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Fenchel-Nilsen Deformation and the Weil-Petersson
Metric
Seddik Gmira
To cite this version:
Fenchel-Nilsen Deformation and the
Weil-Petersson Metric
Seddik Gmira
First, we de…ne the Poincaré metric and study basic properties especially those concerning geodesics and pants. Next, we recall some results on quasi-conformal mappings, Bers’mappings, harmonic beltrami di¤erentials and Weil-Petersson metric
In this work we will give a nice representation due to S.Wolpert of the Weil-Petersson fundamental form on the Teichmüller space of genusg 2 by using Fenchel-Nilsen coordinates. First, a description of Fenchel-Nilsen deformation will be given by using quasiconformal mapping, and then we calculate the tan-gent vector by a Fenchel-Nilsen deformation in the Teichmüller space.
At the last section of this work, we give a proof of Wolpert’s formula, namely, a representation of the Weil-Petersson fundamental form by Fenchel-Nilsen co-ordinates.
1
Hyperbolic geometry
1.1
Poincaré Metric and Geodesics
On the unit disk of the complexe plane, we consider the Poincaré metric ds2= 4 jdzj2 = 1 jzj2 2. For z1, z22 , we set (z1; z2) = inf C Z C 2 jdzj2 1 jzj2
where C, moves all recti…able curves in which connecting z1 and z2. We
call l (C) = RCds the hyperbolic length of C. This arc C is a geodesic if (z1; z2) = l (C).
Proposition 1 For arbitrary z1, z2 2 , there exists a unique geodesic
con-necting them in . Moreover it is a subarc of the circle or the line segment which passes through z1, z2 and is orthogonal to the boundary @ .
Proof. Since the Poincaré is invariant under the action of automorphisms of , we may assume that z1= 0 and z2> 0, by using the translation
(z) = ei z z1 1 z1z
with a suitable 2 R. Then, for every closed arc C connecting 0 and z2, we
Hence, (0; z2) = l (C) if and only if C is coincident with line segment [0; z2].
Now, we de…ne the axis of a hyperbolic real Möbius transformation . Sup-pose that is conjugate to a canonical form 0(z) = z with > 1 by a real Möbius transformation = 0 1 . The half-line L = fiy j 0 < y < 1g in the upper half-plane H is the geodesic, joining 0 and 1, with respect to the Poincaré metric jdzj2= (Im z)2 on H. The image (L) = A of L under is called the axis of . This axis A is the geodesic joining the two …xed points r and a of which is characterized as a semi-circle which joins them, and is orthogonal to the real axis.
1.2
Hyperbolic Metric on a Riemann Surface
Let R be a Riemann surface whose universal covering surface is biholomorphic to the unit disk . Consider a Fuchsian model of R acting on . Let : ! = = R be the projection. Since the Poincaré metric ds2is -invariant,
we obtain a hyperbolic metric ds2
R on R which satis…es ds2R = ds2:
Now every 2 corresponds to an element [C ] of the fundamental group
1(R; p0) of R. In particular, determines the free homotopy class of C . We
say that covers C .
When is hyperbolic (tr2 > 4), it is seen that the closed curve L = A = h i, the image on R of the axis A by is the unique geodesic belonging to the free homotopy class of C . We call L the closed geodesic corresponding to or to C .
Proposition 2 Let R be a Riemann surface with universal covering surface H, and 1 be a Fuchsian model of R acting on the half-plane. Let (z) =
(az + b) = (cz + d), a; b; c; d 2 R, ad bc = 1 be a hyperbolic element, and L be the geodesic on R corresponding to . Then the hyperbolic length l (L ) satis…es
tr2( ) = (a + d)2= 4 cosh2 l (L ) 2
Proof. Since l (L ) and tr2( ) are invariants under the action of Aut (H), we
may assume that a =p , b = c = 0 and d = 1=p . Then we have:
l (L ) = Z
1
dy
y = log = 2 log a
Let R be a Riemann surface whose universal covering surface is biholomor-phic to the unit disk . Consider a Fuchsian model of R acting on with the projection : ! = = R. Since the Poincaré metric ds2is -invariant, we
obtain a hyperbolic metric ds2
R on R which satis…es ds2R = ds2. We know
that for a given covering surface R; ; R , the universal covering groupe is isomorphic to the fundamental group 1(R; p0) of R. Then every 2
determines the free homotopy class of C . We say that covers C . When is hyperbolic (tr2 > 4), it is seen that the closed curve L = A = h i, the image on R of the axis A by is the unique geodesic belonging to the free homotopy class of C . We call L the closed geodesic corresponding to or to C .
1.3
Pants
Let R be a Riemann surface which admits the hyperbolic metric ds2
R. Consider
cutting this surface by a family of mutually disjoint simple closed geodesics. Let P be a relatively compact connected component of the resulting union of subsurfaces. If P contains no more simple closed geodesics of R, then P shoud be triply connected i.e. homeomorphic to a planar region,
P0= (fjzj < 2g) jz + 1j <
1
2 [ jz 1j < 1 2
P0 can be considered as a one of the smallest pieces for rebuilding the surface
R. Hereafter, we call a relatively compact subsurface P of R a pair of pants of R if it is triply connected and if every connected component of the relative boundary of P in R is a closed geodesic on R.
Fix a pair of pants P of R, with a Fuchsian model of R acting on the unit disc , and : ! R = = the projection mapping. Let eP be a connected component of 1(P). Denote by
e
P the subgroup of consisting
of all elements 2 such that P = ee P. Then Pe is a free group generated by two hyperbolic transformations, and P = eP= Pe.
1.4
Existence and Uniqueness of Pants
Let L1, L2 and L3 be the boundary components, which are simple closed
geo-desics of P. Let 0 be a Fuchsian model of acting on . Then 0 is a free
group generated by two hyperbolic transformations, 1and 2. We may assume that 1and 2 cover L1 and L2, respectively.
Theorem 3 For an arbitrarily given triple (a1; a2; a3) of positive numbers there
exists a triply connected planar Riemann surface such that l (Lj) = aj; j =
1; 2; 3:
Proof. Le C1 be the part of the imaginary axis in . Fix another geodesic
C2 on such that the Poincaré distance between C1 and C2 is equal to a1=2.
On the other hand, geodesics on from which the Poincaré distance to C1
are equal a3=2 form a real one-parameter family (the family of circular arcs C
0
3
tangent to the broken circular arc in Fig. below). Hence there exists a geodesic C3 in this family such that the Poincaré distance between C2 and C3 is equal
to a2=2.
Take, z1 and z2 in uniquely determined by the condition
(z1; z2) =
a1
2; z12 C1; z22 C2 Let L0
1 be the geodesic connecting z1 and z2. Similarly, let fz3; z4g and fz5; z6g
be the pairs of points uniquely determined by the conditions (z3; z4) = a2 2 ; z32 C2; z42 C3 (z1; z2) = a3 2 ; z52 C3; z62 C1 respectively. Denote by L0
2and L03respectively the geodesics connecting z3and
z4; and z5 and z6.
Let D be the closed hyperbolic hexagon bounded by Cj; L0j 3
j=1;2;3. Let j be the re‡exion with respect to Cj (the anti-holomorphic automorphism of
C[ f1g preserving Cj pointwise). Set
1= 1 2; 2= 3 1
Then 1 and 2 are hyperbolic elements of Aut ( ). Let 0 be the group
generated by 1and 2. It is clear that = = 0 is triply connected, and that
Theorem 4 The complex structure of a pair of paints P is uniquely determined by the hyperbolic lengths of the ordered components of P.
Proof. [Kee.] Let aj be the hyperbolic length of the boundary component Lj
(j = 1; 2; 3) of P. Let bP be the Nielsen extension of P, and 0a Fuchsian model
of bP acting on the upper half-plane H. For a given system of generators 1; 2 of 0 we may assume that k covers Lk (k = 1; 2) and that 3 = ( 2 1)
1
covers L3. Next, it su¢ ces to show that 1and 2 are uniquely determined by
fajg3j=1. Taking an Aut (H)-conjugation if necessary, we may assume that 1(z) = 2z, 0 < < 1
2(z) =
az + b
cz + d, ad bc = 1 with 1 as an attractive …xed point of 2, or
a + b = c + d, 0 < b c < 1
Then, we have 2(1) = a=c > 0 and a + d > 0, since the middle-point (a d) = (2c) of two …xed points of 2 has a value less than 2(1). Write
1
3 (z) = eaz + eb
ecz + ed, ea ed ebec = 1 Since 31= 2 1, we may assume that
a = a ; b = b= ; c = c ; d = d=
In particular,ec > 0. Moreover, the middle-point ea b = (2ec) of the …xed point of 31 has a value greater than 31(1) = ea=ec. Then ea + ed < 0. On the other hand Proposition 3 gives
( + 1= )2 = 4 cosh2 a1 2 (a + d)2 = 4 cosh2 a2 2 a + d = 4 cosh2 a3 2 Therefore, 1 and 2 are uniquely determined
Corollary 5 Every pair of pants P has an anti-holomorphic automorphism JP of order two. Moreover, the set
FJP = fz 2 P j JP(z) = zg
of all …xed points of JP consists of three geodesics fDjg3j=1 in P satisfying the
following condition: For every j(j = 1; 2; 3), Dj has the endpoints on, and is
orthogonal to, both Lj and Lj+1, where L4= L1. We call this mapping JP the
Now, recall that The Teichmüller space Tg of genusg consists of all marked
closed Riemann surfaces [R;P] of genus g where P = f[Aj] ; [Bj]ggj=1 is a
marking on R. i.e., a canonical system of generators of the fundamental group
1(R; p0) of R. By using Fuchsian models of surfaces we represent the space Tg
as a subset in the Frick space Fg of real (6g 6)-dimensional Euclidean space.
Theorem 6 The Fricke space Fg is a domain in R6g 6 and homeomorphic to
R6g 6.
1.5
Pants Decomposition
Let R be a closed Riemann surface of genus g ( 0). Consider cutting R along mutually disjoint simple closed geodesics with respect to the hyperbolic metric. When there are no more simple closed geodesics of R contained in the remaining open set, then every piece should be a paire of paints of R. Recall that the complexe structure of each pair of paints of R is uniquely determined by the triple of the hyperbolic lengths of boundary geodesics of it. Moreover, R is reconstructed by gluing all resulting pieces suitably. Hence, we can consider, as a system of coordinates for the Teichmüller space Tg, the pair of the set of
lengths of all geodesics used in the above decomposition into paints and the set of the so-called twisting parameters used to glue the pieces. Such a system of coordinates is called Fenchel-Nilsen coordinates on Tg.
Fix a point [R;P] of Tg. A set L of mutually disjoint simple closed
geo-desics on R is termed maximal if there is no set L0 which includes L properly. We call a maximal set L = fLjgNj=1of mutually disjoint simple closed geodesics
on R a system of decomposing curves, and the family P = fPkgMk=1
consist-ing of all connected components of R [N
j=1Lj the pants decomposition of R
corresponding to L.
Proposition 7 Let L = fLjgNj=1be a system of decomposing curves on a closed
Riemann surface R of genus g ( 0), and let P = fPkgMk=1the pants
decompo-sition of R corresponding to L. Then M and N satisfy N = 3g 3 and M = 2g 2
Proof. Cut R along an element L1 of L. Let n1 be the number of connected
components of R L1, andg1be the sum of genera of all connected components
of R L1. Then we have
g1 n1= (g 1) 1
Clearly, the number of boundary components of R L1 equals two.
Moreover, using the induction we see that, whenever we add a cut along a new element of L, the number of boundary components increases by two, and the sum of genera of all connected components minus the number of connected components decreases by one. Hence, we get
1.6
Geodesic length functions
Let [R;P] be a …xed point in Tg. And L = fLjgNj=1 a system of decomposing
curves on R. For every t in Fg we denote by [Rt;Pt] the point in Tg
corre-sponding to t. Then, we can determine uniquely a system Lt= fLj(t)gNj=1 of
decomposing curves on Rt. Namely, take a marking-preserving homeomorphism
ft : R ! Rt. For every Lj in L, let Lj(t) be the unique closed geodesic in
the free homotopy class of the closed curve ft(Lj) on Rt. It is not di¢ cult to
show that Lj(t) is simple, and that Lj(t) and Lj0(t) are mutually disjoint when
j 6= j0. Hence, L
t= fLj(t)gNj=1is a system of decomposing curves on on Rt.
Let tbe the Fuchsian model of Rt. Denote by lj(t) the hyperbolic length of
Lj(t). In deed, the geodesic length function lj(t) is real analytic on the Fricke
space Fg:
1.7
Twisting Prameters
Let Pj;1and Pj;2be two pairs of pants in P having Lj as a boundary component
(we allow the case Pj;1= Pj;2). Recall that Pj;1 and Pj;2 admit the re‡exion J1
and J2, respectively. Take a …xed cjk (k = 1; 2) point of Jk on Lj for each Pj;k
(k = 1; 2), and …x an orientation on Lj:
Let [Rt;Pt] 2 Tg(R) corresponding to t 2 Fg. Let Pj;1(t) and Pj;2(t) be
the connected components of Rt N
[
j=1Lj(t) corresponding to the pants Pj;1
and Pj;2 respectively. Recall that each cj;k (k = 1; 2) is the endpoint on Lj of
the geodesic Djkjoining Lj and another boundary component Lj;kin Pj;k. Let
Lj;k(t) be the boundary component of Pj;k(t) corresponding to Lj;k. Denote by
Dj;k(t) the geodesic joining Lj(t) and Lj;k(t) in Pj;k(t) with minimal length,
and by cj;k(t) the point of Dj;k(t) on Lj(t). Then each cj;k(t) (k = 1; 2) is a
…xed point of the re‡exion of Pj;k(t).
Let Tj(t) be the oriented arc on Lj(t) from cj;1(t) to cj;2(t). Since Lj(t) has
the natural orientation determined from that of Lj, we can de…ne the signed
hy-perbolic length j(t) of Tj(t) ( j(t) is positive or negative according to whether
the orientation of j(t) is compatible with that of Lj(t) or not). The twisting
parameter with respect to Lj is de…ned by
j(t) = 2 j(t)
2
Analytic Theory of Teichmüller Spaces
2.1
Quasiconformal Mappings
Recall that the existence theorem of a quasiconformal mapping with complex dilatation is valid for a general element in the set
B (C)1= 2 L1(C) j k k1= ess: sup
z2Cj (z)j < 1
Namely, For every Beltrami coe¢ cient 2 B (C)1, there exists a homéomor-phism w of the Riemann sphere bC onto it with complex dilatation . More-over w is uniqiuely determined by the conditions: w (0) = 0, w (1) = 1, w (1) = 1. This w is called the canonical -quasiconformal mapping of bC.
Fix a Fuchsian model of a Riemann surface R acting on the upper half-plane H. We assume that each of 0, 1 and 1 is a …xed point of some element of fidg. Denote by QC ( ) the set of all canonical quasiconformal mappings w of the Riemann sphere bC such that w w 1 is also a Fuchsian group. w
1;
w2 2 QC ( ) are said to be equivalent to each other if w1= w2 on R. Denote
by [w] the equivalence class of w. The Teichmüller space of is the set T ( ) = f[w] j w 2 QC ( )g
A Beltrami di¤erential on H with respect to is a bounded measurable function on H satisfying
= ( ) 0= 0 a.e on H, 2
Now, for a given 2 B (H; )1 set
e (z) = (z) ;0; z 2 Cz 2 HH
From the existence theorem, there exists uniquely a canonicale-quasiconformal mapping w of bC which has the complex dilatation e, and leaves 0, 1 and 1 …xed, respectively.
Lemma 8 For any elements ; 2 B (H; )1, the following are equivalent: 1. w = w on R
2. w = w on H
Proof. If w = w on R, then there is a homeomorphism f : bC ! bC given by
f (z) = (w )
1
w (z) ; z 2 H z; z 2 H [ bR
Since (w ) 1 w is quasiconformal on C, we see that f is ALC (absolutely continuous on lines). Thus f is quasiconformal. Hence g = w f (w ) 1 is a 1-qc mapping on C, i.e., a Möbius transformation. Since g leaves each of 0 1 and 1 …xed, g must be the identity. Therefore, we have w = w on H .
Conversely, if w = w on H , then w = w on H [ R[ f1g. Thus the mapping
h = w (w ) 1 w (w ) 1: H ! H
is quasiconformal. By the same reason as before, it follows that w = w on R:
2.2
Bers’Mappings
Let A2(R) denote the complex vector space of all holomorphic automorphic
di¤erentials on R. A holomorphic quadratic di¤erential corresponds ot a holo-morphic autoholo-morphic form of weight 4 with respect to a Fuchsian model of R acting on the upper half-plane H. Here a holomorphic automorphic form of weight 4 with respect to is by de…nition, a holomorphic function ' (z) on H such that
' ( (z)) 0(z)2= ' (z) ; z 2 H; 2
Denote by A2(H ; ) the complex vector space of holomorphic automorphic
forms of weight 4 with respect to on the lower half-plane H . Since it is identi…ed with the vector space A2(H = ) of holomorphic quadratic
di¤eren-tials on H = , the Riemann-Roch theorem shows that A2(H ; ) is a (3g
3)-dimensional complex vector space.
says that the correspondence [w ] ! [w ] is a bijection of T ( ) onto T ( ). In this way we can identify T ( ) with T ( ) as topological spaces of . We also call T ( ) the Teichmüller space of . Let : B (H; )1 ! T ( ) be a mapping given by ( ) = [ ]. Then by the de…nition of the topology of T ( ), we immediately obtain that the mapping is a continuous surjection. The Bers’embedding is de…ned by
B : T ( ) ! A2(H ; )
[w ] ! '
where ' = fw ; zg, the Schwarzian derivative of w . The Brouwer’s theo-rem on invariance of domains implies the image TB( ) of B is a domain in A2(H ; ) and B : T ( ) ! TB( ) is a homeomorphism. Since A2(H ; ) is
a (3g 3)-dimensional complex vector space, TB( ) inherits the complex man-ifold structure of A2(H ; ). The Bers’projection = B is de…ned by
: B (H; )1 ! A2(H ; )
! '
Then the derivative
: [ ] is given by : [ ] = lim t !1 1 t( ( t) ( ))
where the convergence is norm convergence with respect to the hyperbolic L1 -convergence.
Theorem 9 For every 2 B (H; ), the derivative :0[ ] exists and is given by : 0[ ] (z) = 6 Z Z H (&) (& z)4d d ; z 2 H
Proof. By Theorem 4.37 [Ima], or in [Gmi] we can write w t(z) = z + tw [ ] (z) + (t) uniformly on compact subsets of C as t ! 0, where:
:
w [ ] (z) = 1 Z Z
H
(&) z (z 1) & (& 1) (& z)d d
Since w t is holomorphic on H , from Weirstrass’theorem on double series, we get w0 t = 1 + t : w [ ]0+ (t) w00 t = t : w [ ]00+ (t) w000 t = t : w [ ]000+ (t)
uniformly on compact subsets of H as t ! 0. Thus we have w t = '
t = w t; z = t
:
w [ ]000+ (t)
uniformly on compact subsets of H as t ! 0. Since H = is compact, it follows that :0[ ] =
:
3
Weil-Petersson Metric
Here, we recall some known results [Ima]. A canonical Hermitian inner product is de…ned on the space A2(H; ) of holomorphic automorphic forms with respect
to on H. The area element d = (Im z) 2dxdy is regarded as an area element on the Riemann surface R = H= . For any elements ', 2 A2(H ; ), we
de…ne h'; i (z) = (Im z) 4' (z) (z) ; z 2 H as a function on H invariant under , it is considered as a function on the R. Now, the Hermitian inner product h'; iR by h'; iR = Z Z Rh'; i d = Z Z F (Im z) 4' (z) (z) dxdy
where F is a fundamental domain for in H. Since R is compact, it follows that A2(H; ) becomes a Hilbert space with this Hermitian product.
3.1
In…nitesimal Theory of Teichmüller Spaces
3.1.1 The Tangent Space at the Base Point
Denote by T0(T ( )) the holomorphic tangent space of T ( ) at the base point.
Identifying T ( ) with TB( ), we regard T0(T ( )) as the holomorphic tangent
T0(TB( )) = A2(H ; ) of TB( ) at the base point. As was seen before, the
derivative :0 of Bers’projection at the base point is surjective of the space
B (H; ) to T0(TB( )). Thus setting N ( ) = ker :
0, we have
T0(TB( )) u B (H; ) =N ( )
Moreover, we can identify canonically T0(TB( )) with the dual space A2(H; )
of A2(H; ) as follows. Associate every 2 B (H; ) with an element 2
A2(H; ) de…ned by
(') = h ; 'iR= Z Z
F (z) ' (z) dxdy; ' 2 A 2(H; )
Here, R = H= and F is a fundamental domain for in H. The following mapping is an isomorphism
: B (H; ) ! A2(H; )
! 3.1.2 Harmonic Beltrami Di¤erentials
We use harmonic Beltrami di¤erentials to represent tangent vectors of the Te-ichmüller space T ( ). For every ' 2 A2(H; ), we de…ne an element ['] 2
B (H; ) by
This ['] is called the harmonic Beltrami di¤erential induced by '. On the other hand, we associate every 2 B (H; ) with an element ' [ ] 2 A2(H; )
given by
' [ ] = 2:0[ ] (z)
Now, for every 2 B (H; ) we set H [ ] = (Im z) 2' [ ] (z). This H [ ] is said to be the harmonic Beltrami di¤erential induced by . Denote by HB (H; ) the vector space of all theses harmonic Beltrami di¤erentials.
Theorem 10 The space B (H; ) of Beltrami di¤ erentials for on H is the direct sum of HB (H; ) and N ( ) = ker :0:
B (H; ) = HB (H; ) N ( ) The derivative :0 at the base point induces the isomorphism
:
0: HB (H; ) ! T0(TB( ))
In particular, T0(T ( )) u HB (H; ). Moreover, h ; 'iR = hH [ ] ; 'iR; 2
B (H; ) ; ' 2 A2(H; )
3.1.3 Tangent Space of T ( ) at a General Point
Let T ( ) be the Teichmüller space of = w (w ) 1. Then the translation [w ] : T ( ) ! T ( ) given by [w ] w = hw (w ) 1i induces an isomorphism of Tp(T ( )) to T0(T ( )) [Ima].
3.2
Weil-Petersson Metric
First of all, we give a Hermitian inner product on the holomorphic tangent space T0(T ( )) at the base point. We identi…ed T0(T ( )) with HB (H; ), and hence
we start by giving a Hermitian inner product on B (H; ). This inner product of two elements 1, 2 is de…ned by
h ( 1; 2) =
Z Z
F
(Im z)2 1(z) 2(z)dxdy
where F is a relatively compact fundamental domain for in H. Using the de…nition of H [ ] we can write
h (H [ 1] ; H [ 2]) = h' [ 1] ; ' [ 1]iR; 1; 22 B (H; )
We can also show the following lemma [Ima]
Lemma 11 For every 1; 22 B (H; ), the following hold h (H [ 1] ; H [ 2]) = h (H [ 1] ; 2) = h ( 1; H [ 2])
Now, a Hermitian inner product on T0(T ( )) is induced by h under the
identi…cation T0(T ( )) u HB (H; ). Next, we de…ne a Hermitian inner
prod-uct on the tangent space Tp(T ( )) of T ( ) at an arbitrary point p = [w ].
This inner product on the tangent bundle of T ( ) is called the Weil-Petersson metric on T ( ), and is denoted by hW P. Locally, it is written
ds2W P = 2
3g 3X j;k
hjk(t) dtjdtk
Let gW P be the Riemann metric on T ( ) induced by the Weil-Petersson
metric hW P. The real tangent space of T ( ) at a point p is identi…ed with
the holomorphic tangent space Tp(T ( )) as follows: taking local coordinates
zj= xj+ iyj (j = 1; :::; 3g 3) around p, the identi…cation is given by the real
isomorphism sending (@=@xj)p, (@=@yj)p to (@=@zj)p, i (@=@zj)p, respectively
[Gri]. Then gW P is written in the form
gW P (X; Y ) = 2 Re hW P (X; Y )
Here, X and Y on the left hand side are real tangent vectors, and those on the right hand side are regarded as holomorphic tangent vectors under the iden-ti…cation given before. We call gW P the Weil-Petersson Riemannian metric
on T ( ). The Weil-Petersson form !W P of the Weil-Petersson metric hW P is
de…ned by !W P(X; Y ) = gW P(iX; Y ) and Kählerian.
4
Fenchel-Nielsen Deformation
Let R be a closed Riemann surface of genus g (g 2), a Fuchian model of R. Fix an oriented simple closed geodesic C with respect to the hyperbolic metric ds2
R. Recall that the Fenchel-Nielsen deformation of R with respect to C means
that the set fRtj t 2 Rg of marked Riemann surfaces obtained by cutting R
along C, by twisting by hyperbolic length t and then regluing the borders. We will use quasiconformal mappings to represent such a deformation. Let. Set
WC= fp 2 R j (p; C) < ag
where is the hyperbolic distance on R, and a is a positive constant such that WC becomes a tubular neighborhood. Then we construct a quasiconformal
mapping of R which represents the "twisting" along C by t in WC and is equal
to id on R WC.
First, let be a Fuchsian model of R acting on the upper half-plane H. We may assume that 1 is a …xed point of some element of fidg, and that 0 (z) = z ( > 1) belongs to and covers the curve C. Let fWC be the lift of
WC on H with respect to which contains the axis A 0 of 0. For a suitable
0 with 0 < 0< =2 we can write
Next, for every t 2 R, we de…ne a quasiconformal mapping wt: H ! H by wt(z) = 8 < : z 0 < < 2 0 z exp " 2 + 0 2 0 2+ 0 z exp (2" 0) 2 + 0< <
Here, = arg z, and " = t=2 0. This wt gives a surgery of H along the axis
A 0. Now denote by t the complex dilatation of wt. A simple computation
gives t(z) = i" 2 i" I( ) z z; z 2 H
Here, I is the characyeristic function of I = 2 0;2 + 0 on R.
Further-more the complex dilatation tsatis…es
t 0: 00= 00 = t
Thus, tis a Beltrami coe¢ cient with respect to the cyclic group h 0i. On the
other hand, it is clear that the set C consisting of all elements in which cover
the curve C is 0 1j 2 . By lifting this FN deformation to H, we get a
family of self-mappings of H which give surgeries along the axis of all elements in C. Thus, we can construct a family of quasiconformal mappings of H which
cosets of with respect to h 0i, and set t= X 2h 0in ( t ) 0 0
It is clear from the de…nition that t belongs to B (H; )1. For every t, the canonical t-quasiconformal mapping w t of H exists and determines a point
[w t] 2 T ( ). Thus we obtain a family fw t j t 2 Rg in T ( ) with respect to
the FN deformation of R along the curve C. In particular, we have obtained a curve f tj t 2 Rg in B (H; )1which represents the FN deformation of R along
C. We can construct w t more directly and geometrically by using wt which
"twiwt" along A 0 by t in H [Wol].
Now, for a given element t2 B (H; )1, we set
e = t; z 2 H
0; z 2 bC H
Then, there exists uniquely a canonical e-quasiconformal mapping w t of bC which has the complex dilatation e, and leaves 0, 1, and 1 …xed. Next, we will compute the tangent vector of this FN deformation in the real manifold TB( ). For this let t 2 B (H; )1 de…ned for any complex number t in a
neighborhood of the origin such that t= + t + t" (t), where 2 B (H; ), and k" (t)k1 ! 0 as t ! 0.
For every t, we set wt = w t. We can see that the tangent vector of the
curve f ( t) j t 2 Rg at the base point of TB( ) is equal to
'C =
d3
dz3
@wt
Now, recall that such a tangent vector is considered as an element of A2(H ; ).
Further, an integral formula forw = (@w=@t): t=0as follows
:
w (z) = 1 Z Z
H
(&) z (z 1) & (& 1) (& z)d d where (z) = 0(z)+ P
2h 0i
0( (z))
0(z)
0(z); z 2 H and 0(z) = i=4 0 I(arg z) z=z
[Gmi] :
It is easy to have lim
t !0k t=t k1= 0. We know that
i) w:z = in the sens of distribution, and
ii)w (0) =: w (1) = 0, and: w (z) =: jzj2 as z ! 1
Conversly, theses conditions characterizew in the class of continuous func-: tions on C, which can be shown by Weyl’s lemma
Now, the following lemma gives an integral formula for w = (@w: t=@t)t=0,
and we get a simpler representation of 'C [Ima] Lemma 12 The derivativew is written:
: w (z) = z Z arg z 0 I(t) 2 0 dt + i 2 log z + X 2h 0in F (z)
as for every z 2 C. Here, arg z 2 [ ; [, and
F (z) = (z) 0(z) Z arg z 0 I(t) 2 0 dt + i 2 log (z) + P (z)
where P (z) is a polynomial of degree at most two, and it is uniquely determined by the conditions that F (0) = F (1) and that F (z) = jzj2 as z ! 0. Moreover, the series on the right hand side ofw converges locally on C.: Theorem 13 Under the forgoing circumstaces, we have
'C= i 2
X
2h 0in
0 2
which converges locally uniformly on H . Proof. The Lemma gives
: w = i 2 X 2h 0in 0log P
where Pid = 0. On the other hand a direct computation gives the following
Bol’s equation
0 log 000
=
Thus, diferentiating both sides of (5) three times, we get the assertion.
In general for any closed grodesic C on R and any element 0 2 which
covers C, we can construct similar basic series as in Theorem 2. More precisely, let a and b be the two real …xed points of 0, and set
! 0(z) = (a b)
2
(z a)2(z b)2 Then the following Petersson series for the curve C
C=
X
2h 0in
! 0 : ( 0)2
converges locally uniformly on H, and belongs to A2(H; ).
4.1
A Variational Formula for Geodesic Length Functions
Let C be a …xed simple geodesic on R. For every point p = [S; f] in the Teichmüller space T (R) let Cp be the simple closed geodesic on the surface S
freely homotopic to f (C), and denote by lC(p) the hyperbolic length of Cp.
Recall that lC is a real analytic function on T (R). Here, we compute the
variation of lC at the base point p0= [R; id]. More precisely, take 2 B (H; )
arbitrarily, and let wt be the canonical t -quasiconformal mapping of H for every su¢ ciently small real t. Then wt determines a point pt2 T (R) for such
t. Under the circumstances, we compute the value
(dlC)p0( ) =
d
dtlC(pt)t=0
This is essentially well-known [Gar]. Theorem 14
(dlC)p0( ) = Re ;
2
C
Proof. We may assume that 0(z) = z ( > 1) covers C. Then, for every t a constant t(> 1) is determined by the condition that
wt 0 wt 1(z) = tz (6)
And by the de…nition of lC, we have
Let ft be the canonical t -quasiconformal mapping of bC. Then we know that ft = wt on H. Theorem 37 [Gm] implies that
:
W (z) = lim
t !0
ft (z) z
t ; z 2 C has the integral representation
:
W (z) = 1 Z Z
C
(&) R (&; z) d d (7) On the other hand, (6) shows that
:
W ( z) = d t dt (0) z +
:
w (z) Hence, we have the primary formula:
(dlC)p0( ) = d log t dt (0) = : W ( z) z : W (z) z ; z 6= 0
Now, we replaceW in this equality by the right hand side of (7), and rewrite: the integrals as those on the domains
Next, divide F0 into domains f (F ) j 2 g, where F is a fundamental
domain for . Finally, we conclude that Z Z F0 (&) &2 d d = X 2f 0gn Z Z F (&) 0(&) (&) 2 d d = Z Z F (&) X 2f 0gn 0(&) (&) 2 d d = ( ; C)R
4.2
Wolpert’s Formula
We have computed the tangent vector of the Fenchel-Nielson deformation with respect to C at the base point. By a translation of the base point we can compute the tangent vector …eld on (the real manifold) T (R) associated with the FN deformation with respect to C. Namely, @=@ C is the vector …eld obtained by
applying the FN deformation with respect to C with unit speed with respect to the hyperbolic length. Note that @=@ C is real analytic vector …eld. On
the other hand, from the geodesic length function lC, we have de…ned the real
analytic cotangent vector …eld dlC. We have the duality theorem
Theorem 15 For every simple closed geodesic C on R
i @ @ C
= dlC
where i means the natural almost complex structure on the real manifold T (R), with respect to the Riemannian metric gW P on T ( ) and is the operator given
by taking the dual with respect to gW P : dlC(:) = gW P i@@C; :
Proof. By a translation of the base point, it su¢ ces to show that the former equality at the base point p0 of T (R) :
We have gW P i @ @ C ; = 2 Re hW P i @ @ C ; = 2 Re hW P 1 (Im z) 2 c; = 2 Re ;1 C R
Corollary 16 !W P @@C; : = dlC(:) :
Proposition 17 For all simple closed geodesics C and C0 on R,
@lC0
@ C
= @lC @ C0
Proof. By the foregoing Corollary, we have @lC0 @ C = dlC @ @ 0 = !W P @ @ C0 ; @ @ C = !W P @ @ C ; @ @ C0
Theorem 18 (Wolpert’s Formula) For a system of decomposing curves L = fCjg3g 3j=1 on R. Denote by lC1; :::lC3g 3; C1; ::: C3g 3 the Fenchel-Nielsen
coordinates associated with L. Set Cj = lCj=2 for every j. Then
!W P = 3g 3X
j=1
d Cj^ dlCj
Corollary 19 For every simple closed geodesic C on R,
!W P
@ @lC
; : = d C(:)
Proof. Take a system of decomposing curves which contains C, and apply Theorem 18.
Now, to prove Theorem 18, we take a base of tangent vector …elds
fX1; ::::; X6g 6g = @ @lC1 ; ::::; @ @lC3g 3 ; @ @ C1 ; ::::; @ @ C3g 3
on T (R) (considered as a real manifold). Further we set fx1; :::::; x6g 6g = lC1; ::::; lC3g 3; C1; ::::; C3g 3
Then !W P is written as in the form
!W P =
X
1 j<i 6g 6
aijdxi^ dxj
First, we will show that every aij is invariant under the Fenchel-Nielsen
Lemma 20 For every i,j and,k, @aij
@ Ck
= 0, on T (R)
Proof. Let I (X) be the interior prucduct with respect to X. Then the Corol-lary above implies
d I @ @ Ck !W P = d !W P @ @ Ck ; : = d (lCk) = 0
Let LX be the derivation with respect to X. Then H.Cartan formula gives
L @ @ Ck!W P = d I @ @ Ck !W P + I @ @ Ck
Since !W P is kählerian, we coclude that
L @
@ Ck!W P = 0
Finally, we have
LX!W P (Y; Z) = X!W P(Y; Z) !W P ([Y; Z] ; X) !W P(Y; [X; Z])
for every set …elds X; Y and Z. Note if we take X; Y and Z in fXjg6g 6j=1 , then
[X; Y ] = [X; Z] = 0. In particular setting, X = @@ Ck, Y = Xi and Z = Xj, then we obtain @ @ Ck !W P = @aij @ Ck = 0
We coclude that when we change R to another R0 by applying the FN
deformation with respect to any Cj, the representation
!W P =
X
1 j<i 6g 6
aijdxi^ dxj
remains inchanged.
Now, we may assume without loss of generality that R admits a re‡exion J . Then J induces a mapping J : T (R) ! T (R) as follows: For every point [S; f] 2 T (R), let S be a Fuchsian model of S acting on H. Denote by S
the mirror image H = S of S, where H is the lower half-plane. De…ne an anti-holomorphic canonical mapping by setting jS : S ! S jS([z]) = [z], [z] 2 H= S. Then we have an anti-holomorphic mapping J : T (R) ! T (R)
given by J ([S; f]) = [S ; jS f J ], which …xes the base point [R; id]. We can
easily show from the de…nition of J and the Weil-Petersson metric that gW P is
Lemma 21 Denote by J the pull-back operator induced by J . Then J dlCj = dlCj
J d Cj = d Cj +
nj
2 dlCj; nj2 Z
for every (j = 1; ::::; 3g 3). Furtherer, !W P satis…es
J (!W P) = !W P
Proof. The …rst assertion in the Lemma follows by taking the derivation of both sides of lCj J = lJ (Cj)= lCj:
Next, for every j, we see that, though J (Cj) = Cj as point sets, the
orien-tation of Cj at p is the converse of that at J (p) for every p 2 T (R). It is cleair
that Cj is determined modulo lCj=2. Hence, with a suitable integer njwe have
Cj J = Cj+
nj
2 lj
Finally, let X and Y be arbitrary tangent vector …elds on T (R) :Then we obtain
(J !W P) (X; Y ) = !W P (J X; J Y )
= gW P(iJ X; J Y )
Since J is anti-holomorphic, we see that iJ X = J (iX). Since gW P is
invariant under J , we conclude that
(J !W P) (X; Y ) = gW P (iJ X; J Y )
= gW P (iX; Y )
= !W P(X; Y )
Now, we give the proof of the Wolpert’s formula.
Proof. It su¢ ces to show the formula at the base point. Hence all relations bellow are to be considered at the base point, which is a …xed point of J . First, by the Corollary we have
a3g 3+j;k= !W P @ @ Cj ; @ @ Ck = @lCj @lCk = ik
for all j; k with 1 j; k 3g 3.
Next, taking the dual in J dlCj = dlCj and J d Cj = d Cj+
for every (j = 1; :::; 3g 3). Hence by J (!W P) = !W P, we conclude that a3g 3+j;3g 3+k = !W P @ @ Cj ; @ @ Ck = !W P J @ @ Cj ; J @ @ Ck = (J !W P) @ @ Cj ; @ @ Ck = a3g 3+j;3g 3+k
Thus we get a3g 3+j;3g 3+k = 0 for all j; k, 1 k < j 3g 3. Similarly, for
all such j; k, we see that
aj;k = !W P @ @lCj ; @ @lCk = !W P @ @lCj +nj 2 @ @ Cj ; @ @lCk +nk 2 @ @ Ck = ajk
Hence, we obtain ajk= 0 for all j; k with 1 k < j 3g 3. Therefore we see
References
[Ahl] Ahlfors, L.V.: Advences in the Theory of Riemann Surfaces, Press Princeton, New Jersy 1971.
[Gar] Gardiner, F.P.: Shi¤er’s interior variation and quasiconformal mappings, Duke Math .J.42(1975)
[Gmi] Gmira, S,: Quasiconformal Mappings Hal 2016
[Gri] Gri¢ ths, P.A. and Harris, J.: Principles of Algebraic Geometry, Wiley, New York, 1978.
[Ima] Imayoshi. Y-Taniguchi. M,: An Introduction to Teichmüller Spaces, Springer-Verlag
[Kee] Keen, L,: On Fricke Moduli, 1971.
