Riemannian connections and curvatures on the universal Teichmuller space

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Probability Theory

Riemannian connections and curvatures on the universal Teichmuller space

Hélène Airault

^a,b

aINSSET, université de Picardie, 48, rue Raspail, 02100 Saint-Quentin, France bLaboratoire CNRS UMR 6140 LAMFA, 33, rue Saint-Leu, 80039 Amiens, France

Received 3 June 2005; accepted 20 June 2005 Available online 3 August 2005

Presented by Paul Malliavin

Abstract

We define Riemannian connections on the universal Teichmuller spaceU^∞. For the Levi-Civita’s connection onU^∞, the Riemannian curvature tensor is well defined and the Ricci curvature is finite. We obtain several series of infinite dimensional operators which converge. To cite this article: H. Airault, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Résumé

Connexions riemanniennes et courbures sur l’espace de Teichmuller universel. On définit plusieurs connexions rieman- niennes sur l’espace de Teichmuller universelU^∞. Pour la connexion de Levi-Civita surU^∞, le tenseur de courbure existe et la courbure de Ricci est finie. On obtient plusieurs séries d’opérateurs de l’espace de dimension infinie qui convergent. Pour citer cet article : H. Airault, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Version française abrégée

Soit Diff(S¹)le groupe des difféomorphismes du cercle qui préservent l’orientation et soitHle sous-groupe des transformations homographiques. Un difféomorphisme e^{iγ (θ )}s’identifie avec l’applicationγ:θ→γ (θ )mo- dulo 2π. Soit diff(S¹)l’algèbre de Lie de Diff(S¹). L’algèbre de Lie de Hest notée su(1,1), elle est engen- drée par cosθ, sinθ, 1. Pour k un entier, k0, on pose α(k)=ak³+bk où a 0 et b est un nombre réel. Sur l’espace vectoriel V des séries de Fourier u(θ )=

k0a_k^ucos(kθ )+b_k^usin(kθ ) telles que u²=

k1(a^u_k)²α(k)+(b^u_k)²α(k) <+∞, on considère le produit scalaire

E-mail address: [email protected] (H. Airault).

doi:10.1016/j.crma.2005.06.028

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(u|v)=

k1

a_k^ua_k^vα(k)+b_k^ub_k^vα(k). (1)

Comme(w|u)=

k2(a^u_kb^w_k −a^w_kb^u_k) k α(k), on a(w|u)= −(w|u). Pouru,v∈V,

[u, v](θ )=u(θ )v(θ )−u(θ )v(θ ). (2)

Lorsqueα(k)=k³−ket pouru,v,wdans le sous-espace vectorielV₀engendré par{cos(kθ );sin(kθ )}k2, on démontre l’identité remarquable

J u| [v, w] +

J v| [w, u]

+(J w| [u, v]

=0, (3)

oùJ est la transformation de Hilbert, Jcoskθ =sinkθ etJsinkθ= −coskθ,k1. Soitπ la projection de diff(S¹)sur V₀. Lorsque u, v, w∈diff(S¹), on définit la connexion de Levi-Civita sur le groupe quotient H\

Diff(S¹)par (cf. [12, vol. 2, p. 201]) ΓLC(u, v, w)=

ΓLC(v)u|w

=1 2

[w, u] |π v +

[w, v] |π u

−

[u, v] |π w

. (4)

La courbure de cette connexion est égale à celle de la connexion obtenue dans [6]. On étudie la convergence de séries formées par des opérateurs diagonaux dans la base orthonormale{c_k=√^{cos(kθ )}

k³−k;s_k=√^{sin(kθ )}

k³−k}k2et issues de différentes connexions. SoitΓ₁(v)u= [u, v]. La série d’opérateurs

j2Γ_LC(c_j)²+Γ_LC(s_j)²converge et la série

Λ=

j2

(Γ_LC+Γ₁)(c_j)Γ_LC(c_j)+(Γ_LC+Γ₁)(s_j)Γ_LC(s_j) (5) définit un opérateur de courbure borné.

1. Introduction

Among the difficulties of geometry in infinite dimension are the importance of a good choice of coordinate sys- tem and the divergence of series associated to the operation of contraction. Infinite-dimensional geometry appears naturally in Probability Theory [1,2,7] and in Mathematical Physics [6,5,11]. The universal Teichmuller spaceU^∞ is the space ofC^∞Jordan curves of the complex plane. In our program,U^∞will appear as the skeleton of canon- ical probability measures which will be carried by a larger space thanU^∞(see [4,10]). Toγ ∈U^∞, correspond two univalent functionsf_γ⁺,f_γ⁻ sending the inside and the outside of the unit disk on the two regions delimited byγ; thenf_γ⁺,f_γ⁻areC^∞functions on the circleS¹. Therefore,g_γ(θ ):= [(f_γ⁺)⁻¹ ◦ f_γ⁻]exp(iθ )is a diffeomorphism of the circle. Denote Diff(S¹)the group ofC^∞, orientation preserving diffeomorphisms of the circle. Let Hbe the subgroup of Möbius transformations of the disk considered as operating onS¹; thenf_γ⁺,f_γ⁻are defined up to a Möbius transformation. The previous construction defines an injective mapU^∞→H\Diff(S¹)/Rot(S¹).

The Beurling–Ahlfors theory of conformal welding shows that this map is surjective. The Lie algebra diff(S¹)is the set ofC^∞ vector fields on the circle, they are identified with theC^∞functions on the circle. The Lie algebra su(1,1)ofHis the linear subspace of diff(S¹)having for basis 1, cosθ, sinθ. It has been shown in [3] that there exists a unique scalar product on diff(S¹)which is invariant under the adjoint action of su(1,1). The associ- ated metric defines a canonical Riemannian structure onU^∞. An orthonormal basis for this metric at identity is {√¹

k³−kcoskθ,√¹

k³−ksinkθ}, k >1. In this Note,

(i) we prove the identity (3) which is specific to the metricα(k)=k³−k,

(ii) we introduce different Riemannian connections onU^∞, among them the Levi-Civita connection (4) which commutes with the Hilbert transformJ.

This leads to the Riemannian curvature and Ricci curvature ofU^∞.

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2. The geometry ofH\Diff(S¹)

We calculate on Diff(S¹)modulo composition on the left by homographic transformations. Letγ ∈Diff(S¹) andu(θ )∈V. To obtain right invariant vector fields, we define for small >0, γ(θ )=γ (θ )+u(γ (θ ))= (exp(u)◦γ )(θ )+o(²), thenX_u(γ )=_d^d|=0γ=u◦γ is right invariant since for the right translationR(γ )

=γ◦γ₁, we have dR(γ )[X_u(γ )] =X_u(γ◦γ₁)=X_u(γ )◦γ₁. In the same way, we putγ(θ )=γ (θ+u(θ ))= (γ◦exp(u))(θ )+o(²), thenY_u(γ )(θ )=_d^d|=0γ(θ )=γ(θ )u(θ )whereγ(θ )is the derivative ofγwith respect toθ, is left invariant. For a vector field Z(γ ), the parallel transport to the right is given byZ(γ )(θ )→(Z(γ )◦ γ1)(θ )and the parallel transport to the left is given byZ(γ )(θ )→γ₁(θ )Z(γ )(θ ). LetL(γ )=γ1◦γ be the left translation. We define Ad(γ ):V →V by Ad(γ )(u)(θ )=γ(γ⁻¹(θ ))u(γ⁻¹(θ )). Then Ad(γ⁻¹)(u)(θ )=^{u(γ (θ ))}_γ(θ ) . For a rotation of angle φ, θ+φ=r_φ(θ ), Ad(r_φ)(u)=u(θ +φ). Let u, v∈V and consider the vector fields X_u(γ )=u◦γ andX_w(γ )=w◦γ; the bracket is given by[X_u(γ ), X_v(γ )] = [u, v] ◦γ. ForY_u(γ )=γ(·)u(·), then[Y_u(γ ), Y_v(γ )] =γ(·)[u, v].

3. The Hilbert transform

LetJcoskθ=sinkθ andJsinkθ = −coskθ,k1. The Hilbert transform J allows to sort out the coskθ, sinpθ,. . .according to whetherk > pork < p. With (2), we obtain

[Jcoskθ, Jcospθ] − [coskθ,cospθ] =(p−k)sin(p+k)θ,

[Jsinkθ, Jcospθ] − [sinkθ,cospθ] =(k−p)cos(p+k)θ (6) and forX=sinkθ, or coskθ andY =sinpθ, or cospθ, withp,k2,

[J X, J Y] − [X, Y] =J

[X, J Y] + [J X, Y]

. (7)

SinceJ²= −Id, then (7) can be written[X, J Y] + [J X, Y] =J[X, Y] −J[J X, J Y]. Define A(X, Y )=J[X, Y] − [J X, Y] and B(X, Y )=J[X, Y] − [X, J Y],

{X, Y} =1 2

[X, J Y] + [J X, Y]

=1 2J

[X, Y] − [J X, J Y]

. (8)

WhenX =Y, we haveB(X, Y )=J B(X, J Y )as well asA(X, Y )=J A(J X, Y )andA(X, J Y )+A(J X, Y )=0.

Eq. (6) is equivalent to any of the two followingA(J X, J Y )=A(X, Y )orB(J X, J Y )=B(X, Y ). For{,}, we have the Jacobi identity

X,{Y, Z} +

Y,{Z, X} +

Z,{X, Y}

=0 (9)

and{J X, Y} =J{X, Y} = {X, J Y}. With (7), we deduce J

[J X, J Y] + [X, Y]

−

[J X, Y] − [X, J Y]

=2A(X, Y ), J

[J X, J Y] + [X, Y] +

[J X, Y] − [X, J Y]

=2B(X, Y ). (10)

4. The scalar product on diff(S¹)/su(1,1)

With the scalar product (1), it was remarked in [3] thatα(k)=constant×(k³−k)when Ad(h) is a unitary operator for any homographic transformation h. In fact, let u_2k(θ )=cos(kθ ),u_2k₊₁(θ )=sin(kθ ), the condi- tion([u, u_p] |u_q)+(u_p| [u, u_q])=0, ∀p, q2 andu=1, cos(θ ), sin(θ ) is equivalent to(1−k)α(1+k)+ (2+k)α(k)=0 ifk =2 and−α(3)+4α(2)=0 which determines completelyα(k)=constant×(k³−k). In that case, form, n, p, j∈Z,(m−n)α(p)+(n−p)α(m)+(p−m)α(n)=0 ifm+n+p=0 and

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(j−k)(m+p)α(j+k)−(p−k)(m+j )α(p+k)

=(j−p)

(k+j+2p)α(k+j )+(p−k)α(j+p)

ifm=k+j+p.

Lemma 4.1. Ifα(k)=constant×(k³−k), foru,v,win diff(S¹), we have sinkθ| [cosj θ,cospθ]

+

sinj θ| [cospθ,coskθ] +

sinpθ| [coskθ,cosj θ]

=0, coskθ| [sinj θ,cospθ]

+

cosj θ| [cospθ,sinkθ] +

sinpθ| [sinj θ,sinkθ]

=0. (11)

We verify (11) as follows. letδ_j^pbe the Kronecker symbol, we add 2

sinkθ| [cosj θ,cospθ]

=(j+p)α(k)δ^k_j⁺^p+(j−p)α(k)δ_k^j⁺^p−(j+p)α(k)δ^k_p⁺^j, 2

sinj θ| [cospθ,coskθ]

=(p−k)α(j )δ^k_j⁺^p−(k+p)α(j )δ_k^j⁺^p+(k+p)α(j )δ^k_p⁺^j, 2

sinpθ| [coskθ,cosj θ]

= −(k+j )α(p)δ_j^k⁺^p+(k+j )α(p)δ^j_k⁺^p+(k−j )α(p)δ_p^k⁺^j, 2

coskθ| [sinj θ,cospθ]

= −(j+p)α(k)δ_j^k⁺^p−(j−p)α(k)δ^j_k⁺^p−(j+p)α(k)δ_p^k⁺^j, 2

cosj θ| [cospθ,sinkθ]

=(k−p)α(j )δ^k_j⁺^p+(k+p)α(j )δ_k^j⁺^p+(k+p)α(j )δ^k_p⁺^j, 2

sinpθ| [sinj θ,sinkθ]

=(k+j )α(p)δ^k_j⁺^p−(k+j )α(p)δ_k^j⁺^p+(k−j )α(p)δ^k_p⁺^j.

From (11), we deduce that(u| [J v, J w])+(v| [J w, J u])+(w| [J u, J v])=0 or equivalently (3) is true for u,v,wof the form coskθor sinj θ. We use the linearity in each variable to prove (3) in its generality.

From (3), we obtain more identities, let{u, v} =¹₂([u, J v] + [J u, v])as in (8), then u| {v, w}

+

v| {w, u} +

w| {u, v}

=1 2

J u| [J v, J w] +

J v| [J w, J u] +

J w| [J u, J v] , J u| {v, w}

+

J v| {w, u} +

J w| {u, v}

=1 2

u| [v, w] +

v| [w, u] +

w| [u, v] .

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5. The Levi-Civita’s transfer field and related fields

We follow [12, vol. 1, p. 160 and vol. 2, p. 201], [7] and [2, p. 452]. For u, v, w∈diff(S¹), we define Γ_LC(u, v, w)by (4). We denote Γ_LC(u, v, w)=(Γ_LC(v)u|π w)=(Γ_LC)^w_vu. Since Γ_LC(u, v, w)is antisymmetric in(u, w), the adjointΓ_LC^∗ satisfiesΓ_LC^∗ = −ΓLC. With (3), we deduce that the connectionΓLCpreserves the complex structure,ΓLC(u, v, w)=ΓLC(J u, v, J w). The transfer fieldΓLCis the half sum of two antisymmetric transfer fields

Γ_LC=1

2[Γ₃+Γ₅], withΓ₃(u, v, w)=

[w, u] |π v

, Γ₅(u, v, w)=

[w, v] |π u

−

[u, v] |π w . (13) ForΓ =Γ₃,Γ₅,Γ (v)J =J Γ (v). Foru, v, w∈diff(S¹), we denoteΓ_vu^w=Γ (u, v, w)and we define

Γ₁(u, v, w)=

[u, v] |π w

=

Γ₁(v)u|π w

=(Γ₁)^w_vu, Γ₂(u, v, w)=

[v, w] |π u

=

Γ₂(v)u|π w

=(Γ₂)^w_vu, Γ₄(u, v, w)=

[w, u] |π v +

[w, v] |π u

=

Γ₄(v)u|π w

=(Γ₄)^w_vu,

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thenΓLC=¹₂[Γ4−Γ1]andΓ5= −(Γ1+Γ2). Another connection also preserves the complex structure, see [6], ΓA(u, v, w)=Γ_LC(J u, J v, J w)=

ΓA(v)u|w

=(ΓA)^w_vu. (15)

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LetA(v, w)andB(v, w)as in (8). Then B(v, w)|u

−

B(v, u)|w

=

J v| [w, u] +

[J w, v] |u

−

[J u, v] |w ,

A(v, w)|u

−

A(v, u)|w

=

ΓA(J v)u|w , J Γ_A(J v)u+Γ_A(v)u=2J B(u, v).

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To prove the last identity, we remark with (3) that Γ_LC(J v)u|J w

−

Γ_LC(v)u|w

=2

J[u, J v] + [u, v] |w

= −2

J B(u, v)|w

. (17)

With (3), we obtain thatΓ_{}(u, v, w)=¹₂[({w, u} |π v)+({w, v} |π u)−({u, v} |π w)]satisfies Γ_{{ }}(u, v, w)=1

2Γ_LC(J u, J v, J w)+1 2

[u, w] |J v

. (18)

For Γ_LC, the torsion is zero. Consider ΓA(u, v, w)=Γ_LC(J u, J v, J w), the torsion TΓ_A(u, v) is not zero, (TΓ_A(u, v)|w)=([J v, J u] |J w)−(J[v, u] |J w). The two tensors Γ_LC andΓA have the same curvature. In complex coordinates,ΓLC, forp,k2, is given by

Γ_LC(e^ipθ)e^ikθ=i(2p+k)α(k)

α(p+k) e^i(p⁺^k)θ,

Γ_LC(eîpθ)e⁻îkθ= −i(p+k)1_k_p₊₂×e⁻î(k⁻^p)θ=1_k_p₊₂× [eîpθ,e⁻îkθ], Γ_LC(e⁻îpθ)eîkθ=i(p+k)1_k_p₊₂×eî(k⁻^p)θ=1_k_p₊₂× [e⁻îpθ,eîkθ], Γ_LC(e⁻îpθ)e⁻îkθ= −i(2p+k)α(k)

α(p+k) e⁻^i(p⁺^k)θ.

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6. Composition of transfer fields: expression with symbols

Forj2, let2j(θ )=cj(θ )=^√^cos_{α(j )}^{j θ} and2j+1(θ )=sj(θ )=^√^sin_{α(j )}^{j θ}. IfΓ =ΓLC,ΓAor anyΓ as above, we denoteΓ_k^j_r =Γ (_r, _k, _j)=(Γ (_k)_r |_j). Letδ^p_r be the Kronecker symbol. The symbol

A_{j kp}=δ^k_j⁺^p(j+k)√

√ α(p) α(j )√

α(k) fork, j, p2, (20)

is convenient to calculate in a systematic way the composition ofΓ’s.

1

2[A_{j kp}+A_pkj] =1 2

[c_p, s_j] |c_k +

[c_p, c_k] |s_j

−

[s_j, c_k] |c_p

=(Γ_LC)^c_c^p_k_s_j= −(Γ_LC)^s_c^p_k_c_j, 1

2[A_{j kp}−A_pkj] =1 2

−

[c_p, c_j] |s_k

−

[c_p, s_k] |c_j +

[c_j, s_k] |c_p

= −(Γ_LC)^c_s^p_k_c_j

=1 2

[sj, sp] |sk

+

[sj, sk] |sp

−

[sp, sk] |sj

= −(ΓLC)^ss^p_ks_j.

The following operatorsE_j are diagonal in the basis{c_m, s_m}k2and the series

j2E_j converge, Γ_LC(c_j)²+(Γ_LC)(s_j)²

c_m= −1 2

A²_{mj r}+A²_{rj m} c_m, (Γ_LC+Γ₁)(c_j)Γ_LC(c_j)+(Γ_LC+Γ₁)(s_j)Γ_LC(s_j)

c_m

= −1

2A_{mj r}A_mrjc_m= −(2m−j )(m+j )

2α(m) 1mj+2c_m,

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Γ₄(c_j)Γ₁(c_j)+Γ₄(s_j)Γ₁(s_j) c_m

= 1 2

Aj mrAj rm−ArmjArj m+A²_{rj m} +1

2

A²_{mj r}+Amj rAmrj

cm, Γ4(cj)Γ2(cj)+Γ4(sj)Γ2(sj)

cm= 1 2

A²_{rj m}−A²_{j rm} +1

2

A²_{mj r}−A²_mrj cm, Γ₁(c_j)Γ₂(c_j)+Γ₁(s_j)Γ₂(s_j)

c_m= −1 2

A²_{j rm}+A²_{rj m}

−1

2(A_mrj−A_{mj r})²

c_m, Γ_LC(c_j)Γ₂(c_j)+Γ_LC(s_j)Γ₂(s_j)

c_m= 1

2A²_{rj m}+1

2A_{mj r}(A_{mj r}−A_mrj)

c_m. The operatorΛdefined by (5) is a bounded operator with the metricα(k)=k³−k,

2Λc_m= −

j2

A_{mj r}A_mrjc_m= −

j2

(2m−j )(m+j )

α(m) 1_m_j₊₂c_m

= − 13

6 +a finite number of terms in 1

m. (21)

Λis related to the tangent processes introduced in the works [7,9]. Among all the Driver’s connections, see [8], ΓLC andΓA are the only ones for which

j2Γ²(cj)+Γ²(sj)converge. The series

j2Γ²(cj)+Γ²(sj) converge forΓ_LCandΓ₄, it diverge forΓ =Γ₃,Γ₅andΓ₁,Γ₂.

References

[1] H. Airault, P. Malliavin, Unitarizing probability measures for representation of Virasoro algebra, J. Math. Pures Appl. 80 (6) (2001) 627–667.

[2] H. Airault, P. Malliavin, Integration by parts formulas and dilatation vector fields on elliptic probability spaces, Probab. Theory Related Fields 106 (1996) 447–494.

[3] H. Airault, P. Malliavin, A. Thalmaier, Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows, J. Math. Pures Appl. 83 (2004) 955–1018.

[4] H. Airault, J. Ren, Modulus of continuity of the canonic Brownian motion on the group of diffeomorphisms of the circle, J. Funct.

Anal. 196 (2002) 395–426.

[5] M.J. Bowick, A. Lahiri, The Ricci curvature of Diff(S¹)/SL(2, R), J. Math. Phys. 29 (9) (1988) 1979–1980.

[6] M.J. Bowick, S.G. Rajeev, The holomorphic geometry of closed bosonic string theory and Diff(S¹)/S¹, Nucl. Phys. B 293 (1987) 348–

384.

[7] A-B. Cruzeiro, P. Malliavin, Renormalized differential geometry on path space: structural equation, curvature, J. Funct. Anal. 139 (1) (1996) 119–180.

[8] B. Driver, A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 110 (1992) 272–376.

[9] S. Fang, Integration by parts for heat measures over loop groups, J. Math. Pures Appl. 78 (1999) 877–894.

[10] S. Fang, Canonical Brownian motion on the diffeomorphism group of the circle, J. Funct. Anal. 196 (2002) 162–179.

[11] A.A. Kirillov, D.V. Yurev, Kähler geometry of the infinite-dimensional homogeneous spaceM=Diff₊(S¹)/Rot(S¹). Translated from Funktsional. Anal. i Prilozhen. 21 (4) (1987) 35-46.

[12] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vols. I and II, Interscience Tracts in Pure and Appl. Math., Wiley and Sons, 1963.