Liouville Action for Harmonic Diffeomorphisms

In this paper, we introduce a Liouville action for a harmonic diffeomorphism from a compact Riemann surface to a compact hyperbolic Riemann surface of genus $g\ge 2$. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fixed target Riemann surface.


Introduction
In mathematical physics, the Liouville action has been used as the action functional for the Liouville conformal field theory. In mathematics, this was constructed in the works of Takhtajan-Zograf [12,13]. They also proved several fundamental results of the Liouville action using the Teichmüller theory developed by Ahlfors-Bers. One of main results in [12,13] is that the Liouville action is a Kähler potential of the Weil-Petersson symplectic 2-form on Teichmüller space.
One novelty of the works [12,13] in the construction of the Liouville action is the use of the projective structures on the Riemann surface. The projective structures determined by the geometric uniformizations in [9,12,13] define the bounding noncompact hyperbolic 3-manifolds determined by those uniformizations. In this geometric situation, the Liouville actions were proved to be the same as the renormalized volumes of the bounding hyperbolic 3-manifolds. We refer to [3,4,6,7,9] for the relation of the Liouville action with the renormalized volume.
The harmonic map theory has been one of the main tools in the study of Teichmüller space. The basic fact of this approach is that there exists a unique harmonic diffeomorphism for the hyperbolic metric on the target Riemann surface in the homotopy class of an arbitrary homeomorphism between two compact Riemann surfaces. Given a harmonic map, there is an associated Hopf differential, which is a holomorphic quadratic differential on the source Riemann surface. In [8,11], it was proved that the Hopf differentials of harmonic diffeomorphisms give a natural parametrization of Teichmüller space fixing the source Riemann surface. Another associated object to a harmonic map is its energy, which can be considered as a functional on Teichmüller space varying one of the source or target (hyperbolic) Riemann surface and fixing the other. This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html A modest motivation of this paper was to relate two objects -the Liouville action and the energy of harmonic diffeomorphisms -so that we may have a certain object sharing the interesting properties of these two objects. To explain our approach to this problem, let us recall the construction of the Liouville action by Takhtajan-Zograf. In [12,13], they crucially used a map denoted by J from the Poincaré half plane to the region of discontinuity for a Kleinian group determined by a geometric uniformization. Then the main ingredient in the definition of the Liouville action is given by the pullback of the Poincaré metric by J −1 .
A possible generalization of the Liouville action could be achieved by using other map instead of J −1 . In this paper, we develop this idea using a harmonic diffeomorphism which is canonically associated to the quasi-Fuchsian uniformization for two marked compact hyperbolic Riemann surfaces. By our construction, the Liouville action for a harmonic diffeomorphism contains the holomorphic energy of the harmonic map as a part.
As a first step to this study, we derive a variational formula for the Liouville action for harmonic diffeomorphisms. In this formula, the variation of the Liouville action for harmonic diffeomorphisms is described mainly in terms of the Schwarzian derivative and the Hopf differential of harmonic diffeomorphisms. The precise variational formula is given in Theorem 3.6. A main part of the proof of this theorem is based on the variational formula of the Liouville action for a smooth family of conformal metrics on Riemann surfaces, which generalizes the work of Takhtajan-Teo in [9].
Our approach in this paper may raise several related questions. One of them is a possibility to obtain another Liouville action for harmonic diffeomorphisms modifying the construction of this paper. We take the term given by the holomorphic energy density in the pullback of the hyperbolic metric on the target Riemann surface by a harmonic diffeomorphism. But, we can also take the Hopf differential part possibly among the parts of the pullback metric instead of our choice in this paper. It is interesting to see how this different definition would provide us with a useful functional on Teichmüller space. See Remark 3.4 for more detailed remark on this case.
Another natural question is the second variation of the Liouville action for diffeomorphisms. The second variation of the Liouville action defined by Takhtajan-Zograf in [12,13] gives the Weil-Petersson symplectic 2-form on Teichmüller space. On the other hand, the energy functional of harmonic diffeomorphisms varying the source Riemann surfaces with a fixed target hyperbolic Riemann surface is a strictly plurisubharmonic function on Teichmüller space. This follows from that the Levi form given by the second variation of the energy functional is positive definite. For these, we refer to Tromba's book [10]. Hence, as a common feature of the Liouville action and the energy functional of harmonic diffeomorphisms, one may wonder whether the Liouville action for diffeomorphisms would be a strictly plurisubharmonic function on Teichmïller space. The second variation of the Liouville action for harmonic diffeomorphisms and its possible applications will be studied elsewhere.
Finally let us explain the structure of this paper. We start with the basic definitions and terminologies for the Liouville action in Section 2. This is a quick review of [9, Section 2]. In Section 3, we present the basics of the Liouville action for harmonic diffeomorphisms and derive its variational formula. In Section 4, we prove the variational formula of the Liouville action for a smooth family of conformal metrics following [9, Section 4].

Liouville action for quasi-Fuchsian groups
Let us consider a compact Riemann surface X with genus g ≥ 2. Then the Riemann surface X can be realized by the Fuchsian uniformization. It means that X is given by a quotient space Γ\U, where U is the upper half plane and Γ is a marked, normalized Fuchsian group of the first kind. Here Γ is a finitely generated cocompact discrete subgroup of PSL(2, R) which has a standard representation with 2g hyperbolic generators α 1 , β 1 , . . . , α g , β g satisfying the relation where I is the identity element in Γ.
On the other hand, X can be realized by the quasi-Fuchsian uniformization. It means that X is given by a quotient space by a marked, normalized quasi-Fuchsian group Γ ⊂ PSL(2, C). This group has its region of discontinuity Ω ⊂Ĉ, which has two invariant components Ω 1 and Ω 2 separated by a quasi-circle C. There exists a quasi-conformal homeomorphism J 1 ofĈ such that QF1 J 1 is holomorphic on U and J 1 (U) = Ω 1 , J 1 (L) = Ω 2 , J 1 (R) = C, where U and L are respectively the upper and lower half planes.
Let A −1,1 (Γ) be the space of Beltrami differentials for a quasi-Fuchsian group Γ, which is the where T(Γ i ) is the Teichmüller space of Γ i for i = 1, 2. The deformation space D(Γ, Ω 1 ) is defined using the Beltrami coefficients supported on Ω 1 . By definition, the space D(Γ, Ω 1 ) parametrizes all deformations of X = Γ\Ω 1 with the fixed Riemann surface Γ\Ω 2 so that Hence, it is possible to use the deformation space D(Γ, Ω 1 ) as the model of the Teichmüller space T(Γ 1 ). An advantage of this model is that one can use the holomorphic variation on D(Γ, Ω 1 ) given by the quasi-Fuchsian deformations. In the following two subsections, we review the construction of the Liouville action for quasi-Fuchsian groups in [9]. We refer to [9, Section 2] for more details.

Homology construction
Starting with a marked, normalized Fuchsian group Γ, the double homology complex K •,• is defined as S • ⊗ ZΓ B • , a tensor product over the integral group ring ZΓ, where S • = S • (U) is the singular chain complex of U with the differential ∂ , considered as a right ZΓ-module, and B • = B • (ZΓ) is the standard bar resolution complex for Γ with differential ∂ . The associated total complex Tot K is equipped with the total differential There is a standard choice of the fundamental domain F ⊆ U for Γ as a non-Euclidean polygon with 4g edges labeled by a k , The orientation of the edges is chosen such that where L ∈ K 1,1 is given by There exists V ∈ K 0,2 such that One can verify that it is given by Then ∂Σ = 0.
Finally, we also define W in the following way. Let P k be a Γ-contracting path (see [9,Definition 2.3] for the precise definition of Γ-contracting) connecting 0 to b k (0). Then If Γ is a quasi-Fuchsian group, let Γ 1 be the Fuchsian group such that Γ 1 = J −1 1 • Γ • J 1 . The double complex associated with Ω 1 and the group Γ is a push-forward by the map J 1 of the double complex associated with U and the group Γ 1 . Define where the corresponding chains F , L , V in L are given by the complex conjugation of F , L, V respectively.

Cohomology construction
The corresponding double complex in cohomology C •,• is defined as where A • is the complexified de Rham complex on Ω 1 . The associated total complex TotC is equipped with the total differential D = d + (−1) p δ on C p,q , where d is the de Rham differential and δ is the group coboundary. The natural pairing , between C p,q and K p,q is given by the integration over chains.
Denote by CM(Γ\Ω) the space of conformal metrics on Γ\Ω. That is, every ds 2 ∈ CM(Γ\Ω) is represented as ds 2 = e φ(z) |dz| 2 , where φ is a smooth function on Ω satisfying The Liouville action is a function on the space of conformal metrics. Its construction is as follows. Starting with the 2-form Here c(γ) is the element c in the linear fractional transformation γ = a b c d . Notice thať From the definition ofθ and δ 2 = 0, it follows that the 1-formǔ is closed. An explicit calculation givesǔ

J. Park
For e φ(z) |dz| 2 ∈ CM(Γ\Ω), the Liouville action is defined as Here W 1 = J 1 (W ) and W 2 = J 1 (W ) for the chain W in L given by the complex conjugation of W . We also defině

Liouville action for harmonic diffeomorphisms
For compact Riemann surfaces X and Y with genus g ≥ 2, let h : X → Y denote a harmonic map for the hyperbolic metric e ψ(u) |du| 2 on Y , where u denotes a conformal coordinate on Y . The harmonic condition for h is given by for a conformal coordinate z on X. Note that this condition depends on the conformal structure on X and the metric structure on Y . The pullback metric h * e ψ(u) |du| 2 on X has the following expression h * e ψ(u) |du| 2 = e ψ•h h zhz dz 2 + e ψ•h h zhz |dz| 2 + e ψ•h hzh z |dz| 2 + e ψ•h hzhzdz 2 . which is called a Hopf differential of h. By the harmonicity condition (3.1), one can easily check that Φ(h) is a holomorphic quadratic differential on X. We also put e φ := e ψ•h h zhz .
By [2, Theorem 3.10.1 and Corollary 3.10.1], we have Proposition 3.1. For a compact Riemann surface X and a compact hyperbolic Riemann surface Y with same genus g ≥ 2 and a continuous map g : X → Y of degree 1, there is a unique harmonic diffeomorphism h : X → Y in the homotopy class of g. In this case, e φ = e ψ•h h zhz never vanishes on X, where e ψ(u) |du| 2 is the hyperbolic metric on Y .
By Proposition 3.1, for a harmonic diffeomorphism h : X → Y for a hyperbolic metric on Y , e φ = e ψ•h h zhz defines a metric on X. Similarly |Φ(h)| defines a singular flat metric on X since the holomorphic quadratic differential Φ(h) should have −2χ(X) number of zeros.
Proposition 3.2. For a harmonic diffeomorphism h : X → Y for a hyperbolic metric on Y , In particular, K φ never vanishes.
Proof . By the equality (3.1), Here the second equality in (3.5) follows by the Liouville equation for e ψ , Hence we have For the harmonic diffeomorphism h : X → Y , its Jacobian J h = |h z | 2 − |hz| 2 never vanishes. Hence K φ never vanishes by the above equality.
For two marked compact Riemann surfaces of genus g ≥ 2, there exists a marked, normalized quasi-Fuchsian group Γ such that X = Γ\Ω 1 and Y = Γ\Ω 2 for a region of discontinuity Ω 1 Ω 2 by Bers' simultaneous uniformization theorem. We also assume that Y is realized by the Fuchsian uniformization with a Fuchsian group Γ Y acting on the upper half plane U such that Y = Γ Y \U. By Proposition 3.1, for these marked compact Riemann surfaces X and Y , there exists the unique harmonic diffeomorphism h : X → Y for the hyperbolic metric e ψ(u) |du| 2 on Y such that h maps the marking of X to the marking of Y . This also induces a harmonic map from Ω 1 to U, denoted by the same notation h, such that for a given γ ∈ Γ, there is a γ Now we define a metric e φ(z) |dz| 2 on Ω 1 Ω 2 by the pullback of the hyperbolic metric e ψ(u) |du| 2 on U L by h : Ω 1 → U and J −1 2 : Ω 2 → L respectively. More precisely we have e φ(z) = e ψ•h(z) |h z (z)| 2 for z ∈ Ω 1 ,

J. Park
Note that we take only the second part of the pullback metric given in (3.2) for z ∈ Ω 1 in the above definition of e φ(z) . By the definition, it follows that e φ(γ(z)) |γ z | 2 = e φ(z) for any γ ∈ Γ as in (2.3). Now the Liouville action for the harmonic diffeomorphism h is defined by and its modification is defined by The holomorphic energy of h is defined by From the definitions we have  [5].
Given a harmonic Beltrami differential µ ∈ B −1,1 (Γ), let f ε = f εµ be the unique quasiconformal map satisfying (2.1) with the Beltrami differential εµ. Notice that f ε varies holomorphically with respect to ε and thus It follows from the definition f ε z = εµf ε z thaṫ fz = µ.
Let us remark that the equation (3.1) for the harmonic map condition is the Euler-Lagrange equation for the holomorphic energy functional given by This can be checked easily as in the proof of Theorem 3.5. This will be also used crucially in the proof of the following theorem.
Remark 3.7. When X and Y are the same Riemann surface, the harmonic diffeomorphism h : X → Y is induced by J −1 1 : Ω 1 → U so that its Hopf differential Φ(h) vanishes. Hence, the variation formula (3.8) simplifies at the origin point X = Y in D(Γ, Ω 1 ) T(Γ 1 ). This may suggest that the second variation formula for S[h] would be simpler at the origin X = Y than other points in D(Γ, Ω 1 ). This is the case of the energy functional of harmonic diffeomorphisms whose second variation gives the Weil-Petersson symplectic 2-form at X = Y (see [11,Corollary 5.8] and [10, Theorem 3.1.3]).

Variation of Liouville action
In this section, we compute the variation of the Liouville action defined for any smooth conformal metric. Most of the computations are similar to the one given in [9], where a smooth family of conformal metrics is given by the hyperbolic metrics. However, we will have some additional terms since we do not assume the hyperbolic metric condition. On the other hand, we will also see that the variational argument developed in [9] works well for a smooth family of conformal metrics and these additional terms can be nicely organized. Now we decompose the Liouville action S = S[φ] into two parts by For a harmonic Beltrami differential µ ∈ B −1,1 (Γ), let f ε = f εµ : X → X ε denote the quasiconformal map satisfying the Beltrami equation (2.1).
Theorem 4.1. For a smooth family of conformal metrics e φ εµ (z ε ) |dz ε | 2 on X ε , Most of the remaining part of this section is a proof of Theorem 4.1. By definition, To deal with the first term on the right hand side of (4.1), we start with some lemmas.
Proof . The proof is just a straightforward computation as follows.
Remark 4.3. The term λ =φ + φ zḟ +ḟ z in Lemma 4.2 vanishes when the metrics e φ(z ε ) |dz ε | 2 are the hyperbolic metrics on X ε by the work of Ahlfors in [1]. By Lemma 4.2, the equality (4.1) can be rewritten as follows: where the third equality follows from ∂ F i = ∂ L i for i = 1, 2. To deal with terms in the last line of (4.3) together, let us put Proof . The second equality follows easily by To show the first equality dχ = 0, we start with some equalities. For the following equality we take derivative with respect to ε to obtaiṅ We also take derivative with respect to z and put ε = 0 for the equality (4.5) to get Similarly taking derivative with respect to ε for f εµ • γ = γ εµ • f εµ , we havė Using (4.6), (4.7), and (4.8), we observe that λ satisfies Hence, λ is Γ-invariant and this implies Recalling the definition of χ in (4.4) and using the equalities (4.2) and (4.9), This completes the proof.