) )

-~ A ~ ( A )

given by

o 0

n r n = 2

R e m a r k 2.12. Lemmas 2.6, 2.8 and Theorem 2.9 imply that the tangent vectors at the origin of

Homeoqs(S1)/S 1

have Fourier series v" -

z.,n cn~

_i,~o

which converge absolutely and uniformly and belong to the Sobolev class H * for all s < 1. Here the Sobolev space

H*(S 1)

is defined as

Hs(S1)= {u(ei~

~ anei'~~ : ~[nl2"[a,d2 < oo}.

nEZ nEZ

anz '~+1 ToO

is In light of Theorem 2.9, we say that a tangent vector u = ~ n = i E in H s if it is the image of a

H s

vector ~,~

e,~e i'~~

under the map DoW.

2.3.1 M o r e o n c o m p l e x s t r u c t u r e s . The almost complex structure J at the origins of

Diff+(S1)/S 1

and Diff+(S1)/M6b(S 1) is defined by the linear map J : To --4 To given by

9 ~ v ' , ino 0

(2.14)

av = i Z sgn(n)cne~nOou'

where v = L c,~e -~.

n n

See references in [NV90]. (Note that we differ from the definition in [NV901 by a negative sign.) By Remark 2.12, J extends to almost complex structures on Home%,

(S1)/S 1

and Horne%s

($1)/M6b(S1).

In [NV90], Nag and Verjoysky proved that the almost complex structure J on Diff+(S1)/MSb(S 1) is integrable and the corresponding complex structure is the pull-back of the complex structure on T(1), induced by the complex structure of L ~ ( A ) I . Adapting their proof to our convention, we immediately see that the complex structure J on Homeoq,

(S 1)/S 1

coincides with the complex structure induced from T(1).

Under this convention, the holomorphic tangent vectors are of the form

v - iJv

.

w = ~

- ~ cne 'n~

n > 0

and the antiholomorphic tangent vectors are of the form ~ v = ~ = v

+ i Jr

Z enein~

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2 8 6 L. E TEO

2.4 M e t r i c s . We are interested in homogeneous Hermitian metrics, i.e., Hermitian metrics which are invariant under the right group action on the homoge- neous spaces of Horneoq~ ($1), In [Kir87] and [KY87], Kirillov and Yuriev studied K~ihler metrics on Diff+ (S 1)/S 1 . It is known that the homogeneous K~ihler metrics

o n Diff+(S1)/S 1 must be of the form

~(a,~

b,~)lc.l ~,

(2.15) II-II~,b = +

n > 0

where v = ~,~ezc,~e~n~ E To Diff+(S1)/S 1. The metric 11 9 11o,1 is called the

Kirillov metric.

On the other hand, since the vector fields e-i~ 0/00, ei~ generate the PSU(1, i) action on S ' , (2.15) defines a metric on Diff+(S1)/MSb(S 1) if and only

if an 3 + bn = 0 for n = - 1 , 0 , 1 . This implies that, up to a constant, there is a

unique homogeneous Kahler metric on Diff+ (S 1)/MSb(S 1) given by (2.16) I1" II 5= ~ ( n 3 - n)lc-I ~-

n > 0

Let F be a Fuchsian group realized as a subgroup of PSU(1, 1) acting on A *. Let L ~176 (A* F) be the space of Beltrami differentials for F, i.e.,

L ~ ( A ' , F ) = # E L e C ( A *) : # o v - - ; = p , VVei? .

The Teichmtiller space T(F) of F is the subspace of the universal Teichmiiller space

where

T(I?) = LCC(A *, F)I/,-%

L~176 = L~176 A L~(A*,F),

and ,-, is the same equivalence relation we use to define T(1). The tangent space at the origin ofT(i?) is identified with the space of harmonic Beltrami differentials of F

f~-l'l(A*, F) = f / - l ' l (A*) n L~176 *, F).

When I? is a cofinite Fuchsian group, i.e., when the quotient Riemann surface F\A* has finite hyperbolic area, there is a canonical Hermitian metric on T(I?) given by

v> =

[[

~p, ~, v e f~-~,x(A,, I?),

<,,

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T H E V E L L I N G - K I R I L L O V M E T R I C 287

where p is the area form of the hyperbolic metric on A*. This metric is called the Weil-Petersson metric. The notation T(1) for the universal TeichmiJller space indicates that it corresponds to the case F = {id}. This suggests defining the Weil-Petersson metric on T(1) by

A*

However, this integral does not converge for all #, v E Ft-I'I(A*). In particular, it diverges when both #, v are Beltrami differentials of a Fuchsian group which contains infinitely many elements. However, it was proved by Nag and Verjoysky in [NV90] that the integral is convergent on Sobolev class H a/~ vector fields, which contains the C 2 class vector fields. More precisely, they proved that the pull-back o f the Weil-Petersson metric on T(1) to Diff+(S1)/MSb(S 1) coincides with the unique homogeneous KShler metric (2.16) on Diff+ S 1 / MSb(S 1) (up to a factor 4). Henceforth, when we say the Weil-Petersson metric on T(1), we understand that it is only defined on tangent vectors in the Sobolev class H 3/2.

Under the Bers embedding, the Weil-Petersson metric on T(1) induces a metric on Am(A). It is given by

T h e o r e m 2.13. For Q = Uz=z E Am (A), identified as a tangent vector to T(1)

O 0

at the origin such that u = ~ n = I anzn+l E H 3/2, the Weil-Petersson metric has the f o r m

~r )--~(n a _ n)lanl 2 = 4 iQ(z)t2( 1 _ izl2)Zdxdy"

11 Q II~vP =

n = 2 A

P r o o f . The first equality follows immediately from the identification of tan- gent spaces given by Theorem 2.11. The second equality is an explicit computation

of the integral. []

R e m a r k 2.14. The derivative of the map 7) ~ Am(A) at the origin, ] ~ ]zzz can be viewed as a linear mapping sending vector fields to quadratic differentials. The theorem states that the Weil-Petersson metric on A ~ ( A ) given by the Bers embedding T(1) ,--+ Am(A) is the usual Weil-Petersson metric defined on the space of quadratic differentials. This can also be proved directly by using the isomorphism (2.3). In particular, we have

II Q o-r(7') ~ II~vP=ll Q

II~vP, for all 7 E PSU(1,1).

R e m a r k 2.15. Analogues of Theorems 2.9, 2.11 and 2.13 hold for finite dimensional Teichmfiller spaces T(F) embedded in the universal Teichmiiller space T(1).

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2 8 8 L . P . T E O

According to Remark 2.12, the Kirillov metric on Diff+(S1)/S 1 extends to T(1). Namely, at the origin, it is of the form

(2.17)

II- II 5--

~-~nlcnl 2,

n > 0

where v = ~ n cnei'~~ is the corresponding tangent vector. The series (2.17) is convergent. Using the right translations, we define a homogeneous K~ihler metric on T(1).

Since every homogeneous Kahler metric on Diff+(S1)/S 1 can be written as a linear combination o f the metric (2.17) and the Weil-Petersson metric, and only the former is convergent for all the tangent vectors o f 7-(1), we have

T h e o r e m 2.16. Every homogeneous Kgihler metric on T(1) is a multiple of the metric (2.17).

3

Velling's H e r m i t i a n f o r m a n d the V e l l i n g - K i r i l l o v

m e t r i c

3.1 S p h e r i c a l a r e a t h e o r e m . The spherical area o f a domain fi in C is

/ 4dxdy

As(f~) = "(1 + Iz12) ~"

f]

It is invariant under rotation, i.e., As (f~) = As (e w (f~) ).

Following Veiling [Vel], for Q E A ~ ( A ) and t small, we consider the one- parameter family of functions ftQ E 79 satisfying S ( f tQ) = tQ and the spherical areas o f the domains f~t = ftQ(A),

f f 4dxdy As(fit) = (1 + [z[2) 2 f~t

ff IdftQI2

= 4 (1 + If*QI2) 2" A

Velling's spherical area theorem is the following.

T h e o r e m 3.1 (Veiling [Vel]). For Q E A ~ ( A ) , we have

d As(ftQ(/X))lt=o = O,

f-~As(ftO(m))lt=o >_ O,

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THE VELLING-KIRILLOV METRIC 289 This follows from another result, proved by applying the classical area theorem.

T h e o r e m 3.2 (Felling [Fell). Let f : A --+ C be a univalent function

(perhaps meromorphic) with Taylor expansion f(z) = z(1 + a2z 2 + a 3 z 3 q - " . . ) at

the origin. Then the spherical area A s ( f (A)) satisfies

As(f (A)) >> 2~r,

with equality if and only if f = id.

The second inequality in Velling's spherical area theorem implies that

~tAs(ftQ(A))lt=o is a Hermitian form on A ~ ( A ) . Our goal is to compute this

form explicitly.

The following lemma is very useful for computations.

L e m m a 3.3 ( [ Z y g 8 8 ] ) . L e t f ( z ) = ~n=O anz be an analytic function on A oo n

and r an integrable function on [0, 1). Then

g

r

= 2~rRe (a0)

~01

r

A

/ /

-_

,oo, /o

& n = 0

3.2 V e l l i n g ' s I - l e r m i t i a n f o r m . Now we compute Velling's Hermitian form ~ t A s (ftQ (A))It=0- For t small, we write the perturbative expansions

I~Q(z) = z + t~(z) + t2v(z) + o(t3),

( 3 . 1 )

u(z)

=

z(a2 z2 -4- a3 z3 + ' " ) = ~ an zn+l,

rt=2

v(z) = z(b2z 2 + b3 z3 + . . . ) = Z bnzn+l"

rt=-2

Taking the t derivative of the equation 3 ( f Q ) = tQ and setting t = 0, we get the

relation

0 3

Oz3 u(z) = O,(z),

Using the expansion IftQI 2 (1 + lYtQI2) = i.e., Q(z) = ~ ( n ~ - n ) a , z "-~. n = 2 [I + tuz + t2Vz[ 2 (1 + ]z + tu + t2v[2) 2 + O(t3)'

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2 9 0 L.P. T E O we obtain

f/X(z)dxdy

As(f (a))J o = S

dt

= (1 + Iz{2) 2' A 1 + lz[ 2 +5(1+1zl2) 2" Using the series expansion (3.1) and v(0) = v'(0) = 0, we see that v drops out from the integration. Applying Lemma 3.3, we get

oo

d2 A

tQ A

(3.2)

dt 2 s(f (

))lt=o = 167r E

entail2'

n = 2 where f01 ( 6r 2n+4 (4n+6)r2n+2 ( n + 1 ) 2 r 2 ' ~ cn = [(]-~r-~) 4 (1 + r2) a + (1

+ r2) 2 ] rdr.

We compute cn by repeatedly using integration by parts:

1 ~ 1 ( 6r n+2

( 4 n + 6 ) r n+l

(n+ l)2r n)

C n = ~ ( I + r ) 4 ( l + r ) 3 + ( l + r )

2

dr,

f l rn+2 2n2 + 7n + 7

n(n + l)(n T 2) jfol rn-1

]o

r

(1 +

dr

= 24 + 6

l ~ r dr'

fo 1 r n+l 2 n + 3

n(n+i) jfolrn-~

(1 +

r)

"---~ dr =

8

+

2

l ' ~ r dr'

fo i r a

1 f o l r n - 1

When we substitute into c~, all the terms with integrals cancel; and we are left with

c . = n / S .

Therefore, we have

T h e o r e m 3.4. Let Q E A~(A).

Then

A s ( f Q ( A ) ) t=o = 2~ ~

n[ant 2.

r~=2

Remark 2.I2 implies that the series is convergent for all Q E A~(A). Hence, ~,e can define a Hermitian form on A ~ ( A ) by

[} O []~=

l d~As(f'Q(A))],-o = E n]an[2'

zu dt

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T H E V E L L I N G - K I R I L L O V M E T R I C 291

where

o o

Q(z)

= E ( n 3 -

n)anz~-2;

n = 2

we call this Velling's Hermitian form.

R e m a r k 3.5. T h e first half o f the computation above is reproduced from Velling's unpublished manuscript [Vel]. Veiling gave the result in terms o f (3.2). Our observation is that cn can be computed explicitly.

Note that in evaluating the Hermitian form, we have chosen a particular nor- malized solution

ftQ

to the equation

S ( f tQ) = tQ.

Any other choice will differ from this one by post-composition with a PSL(2, C) transformation. However, the spherical area o f a d o m a i n

A s ( f (A))

is not invariant if f is p o s t - c o m p o s e d with a PSL(2, C) transformation. If we choose different normalization conditions to identify T(1) as a subgroup o f T(1), we get a different right invariant metric on T(1). Hence the Hermitian form [I 9 Ils does not naturally define a right invariant metric on T(1).

On the other hand, since the correspondence between 3' C

Homeoqs(S1)/S 1

and f ~ ~3 is canonical, we can use the same approach to define a metric on T ( 1 ) =

Homeoqs(S1)/S 1.

Namely, given the tangent vector v = ~ n e 0

c,~ein~

at the origin with the associated one-parameter flow 3, t = (gt)-~ o

ftlsl ,

we define a Hermitian form by

1 c~ it=oAs(ft(A)).

II v II 2 - 2~r

at 2

T h e p r o o f above holds with an extra term n = 1 (notice that we only need the fact there are no constant terms and terms linear in z in the first and second order perturbations), and we get

1 d 2 oo

II

v ll2=

2---~

dt----~lt=oAs(ft(A)) = E n]an[ 2 = E nlcn[2'

n----I n = l

which coincides with the metric (2.17) at the origin. It is quite remarkable that this metric, introduced by Veiling using classical function theory, coincides with the metric introduced by Kirillov using the orbit method. Henceforth, we call this metric on T(1) the Velling-Kirillov metric.

4 M e t r i c s o n T e i c h m i i l l e r s p a c e s

4.1 U n i v e r s a l T e i c h m i i l l e r s p a c e . Let ~ be the symplectic form o f the

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292 L.P. TEO

vertical integration of the (2, 2) form ~ An. Namely, let

fiber

and define a Hermitian metric on T(t) such that w is the corresponding symplectic form. 3 Since ~; defines a right invariant metric, w also defines a right invariant metric. Hence we only have to compute the form w at the origin of T(1). We

identifythetangentspaceofT(1)attheoriginwithAoo(A)@C. The vertical tangent

space is spanned by O/Ow and 0/0~-, where w is the coordinate on C. Observe

that the horizontal and vertical tangent spaces are orthogonal with respect to the Velling-Kirillov metric. Hence, given a holomorphic tangent vector Q E Aoo (A), we have

A*

where O is the horizontal lift of (O, 0) to every point on the fiber. Using the fight invariance of the Velling-Kirillov metric, we see at once that ~(8/8w, 8 / 8 ~ ) d w A d ~

is the area form of a fight invariant metric on A *. Hence, up to a constant, it is the hyperbolic area form d A u . Checking at the origin, we find that

d w ^ (i - Iw12) 2

~ ] a ~ = - - = dAH, w = x + i y .

Via the identification (2.12) and the isomorphism (2.3), Q at (0,w) is identified with Q o a~ i ((a~ 1)')2 at the origin. Hence

i Q o a - i a - 1 , 2

= II , , ( ( , , ) ) II .

Under the change of variable w ~ 1/w, a~ i is changed to 7,,, where modulo S i,

z + w

7,,(z) = 1 + z~"

Since pre-composing Q with a rotation does not change the Hermitian form [I Q II ~, we finally get

w(Q, Q) = -~ II Q,, tl2s dAH, O,, = Q o 7,, (7~,) 2.

A

3Since the fiber is not compact, it is not a priori clear that w e get a well-defined symplectic form on T(1).

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THE V E L L I N G - K I R I L L O V M E T R I C 293 Thus our approach to defining a Hermitian metric on T(1) coincides with Velling's suggestion [Veil of averaging the Hermitian form [[ 9 II ~ along the fiber to define Hermitian metric on T(1), i.e.,

1

f/

4dxdy

(4.1) il Q [l~: = -2iw(Q,O) = ~ II Q~ tl~ ( l : ~ - ~ ) e .

R e m a r k 4.1. I am grateful to m y advisor L. Takhtajan for his suggestion o f using vertical integration to obtain a metric on T(1).

Since the Hermitian form tl Q II~ is expressed in terms of the norm square of the corresponding coefficients lanl 2, to compute (4.1) it is sufficient to average ta~t 2 for n > 2. We set oO Q ~ ( z ) = Q o . r ~ ( ~ ' ) ~ ( ~ ) = ~ ( n 3 - ~ a ~ z ~ , ~ ( z ) = l + n = 2 Then 1 (Q o 7to (7")2)(n-2) (0), (4.2) anW = (n 3 - n) (n - 2)! and n ~ - 2 P r o o f . Using (4.2), we set 1 (Q ~ 7"(%)~)(J-2) (0) =

caw)

a~' = (j3 _ j) (j - 2)!

and introduce the generating function for the cj (w)'s,

o o f ( u , ~ ) : ~ ~j(w)~,J -~ j=2 ~ , (Q o .y~(./)2)(j-2) (0)u j-~ = O o-y~(~)(-y'(~))t (j3 _ j ) ,

T h e o r e m 4.2. Let u(z) = ~n=l a, z~+l 6 oo Ha~2 and Q = u~z~. Then

4dxdy _ 2 f f [Q(w)[~( I _ lwl~)2dxdy

If laTl~

(a - I~,l~)~ 3(2 3 - j ) A

OO

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294 L. P. T E O

Writing u =

pe'%

we have

1 f02"

Icr

= 2---~

If(Pei~'

w)12da

j = 2

and

f f

icj(w)12

dxdy

p2~-4

= 1 f~ f f

if(pei~,w)[ e

~d~ ,

(1 -Iw12)

2

2~r Jo JJ

(1

-q-w-i-2) 2 a a

j = 2 A A

= 2--~

IQ o 7~(pe~")(7~(pei"))212

(1 -~-~2)2 aa.

A

Denoting this integral by 77, substituting the series expansion of Q and using polar coordinates w =

re i~

we get (4.3) 1 2. 1 2~ (1 - r 2) 2

Z=2-~o

~0 fo dOrdrdOl(l+-'r~i-~--~ 4

~ ( n 3 - n ) a n \ l + r p e , ( . _ o

)

~ (m3 - m)a-~ \ i - ~ - r r p ~

n = 2 m = 2 r a - 2 We do some "juggling",

pei. + redo .-~ ( pe_i. + re_iO

~ m-2

GTF,~))

\ ~u

/

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THE VELLING-KIR1LLOV METRIC 295

and make a change o f variable a ~ (c~ + 0) to get

I

n,rn~2

) l

\ 1 + rpe -icx

(1 +

rpeiC~) 4

L2~Llrdrd~

s

n)21anl2 (_p~a_+__r'~ n-: pe-ia +r ~ n-2

( 1 - r 2) 12

n:2

\1

+rp<'=]

(1--+-~rp~'~J

1(1

+rpei=) '

L2. L i rdrd~

oo

( p + rei~ , t

( n3

i ~ ) 2 I a~ 12 t f ~ ] n = 2 A = (n3

- n)21a"12

\ l + p w )

\ 1 + p ~ )

I(l; p-~ 4

where we have d o n e m o r e juggling to get the second to last equality. Observe that

p + w _ 7 p ( W ) ,

1

+pw

1 __ 7~(w) 2

( l + p w ) 4 (1 _ p 2 ) 2 "

H e n c e we have

S / ( "+ w ~o_2(. + ~ ~o_21_ lwi ~ 2

A

, 2" (1 -Iw12) ~

= i S ((~"-~) o ~.(~;)~1 (~)((z~-~) o ~.(~p )(~) Ti- p--~ e~e~

A

dJ

t i - P ) A

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2 9 6 L . P . T E O

using PSU(1, 1)-invariance of the Weil-Petersson metric. This gives

o o

z = f f

Z ( n 3 -

n~2rajl n,12wn-2~

( 1 ~

~f-_-p-~)4-

lw12)2 dxdy

A n : 2 _ 1 f / i Q ( w ) l Z ( 1 _

iw?)2dxdy

(1 _p2)4

A

= ~ j3_.__~_ jo2j_4

H"

l!

[ Q(w)12(1 -

I wl2)2dxdy"

j : 2 A

Comparing coefficients, we get

= ~ 3 _ j

fflo(w)12(l_l l ) axa '

(1

-}wl2)

2 6

A

(1 - {wp) ~ = 3(j 3"- j) ]Q(w)l~(1 -

lwl2)2dxdy'

A A

which finishes the proof. []

T h e o r e m 4.3.

Let Q = Uzzz E Aoo

(A)

be a tangent vector to

T(1)

at the origin

such that u E H 3/~. Then

A A

which is the Weil-Petersson metric.

P r o o f .

/ l l O w II~

&

This is just a simple sum of the telescoping series:

2dxdy

c~ f /

2dxdy

(1 - [ w l 2 ) 2 : ~ J la~]2 (1 -Iw[2) 2 = A

ff

= ~ 3(j - 1)(j + 1) IQ(w)[2(1 -lwl2)Zdxdy j = 2 A

= ~ g ]O(w),~(1- [w[2)2dxdy.

A [3 4.2 F i n i t e - d i m e n s i o n a l T e i e h m i i l l e r s p a c e s . Let F be a

Fuchsian group. The tangent space to T(F) at the origin is identified with

Aoo(A, r)

= {Q e Ace(A): Q o 7(7') 2 = Q,V-y e F } ,

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THE V E L L I N G - K I R I L L O V METRIC 297

and the Weit-Petersson metric is given by I f f

(4.4) I1Q ll~'P = " ~ ] ] IQ(w)12( 1 - lwl2)2dxdy.

r\A

The inverse image of T(F) under the projection map T(1) ~ T(F) is the Bers fiber space B,T(F). The quasi-Fuchsian group F ~' = w ~' oFo ( w , ) - I acts on the fiber w" (A*) at the point [#] E T(F). The quotient space of each fiber is a corresponding Riemann surface. They glue together to form the fiber space 5 ( F ) over T(F), which is called the TeichmtiUer curve of P. First we have

L e m m a 4.4. Let F be a Fuchsian group. The symplectic form ~ on 7-(1) restricted to BSc(F) is equivariant with respect to the group action on each fiber

P r o o f . We only need to check this statement on the fiber at the origin. The form ~; restricted to the vertical direction is clearly equivariant. We are left to verify that if w E A*,-~ E F a n d Q e A ~ ( A , P ) , then

^ - % - ~ " % - " I

a(O, Q)(w) = a(O, Q)(w ),

where w' = "/(w). Note that the PSU(1, 1) transformation aw, o O' o a~ 1 fixes oc, hence is a rotation. Using the fact that the Hermitian form II Q 112 is invariant if Q is pre-composed with a rotation, we have

I[ Q o a ; , ~ ((cry1)'): 112 =tl (Q o 7(7') 2) o a ; ~ ( ( o ' w l ) ' ) 2 112

=ll Q o O'~ 1 ((O'wl)') 2 II 2 . []

The lemma implies that ~; descends to a well-defined symplectic form on 5r(F). We vertically integrate the (2, 2)-form a A ~ on ~(F) to define the Hermitian metric on T(F). Using the same reasoning as in Section 4.1, we get

if/

(4.5) 11Q II~ "= ~ II Qw 112 dAn, Q e A~(ax, r).

r\Lx

We want to compute this integral using a regularization technique suggested by J, Veiling [Veil.

T h e o r e m 4.S. Let F be a cofinite Fuchsian group and h E L ~ ( A ) be F- automorphic. Then

f f Areatt(F\A) f f A , h(w)dAH

h(w)dAH = lim

r,-~l-

ffAo,

d A n

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2 9 8 L . P . T E O

where AreaH(F\A) is the hyperbolic area o f the quotient Riemann surface F\A a n d A r , = {z: [zl < r'}.

P r o o f . We use the fact that for any z C A, the number of elements 7 E F such that 7(z) is in the disc Ar,, is given asymptotically in terms of r' by

If

(1

(4.6) AreaH(F\A) dAH + o(1)), as r'

A i

where the o(1) term is uniform for all z in a compact set (see [Pat75]).

Let F be a fundamental domain of F. Given E C F, let E ' = [,J-~cr 7(E). Let

XA be the characteristic function of the set A. Since F is cofinite, using (4.6), we

have

f f AreaH(F\A) ffa~, XE, dAH

x E d A H =

ffA

, dAH

F

+o(1).

Here the o(1) term is uniform for all the sets E C F. Since

sup Ih(w)[ < oo,

w E A

standard approximations of h by bounded step functions give our assertion. [] C o r o l l a r y 4.6.

II Q

II~ve = lim

r l - - - + l -

AreaH(F\A) f l A . IQ(w)[ 20-1~12)2dxdy ffA.., dAH

P r o o f . Take h(w) = IQ(w)(1-

Iwt2)212.

Since Q e Aoo(A), h is in

L ~ ( A ) . []

L e m m a

4.7. Let Q E A ~ (A). Then

sup II Qw [l~< oo. w E A

P r o o f . Let (Q o 7~(7~) 2) (z) = Qw(z) = E~=2 ~o . - 2 a n z . The proof of Lemma

2.6 with a = 0 implies that o o

II Q~ ][~-- ~-~n[a~l 2 < C / / IQw(z)(1 -Jzl2)2[2dxdy,

n = 2 A

(29)

T H E V E L L I N G - K I R I L L O V M E T R I C 299

where C is a constant independent o f Q E Aoo(A). After the change of variable z ~ 7~ ! (z), the integral on the right hand side becomes

f f

lQ(z)O -Iz?)=l

= =

d dy

A

Since Q

[Kra72])

C Ao~(A),

]Q(z)(1 -Izl2)21=

is bounded on A; thus the formula (see

/ / ](.y~l),(z)[2 dxdy = f f (1_ -Iwl2)2dxdy

[1 - z~t 4 = 27r

A A

concludes the proof o f the lemma. []

Theorem 4.5 and L e m m a 4.7 imply that our approach to defining a Hermitian metric on T(F) agrees with J. Velling's original suggestion o f using regularized integrals. Namely, one has from Theorem 4.5 and L e m m a 4.7

C o r o l l a r y 4.8.

1 lim II Q

II~,-- ~

r ' - + l -

A r e a g ( F \ A )

ff, x~,

II

Q~ I1~ dAg

flA., dAH

Now we start to compute

II Q I1~.

First we have

T h e o r e m 4.9.

Let F be a cofinite Fuchsian group, Q E

A ~ ( A , F).

Then

A r e a u ( F \ A )

ffA , laTI2dAH

8

lim =

II

Q 11,2~

~'--'~-

ffA,

dAH

3(j 3 - j ) "*'P " P r o o f . The proof is almost the same as that o f Theorem 4.2. We have

~~2 / /

dxdy

t~2j_4

Z = Icj(w)12 (1 - Iwt2) 2 "

=

~-~(n3 -n)21a"12

\ l + p w ]

A , n = 2

(

p + ~ , ~ n - 2 1 - l w l 2

2dxdy"

Now observe that if 7 E PSU(1,1) and Q E Aoo(A,F), then Q o 7(7') 2

E Aoo(A,7-~FT), and

(30)

300 L. E TEO

In particular, for any u =

pem

E A,

AreaH(r\a) ffa I(Q

o -Mq)2)(w)l +-

!s --!~,12~:

dxdy

II Q tl~vr ,=

lim

"'

,'-,1-

f l a .

d A H '

since A r e a • ( r \ A ) = Arean(v~-iFT,,\A). It follows that

11 Q ll~,P = i..s ..+ 1 -- lim

a r e a H ( r \ A ) ~ Jo 2" fla. I(Q ~ ~ I l l , f l u # c~+, ~,~(w~! '-> I t It

('-,t'<'li?dxd.,da

4 "9~

ffA ,

dAtt

But

'])ff

2. I(Q ~ ~(g)2)(w)l~(1 - w2)2

dxdyda

4 A r t

'

]:':

s fo"

I

(1-'.)('- r '

=2-7

do.~.d. 7(i-~m-~7+1

( Pem +re+ 0 )~-'2

Z (na - n)an \ 1" ~ rp~+(-O__o)

n ' ~ 2

pe_ia + re_iO . m-2

F_, ( m~

- , , , ) ~ \

: i 7 ~ 7 - ~

)

m = 2

This is similar to the integral (4.3) with the role of 0 and a interchanged, so it is equal to

(1-P2)4 f f ~

~1" ('13

--'~)21anl'2 \~ j l

(P+W ~n-2 ( p+w ~ "-2

k ~ )

I ( l + p w ) 4

1-,wl2 12

dxdy

(1 - p2)4 4 Hence o O

Z

lim

r ~ - + l -

j=2

A r e a H ( F \ a )

fla.

IcJ(w)I ~ { l - l w l 2 ) ~ p 2 j - 4 : 4

fla.,

dAH

(1 - p~)* It Q II~'p 9

Comparing coefficients, we have

lim

r ' - + l -

A r e a H ( r \ A )

ff~., I c j ( w ) 1 2 ~

2(js _ j)

ffA , dAH

- -

3

II Q ll~vp

and

AreaH(F\A)

fla., la~l 2dAH

S

lim

(31)

THE VELLING-KIRILLOV METRIC 301

As in the case o f Theorem 4.3, this immediately implies

T h e o r e m 4 . 1 0 . Let F be a cofinite Fuchsian group and Q E A ~ ( A , F) a

tangent vector to T(F) at the origin. Then

II Q I1~=11 Q II~vp 9

For a general Fuchsian group F and Q E A ~ ( A , P), we can define [1Q I1~= lim ffa,,nF(r) dAB

ff~,

II Qw I1~ dAH

r'-~:-

flat,

d A n '

whenever the limit is finite. Here F ( F ) is a fundamental domain o f f on A. When F is the trivial group, this reduces to integrating over the whole disc, which coincides with our original definition.

Appendix A

Embedding of T(1)

Consider the Banach space

A ~ ( A ) = ( r holomorphic on A : sup 1r -Izl~)l

< ~

l zEA )

Analogous to the Bers embedding T(1) ~_ 79 ~-~ Ao~(A) (defined in Section 2.1), which is achieved by the mapping f E 79 ~ S ( f ) E A ~ ( A ) , w e prove that there is an embedding T(1) _ 7) r A ~ ( A ) , achieved by the mapping f E 7) ~ O(f), where

d .Lz

o(f)

= ~ log A = A "

B y the classical distortion theorem (see, e.g., [Ah173]), f E 79 implies that

fzz 2~ I 4

A (: --]zl 2) , -< - - - z " : - I z l

Hence O(f) E A ~ ( A ) , and the map 0 : 7) --+ A ~ ( A ) is well-defined. We claim that this map is an embedding, and the image contains an open ball.

L e m m a A . 1 . The map 0 is injective.

P r o o f . If f, 9 E 79 are such that O(f) = 0(9), then

d log f z = d log 9~.

This implies that f = clg + c2 for some constants ca and c2. The normalization conditions f(0) = 9(0) = 0, f'(0) = 9'(0) = 1 (from the definition o f f , 9 E 9 )

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3 0 2 L. P, TEO

We use the following notation for the sup-norms o f Aoo(A) and A ~ ( A ) :

II ~0 It~,~= sup I~(z)(1 - Iz12)l, r c A ~ ( A ) ; zEA

II r 11~,2-- sup Ir - Iz12)=l, r ~ A ~ ( A ) . zEA

Notice that S ( f ) = a(f)= - 89 2. For r E A ~ ( A ) , we define

We claim that this is a m a p from .A~(A) to A ~ ( A ) . First, we have the following continuity theorem.

T h e o r e m Ao2. For any E > O, there exists 5 > 0 such that if ~p E .A~( A ) satisfies II ~ II~,l< 5, then tt,(~) E A ~ ( A ) and

II ,I,(r I1~,~<

~.

P r o o f . Fix 5 > 0 and assume that r E .A~(A) satisfies II ~ I1~,~< 5. We use the Cauchy formula

1 ~llw ~_~w) .dw,

~ ( z ) = 7~/ ,=r (w - z)~ Izl < r < 1,

to estimate ~,~(z). Since s u p ~ e a Ir - Iwlm)l < 5,

Ir _< 2:~(1- r=) I--r I w - zl 2"

Elementary computation gives

1 f

Idwl

r

2S ~l~l=r t t J - z l 2 - r2 - I z l e"

C h o o s i n g r = (1 +

Izl)/2,

after some e l e m e n t a r y computations, we obtain 4

Ir - 1zt2)21 _< 85(1 + Izl) 3 < 645 for lzl _< 1. (Izl + 3)(1 + 31zl) - 3

H e n c e

~ ( z ) - (r 2 (1 - < --5- + Y"

Given e > 0, we can always find 5 > 0 such that 645/3 + 52/2 < e. This proves our

assertion. []

(33)

T H E V E L L I N G - K I R I L L O V METRIC 303

Corollary

A . 3 . ~, ~ ~(~/)) is a holomorphic map from .4~( A ) to Aoo( A ). P r o o f . T h e m a p ~b ~ ~: is linear. From the p r o o f o f the theorem above, we see that it is a continuous map from .Aoo(A) to Aoo(A). T h e map ~/, ,--r -89 is clearly a continuous map from .Ao~(A) to A ~ ( A ) . Hence tI, is a continuous map from .A~(A) to A ~ ( A ) .

To prove holomorphy, it is sufficient to note that for any ~/~,~p E .A~(A) and E C in a neighbourhood o f 0, the Frechet derivative

lim ~(~/) + ~ ) - ~(~b)

~--~0 r

= ~p. - ~/)~p exists in the l] " 11~,2 norm, since

~,(r + ~ ) - ~ ( r

l

1 2

1

tends to 0 as e ~ 0. []

T h e o r e m A . 4 . The image of 0 contains an open ball about the origin o f .4~( A ).

P r o o f . By T h e o r e m A.2, there exists a such that if II ~b II ~ , , < a , then ~b = tI, (g,) satisfies [i 0 [1~,2< 2. By the Ahlfors-Weill theorem, there exists a univalent function f l : A ~ C such that S ( f l ) = ~b and f l has a quasiconformal extension to C. On the other hand, there exists a unique holomorphic function f : A ~ C which solves the ordinary differential equation

d

d--z log fz = r f ( 0 ) = 0, if(0) = 1.

Obviously, S ( f ) = q(~b) = ~. Hence f and f~ agree up to post-composition with a PSL(2,C) transformation. This implies that f also has a quasiconformal extension to C and f E 7). Hence the image o f 0 contains the open ball o f radius

a. []

From L e m m a A. 1 and T h e o r e m A.4. it follows that there is an e m b e d d i n g o f 79 into .A~(A) w h o s e image contains an open ball about the origin. This implies that /5 has a Banach manifold structure modeled on .A~(A). We want to c o m p a r e this structure with the structure induced from the embedding 7~ ~_ T ( 1 ) ~ Aoo(A) 9 C. We define a map ~ : .Aoo(A) ~ A ~ ( A ) ~ C by

(34)

304 L. E TEO

T h e o r e m A.5. The map ~P is holomorphic and one-to-one.

P r o o f . Holomorphy follows directly from Corollary A.3. To prove injectivity, suppose @(r = ~(r For j = 1, 2, let

/o

fj(z) = efg '~(~')e'~dw.

Then ~ l o g f ~ = Cj, fs(0) = 0, f~(0) = 1. This implies that ,-q(fx) = ~(r --

~(r = S(f2). Hence fl = o- o f2 for some a E PSL(2, C). Now f3(0) = 0, f~(0) --

1, j = 1, 2 implies that a = ( ~ 0 ) for some c E C. We also have

d log f~ = Tzz d (log a' o f2 + log f~) Setting z = 0 gives

r (0) • --2C "[- r (0).

Thus ~(r = ~(r implies c = 0 and fl = f2, r = r [] T h e o r e m A.6. The Banach spaces Aoo( A ) and A ~ ( A ) ~ C induce the same

Banach manifold structure on f).

P r o o L From our discussion in Section 2.2 (in particular (2.9), (2.8), (2.7)), we know that the embedding 79 ~ Aoo(A) @ C factors through the map ~, i.e., it is given by f ~ O(f) ~ (S(f), 10(f)(0)). Let U (resp., V) be the image of D in .A~(A) (resp., Ao~(A) ~ C). Bers proved that V is open in Ao~(A) @ C ([Ber73]) using a theorem of Ahlfors ([Ah163]) which says that the image of T(1) in A o~ (A) is open. The continuity of the map ~ implies that U is open in .A~ (A). Hence we have a holomorphic bijection ~]u : U ~ V. In order to conclude that this is a biholomorphic map between open subsets of Banach manifolds, by the inverse mapping theorem (see, e.g., [Lan95]), we only have to show that for any r E U, the derivative of ~ at r De ~, is a topological linear isomorphism between .Ao~ (A) and Aoo ( A ) @ C.

From the proof of Corollary A.3, the linear map D r : Aoo(A) --+ Ao~(A) @ C is given by

D ~ ( ~ ) = (qoz - r 2~P(0) ) 9

From the theory of ordinary differential equations, it is easy to prove that this map is injective. To prove surjectivity, let f E D be such that O(f) = r Given (r c) E Aoo (A) @ C, consider

lz =,lzl(/o z lul. 2c)

(35)

T H E V E L L I N G - K I R I L L O V M E T R I C 305

It is straightforward to check that ~ is the unique holomorphic function on A that

satisfies 1

qOz - r = r and 2~P(0) = c. What remains to be proved is that ~ c Aoo (A).

Let f~ = f ( A ) and f~* = ~2 \ fl be the exterior of the domain fL Let

,~(w)ldw I

be the Poincar6 metric on t , which is given by

1

A o f(z)lf'(z)l =

(1 - i z l 2 ) "

For w E t , let ~(w) denote the Euclidean distance from w to the boundary of fL The Koebe one-quarter theorem (see, e.g., [Nag88, Leh87]) implies that

1 (A.1) ~ < ~(w)~(w) < 1. 1 2 Let r = r o

f - l ( w ) (f~ (w)) .

Then f0 z r d

[

w = ~o ~ ( v ) a v =

~(~),

1L where w =

f(z).

Since

s u p 1,X-e(w)~(w)l = sup I(1 - Izl2)er < cr

wEf~ z ~ A

by a theorem o f Bets ([Ber66]), there is a bounded harmonic Beltrami differential on f~*,/z : f~* ~ C, sup~e n. I~(w)l = ~ < ~ , such that

6

This implies that

w E l L

2

~,(v)

Cl,

ff (v_ )31av d l+

f~*

where C1 is a constant such that ~(0) = 0. Since every point v E f~* is of distance at least ~(w) away from w, we have the following estimate:

,~(w), < f f , v = ~ 3'#(v)lldv2d---~

1 +C1

l~-wl>_a(~)

- T

(w) -fi pdpdO + C1

4~

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