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 ≥ 2. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fix...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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| Цитувати: | Liouville Action for Harmonic Diffeomorphisms. Jinsung Park. SIGMA 17 (2021), 097, 16 pages |
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| citation_txt | Liouville Action for Harmonic Diffeomorphisms. Jinsung Park. SIGMA 17 (2021), 097, 16 pages |
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| description | 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 ≥ 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 097, 16 pages
Liouville Action for Harmonic Diffeomorphisms
Jinsung PARK
School of Mathematics, Korea Institute for Advanced Study,
207-43, Hoegiro 85, Dong-daemun-gu, Seoul, 130-722, Korea
E-mail: jinsung@kias.re.kr
URL: http://newton.kias.re.kr/~jinsung/home.html
Received May 25, 2021, in final form October 27, 2021; Published online November 02, 2021
https://doi.org/10.3842/SIGMA.2021.097
Abstract. 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 ≥ 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.
Key words: quasi-Fuchsian group; Teichmüller space; Liouville action; harmonic diffeomor-
phism
2020 Mathematics Subject Classification: 14H60; 32G15; 53C43; 58E20
Dedicated to Professor Leon Takhtajan
on the occasion of his 70th birthday
1 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 homeo-
morphism 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 ob-
ject 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 Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
mailto:jinsung@kias.re.kr
http://newton.kias.re.kr/~jinsung/home.html
https://doi.org/10.3842/SIGMA.2021.097
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 J. Park
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 differ-
ential 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 func-
tional 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 Teich-
mı̈ller 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].
2 Liouville action for quasi-Fuchsian groups
Let us consider a compact Riemann surface X with genus g ≥ 2. Then the Riemann sur-
face 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
Liouville Action for Harmonic Diffeomorphisms 3
has a standard representation with 2g hyperbolic generators α1, β1, . . . , αg, βg satisfying the
relation
α1β1α
−1
1 β−1
1 · · ·αgβgα−1
g β−1
g = I,
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 Ω1 and Ω2
separated by a quasi-circle C. There exists a quasi-conformal homeomorphism J1 of Ĉ such that
QF1 J1 is holomorphic on U and J1(U) = Ω1, J1(L) = Ω2, J1(R) = C, where U and L are
respectively the upper and lower half planes.
QF2 Γ1 = J−1
1 ◦ Γ ◦ J1 is a marked, normalized Fuchsian group.
Since J1 is holomorphic on U, the Riemann surface X = Γ1\U is biholomorphic to the one given
by the quasi-Fuchsian uniformization Γ\Ω1. There is also a quasi-conformal homeomorphism J2
of Ĉ, holomorphic on L with a Fuchsian group Γ2 = J−1
2 ◦ Γ ◦ J2 so that Γ2\L has the quasi-
Fuchsian uniformization by Γ\Ω2.
Let A−1,1(Γ) be the space of Beltrami differentials for a quasi-Fuchsian group Γ, which is the
Banach space of µ ∈ L∞(C) satisfying
µ(γ(z))
γ′(z)
γ′(z)
= µ(z) for all γ ∈ Γ
and
µ
∣∣
C = 0.
Denote by B−1,1(Γ) the open unit ball in A−1,1(Γ) with respect to the ∥·∥∞ norm. For each Bel-
trami coefficient µ ∈ B−1,1(Γ), there exists a unique quasi-conformal map fµ : Ĉ → Ĉ satisfying
the Beltrami equation
fµz̄ = µfµz (2.1)
and fixing the points 0, 1 and ∞. Set Γµ = fµ ◦ Γ ◦ (fµ)−1 and define the deformation space of
quasi-Fuchsian group by
D(Γ) = B−1,1(Γ)/∼,
where µ ∼ ν if and only if fµ = fν on C. The space D(Γ) is a complex manifold of dimension
6g − 6. It is known that
D(Γ) ≃ T(Γ1)× T(Γ2),
where T(Γi) is the Teichmüller space of Γi for i = 1, 2. The deformation spaceD(Γ,Ω1) is defined
using the Beltrami coefficients supported on Ω1. By definition, the space D(Γ,Ω1) parametrizes
all deformations of X = Γ\Ω1 with the fixed Riemann surface Γ\Ω2 so that
D(Γ,Ω1) ≃ T(Γ1).
Hence, it is possible to use the deformation space D(Γ,Ω1) as the model of the Teichmüller
space T(Γ1). An advantage of this model is that one can use the holomorphic variation on
D(Γ,Ω1) 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.
4 J. Park
2.1 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 ∂ = ∂′ + (−1)p∂′′ on Kp,q.
There is a standard choice of the fundamental domain F ⊆ U for Γ as a non-Euclidean
polygon with 4g edges labeled by ak, a
′
k, b
′
k, bk; 1 ≤ k ≤ g satisfying αk(a
′
k) = ak, βk(b
′
k) = bk,
1 ≤ k ≤ g. The orientation of the edges is chosen such that
∂′F =
g∑
k=1
(ak + b′k − a′k − bk). (2.2)
Set ∂′ak = ak(1)− ak(0), ∂
′bk = bk(1)− bk(0), so that ak(0) = bk−1(0), 2 ≤ k ≤ g.
According to the isomorphism S• ≃ K•,0, the fundamental domain F is identified with F ⊗ [ ]
∈ K2,0. We have ∂′′F = 0, and it follows from (2.2) that
∂′F =
g∑
k=1
(
β−1
k (bk)− bk − α−1
k (ak) + ak
)
= ∂′′L,
where L ∈ K1,1 is given by
L =
g∑
k=1
(
bk ⊗ [βk]− ak ⊗ [αk]
)
.
There exists V ∈ K0,2 such that
∂′L = ∂′′V.
One can verify that it is given by
V =
g∑
k=1
(
ak(0)⊗ [αk|βk]− bk(0)⊗ [βk|αk] + bk(0)⊗
[
γ−1
k |αkβk
])
−
g−1∑
k=1
bg(0)⊗
[
γ−1
g · · · γ−1
k+1|γ
−1
k
]
,
where γk = [αk, βk] = αkβkα
−1
k β−1
k . Define
Σ = F + L− V.
Then
∂Σ = 0.
Finally, we also define W in the following way. Let Pk be a Γ-contracting path (see [9,
Definition 2.3] for the precise definition of Γ-contracting) connecting 0 to bk(0). Then
W =
g∑
k=1
(
Pk−1 ⊗ [αk|βk]− Pk ⊗ [βk|αk] + Pk ⊗ [γ−1
k |αkβk]
)
−
g−1∑
k=1
Pg ⊗
[
γ−1
g · · · γ−1
k+1|γ
−1
k
]
.
Liouville Action for Harmonic Diffeomorphisms 5
If Γ is a quasi-Fuchsian group, let Γ1 be the Fuchsian group such that Γ1 = J−1
1 ◦ Γ ◦ J1.
The double complex associated with Ω1 and the group Γ is a push-forward by the map J1 of the
double complex associated with U and the group Γ1. Define
Σ1 = F1 + L1 − V1,
where F1 = J1(F ), L1 = J1(L), V1 = J1(V ). Similarly we define
Σ2 = F2 + L2 − V2.
Here F2 = J1(F
′), L2 = J1(L
′), V2 = J1(V
′), where the corresponding chains F ′, L′, V ′ in L are
given by the complex conjugation of F , L, V respectively.
2.2 Cohomology construction
The corresponding double complex in cohomology C•,• is defined as Cp,q = HomC (Bq,A
p),
where A• is the complexified de Rham complex on Ω1. The associated total complex TotC is
equipped with the total differential D = d+(−1)pδ on Cp,q, where d is the de Rham differential
and δ is the group coboundary. The natural pairing ⟨ , ⟩ between Cp,q and Kp,q is given by the
integration over chains.
Denote by CM(Γ\Ω) the space of conformal metrics on Γ\Ω. That is, every ds2 ∈ CM(Γ\Ω)
is represented as ds2 = eϕ(z)|dz|2, where ϕ is a smooth function on Ω satisfying
ϕ ◦ γ + log |γ′|2 = ϕ ∀ γ ∈ Γ. (2.3)
The Liouville action is a function on the space of conformal metrics. Its construction is as
follows. Starting with the 2-form
ω[ϕ] =
(
|ϕz|2 + eϕ
)
dz ∧ dz̄ ∈ C2,0,
we have
δω[ϕ] = dθ̌[ϕ],
where θ̌[ϕ] ∈ C1,1 is given explicitly by
θ̌γ−1 [ϕ] =
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)(
γ′′
γ′
dz − γ′′
γ′
dz̄
)
.
Here c(γ) is the element c in the linear fractional transformation γ =
(
a b
c d
)
. Notice that
θ̌γ−1 [ϕ] = 0 if c(γ) = 0.
Next, set
ǔ = δθ̌[ϕ] ∈ C1,2.
From the definition of θ̌ and δ2 = 0, it follows that the 1-form ǔ is closed. An explicit calculation
gives
ǔγ−1
1 ,γ−1
2
=−
(
1
2
log |γ′1|2 + log
|c(γ2)|2
|c(γ2γ1)|2
)(
γ′′2
γ′2
◦ γ1 γ′1 dz −
γ′′2
γ′2
◦ γ1 γ′1 dz̄
)
+
(
1
2
log |γ′2 ◦ γ1|2 + log
|c(γ2γ1)|2
|c(γ1)|2
)(
γ′′1
γ′1
dz − γ′′1
γ′1
dz̄
)
.
6 J. Park
For eϕ(z)|dz|2 ∈ CM(Γ\Ω), the Liouville action is defined as
S[ϕ] =
i
2
(
⟨ω[ϕ], F1 − F2⟩ −
〈
θ̌[ϕ], L1 − L2
〉
+
〈
ǔ,W1 −W2
〉)
.
Here W1 = J1(W ) and W2 = J1(W
′) for the chain W ′ in L given by the complex conjugation
of W . We also define
Š[ϕ] =
i
2
(〈
ω̌[ϕ], F1 − F2
〉
−
〈
θ̌[ϕ], L1 − L2
〉
+ ⟨ǔ,W1 −W2⟩
)
, (2.4)
where
ω̌[ϕ] = |ϕz|2dz ∧ dz̄ ∈ C2,0.
Note that δω[ϕ] = δω̌[ϕ] since eϕ dz ∧ dz̄ is Γ-invariant.
3 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 eψ(u)|du|2 on Y , where u denotes a conformal coordinate on Y .
The harmonic condition for h is given by
hzz̄ +
(
ψu ◦ h
)
hzhz̄ = 0 (3.1)
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∗
(
eψ(u)|du|2
)
on X has the following
expression
h∗
(
eψ(u)|du|2
)
= eψ◦hhzh̄zdz
2 + eψ◦hhzh̄z̄|dz|2 + eψ◦hhz̄h̄z|dz|2 + eψ◦hhz̄h̄z̄dz̄
2. (3.2)
We denote the (2, 0)-component of h∗
(
eψ(u)|du|2
)
by
Φ(h) = eψ◦hhzh̄z,
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
eϕ := eψ◦hhzh̄z̄.
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 sur-
face 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, eϕ = eψ◦hhzh̄z̄
never vanishes on X, where eψ(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 ,
eϕ = eψ◦hhzh̄z̄ 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 ,
Kϕ := −2ϕzz̄e
−ϕ =
|hz̄|2
|hz|2
− 1. (3.3)
In particular, Kϕ never vanishes.
Liouville Action for Harmonic Diffeomorphisms 7
Proof. By the equality (3.1),
ϕz = (ψu ◦ h)hz +
hzz
hz
, (3.4)
and
ϕzz̄ = (ψuū ◦ h)
(
|hz|2 − |hz̄|2
)
=
1
2
eψ◦h
(
|hz|2 − |hz̄|2
)
. (3.5)
Here the second equality in (3.5) follows by the Liouville equation for eψ,
ψuū =
1
2
eψ. (3.6)
Hence we have
Kϕ = −2ϕzz̄e
−ϕ = −eψ◦h
(
|hz|2 − |hz̄|2
)
· e−ψ◦h|hz|−2 =
|hz̄|2
|hz|2
− 1.
For the harmonic diffeomorphism h : X → Y , its Jacobian Jh = |hz|2 − |hz̄|2 never vanishes.
Hence Kϕ never vanishes by the above equality. ■
Proposition 3.3. For a harmonic diffeomorphism h : X → Y for a hyperbolic metric on Y ,
eϕ satisfies the following equality:
ϕzz −
1
2
ϕ2z =
((
ψuu −
1
2
ψ2
u
)
◦ h
)
h2z +
1
2
Φ(h) + S(h) on U,
where z is a conformal coordinate on an open set U ⊂ X and S(h) = hzzz
hz
− 3
2
(
hzz
hz
)2
.
Proof. From (3.4), we have
ϕzz = (ψuu ◦ h)h2z + (ψuū ◦ h)hzh̄z + (ψu ◦ h)hzz +
hzzz
hz
−
(
hzz
hz
)2
. (3.7)
Then the claimed equality follows by (3.4), (3.6), and (3.7). ■
For two marked compact Riemann surfaces of genus g ≥ 2, there exists a marked, normalized
quasi-Fuchsian group Γ such that X = Γ\Ω1 and Y = Γ\Ω2 for a region of discontinuity
Ω1 ⊔ Ω2 by Bers’ simultaneous uniformization theorem. We also assume that Y is realized by
the Fuchsian uniformization with a Fuchsian group ΓY acting on the upper half plane U such
that Y = ΓY \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 eψ(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 Ω1 to U, denoted by the same notation h, such that for a given γ ∈ Γ, there is a γY ∈ ΓY
with
h ◦ γ = γY ◦ h.
Now we define a metric eϕ(z)|dz|2 on Ω1⊔Ω2 by the pullback of the hyperbolic metric eψ(u)|du|2
on U ⊔ L by h : Ω1 → U and J−1
2 : Ω2 → L respectively. More precisely we have
eϕ(z) =
{
eψ◦h(z)|hz(z)|2 for z ∈ Ω1,
eψ◦J
−1
2 (z)
∣∣(J−1
2
)
z
(z)
∣∣2 for z ∈ Ω2.
8 J. Park
Note that we take only the second part of the pullback metric given in (3.2) for z ∈ Ω1 in the
above definition of eϕ(z). By the definition, it follows that eϕ(γ(z))|γz|2 = eϕ(z) for any γ ∈ Γ as
in (2.3).
Now the Liouville action for the harmonic diffeomorphism h is defined by
S[h] =
i
2
(
⟨ω[ϕ], F1 − F2⟩ −
〈
θ̌[ϕ], L1 − L2
〉
+
〈
ǔ,W1 −W2
〉)
and its modification is defined by
Š[h] =
i
2
(〈
ω̌[ϕ], F1 − F2
〉
−
〈
θ̌[ϕ], L1 − L2
〉
+
〈
ǔ,W1 −W2
〉)
.
The holomorphic energy of h is defined by
E(h) =
∫
Γ\Ω1
eϕd2z =
∫
Γ\Ω1
eψ◦hhzh̄z̄ d
2z.
From the definitions we have
S[h] = Š[h] + E(h) + 2π(2g − 2).
Remark 3.4. In the above definition of the Liouville action for diffeomorphisms S[h], one may
use |Φ(h)| =
∣∣eψ◦h(z)hzh̄z∣∣ instead of eϕ = eψ◦h(z)hzh̄z̄. Since |Φ(h)| defines a singular flat metric
on X, the corresponding term ω̌[ϕ] defined by |Φ(h)| is singular where |Φ(h)| has a zero. To deal
with these singularities, we need to regularize the integral ⟨ω̌[ϕ], F1 − F2⟩ at the singular points
as in [5].
Given a harmonic Beltrami differential µ ∈ B−1,1(Γ), let f ε = f εµ be the unique quasi-
conformal map satisfying (2.1) with the Beltrami differential εµ. Notice that f ε varies holomor-
phically with respect to ε and thus
∂
∂ε̄
∣∣∣∣
ε=0
f ε = 0.
Let
ḟ =
∂
∂ε
∣∣∣∣
ε=0
f ε.
It follows from the definition f εz̄ = εµf εz that
ḟz̄ = µ.
For any linear fractional transformation γ, let
γεµ = f εµ ◦ γ ◦ (f εµ)−1.
Then γεµ varies holomorphically with respect to ε. The Lie derivative of the smooth family
of (l,m) tensors ωεµ on D(Γεµ) for Γεµ = f εµ ◦ Γ ◦ (f εµ)−1 is defined as
Lµω =
∂
∂ε
∣∣∣∣
ε=0
ωεµ ◦ f εµ(f εµz )l
(
f εµz
)m
.
For a harmonic diffeomorphism h : X → Y , we consider the situation of varying harmonic
diffeomorphisms along a variation of X with a fixed Y . For this purpose, we put B−1,1(Γ,Ω1)
to be the subspace of B−1,1(Γ) consisting of µ ∈ B−1,1(Γ) whose support lies in Ω1. Hence,
Liouville Action for Harmonic Diffeomorphisms 9
a Beltrami differential µ ∈ B−1,1(Γ,Ω1) represents an element in D(Γ,Ω1). Now we have the
following commuting diagram with f ε = f εµ for µ ∈ B−1,1(Γ,Ω1),
X
fε−−−−→ Xεyh yhε
Y
gε−−−−→ Y,
where Xε = f ε(X) and gε := hε ◦ f ε ◦ h−1 : Y → Y .
For the Lie derivative LµS[h] = LµŠ[h] + LµE(h), first we consider LµE(h).
Theorem 3.5. For a harmonic Beltrami differential µ ∈ B−1,1(Γ,Ω1),
LµE(h) = −
∫
Γ\Ω1
Φ(h)µd2z.
Proof. By the definition of the Lie derivative Lµ, we have the following equalities.
∂
∂ε
∣∣∣∣
ε=0
(
eψ◦h
ε◦fε(hεz ◦ f ε)
(
h̄εz̄ ◦ f ε
)
df ε ∧ df̄ ε
)
= eψ◦h
(
(ψu ◦ h)
(
ḣ+ hz ḟ
)
+ (ψū ◦ h)
( ˙̄h+ h̄z ḟ
))
hzh̄z̄ dz ∧ dz̄
+ eψ◦h
((
ḣz + hzz ḟ
)
h̄z̄ + hz
( ˙̄hz̄ + h̄z̄z ḟ
))
dz ∧ dz̄ + eψ◦hhzh̄z̄ ḟz dz ∧ dz̄.
Hence,
LµE(h) =
∫
Γ\Ω1
eψ◦h
(
(ψu ◦ h)hzh̄z̄
(
ḣ+ hz ḟ
)
+
(
ḣ+ hz ḟ
)
z
h̄z̄
)
d2z
+
∫
Γ\Ω1
eψ◦h
(
(ψū ◦ h)hzh̄z̄
( ˙̄h+ h̄z ḟ
)
+ hz
( ˙̄h+ h̄z ḟ
)
z̄
)
d2z
−
∫
Γ\Ω1
eψ◦hhzh̄z ḟz̄ d
2z.
By integration by parts, we have
LµE(h) =
∫
Γ\Ω1
eψ◦h
(
(ψu ◦ h)hzh̄z̄
(
ḣ+ hz ḟ
)
− (ψu ◦ h)hzh̄z̄
(
ḣ+ hz ḟ
))
d2z
−
∫
Γ\Ω1
eψ◦h
(
(ψū ◦ h)h̄zh̄z̄
(
ḣ+ hz ḟ
)
+ h̄zz̄
(
ḣ+ hz ḟ
))
d2z
+
∫
Γ\Ω1
eψ◦h
(
(ψū ◦ h)hzh̄z̄
( ˙̄h+ h̄z ḟ
)
− (ψū ◦ h)hzh̄z̄
( ˙̄h+ h̄z ḟ
))
d2z
−
∫
Γ\Ω1
eψ◦h
((
ψu ◦ h
)
hzhz̄
( ˙̄h+ h̄z ḟ
)
+ hzz̄
( ˙̄h+ h̄z ḟ
))
d2z
−
∫
Γ\Ω1
eψ◦hhzh̄z ḟz̄ d
2z = −
∫
Γ\Ω1
eψ◦hhzh̄z̄ ḟz̄ d
2z.
Here the last equality holds by the equality (3.1). ■
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
E(gε ◦ h) = E(hε ◦ f ε) =
∫
Γ\Ω1
eψ◦h
ε◦fε |(hε ◦ f ε)z|2 d2z.
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.
10 J. Park
Theorem 3.6. For a harmonic Beltrami differential µ ∈ B−1,1(Γ,Ω1),
LµS[h] =
∫
Γ\Ω1
(
2S(h)−KϕΦ(h)
)
µd2z. (3.8)
Proof. Putting λ := ϕ̇+ ϕz ḟ + ḟz with ϕ̇ = d
dε
∣∣
ε=0
ϕεµ, by Theorems 3.5 and 4.1,
LµS[h] = LµŠ[h] + LµE(h) =
∫
Γ\Ω
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
d2z −
∫
Γ\Ω1
Φ(h)µd2z.
For µ ∈ B−1,1(Γ,Ω1), µ vanishes over Ω2. Moreover the term λ vanishes over Ω2 since the metric
eϕ(z)|dz|2 is hyperbolic on Ω2. Hence, by Proposition 3.3 we have
LµS[h] =
∫
Γ\Ω1
(
(2ϕzz − ϕ2z)µ− 2ϕzz̄λ
)
d2z −
∫
Γ\Ω1
Φ(h)µd2z
=
∫
Γ\Ω1
((
2
((
ψuu −
1
2
ψ2
u
)
◦ h
)
h2z +Φ(h) + 2S(h)
)
µ− 2ϕzz̄λ
)
d2z
−
∫
Γ\Ω1
Φ(h)µd2z =
∫
Γ\Ω1
(
2S(h)µ− 2ϕzz̄λ
)
d2z.
Here, for the last equality, we used the fact that ψuu− 1
2ψ
2
u ≡ 0 on U for eψ = (Im(u))−2, where u
denotes the global coordinate on U. Now we analyze the term λ = ϕ̇+ϕz ḟ+ ḟz as follows. First,
by definition,
ϕ̇+ ϕz ḟ + ḟz = ψu ◦ h
(
ḣ+ hz ḟ
)
+ ψū ◦ h
( ˙̄h+ h̄z ḟ
)
+
ḣz+ hzz ḟ+ hz ḟz
hz
+
˙̄hz̄+ h̄z̄z ḟ
h̄z̄
. (3.9)
On the other hand, recalling that the harmonic diffeomorphism h : X → Y is a critical point of
the holomorphic energy functional along the variation gε ◦ h,
0 =
∂
∂ε
∣∣∣∣
ε=0
(
eψ◦g
ε◦h|(gε ◦ h)z|2
)
=
∂
∂ε
∣∣∣∣
ε=0
(
eψ◦h
ε◦fε |(hε ◦ f ε)z|2
)
,
so that
0 = eψ◦h|hz|2
(
ψu ◦ h
(
ḣ+ hz ḟ
)
+ ψū ◦ h
( ˙̄h+ h̄z ḟ
)
+
ḣz + hzz ḟ + hz ḟz
hz
+
˙̄hz̄ + h̄z̄z ḟ + h̄z ḟz̄
h̄z̄
)
. (3.10)
Hence, by (3.9) and (3.10) we have
ϕ̇+ ϕz ḟ + ḟz = − h̄z ḟz̄
h̄z̄
. (3.11)
Finally, by (3.3), (3.5), and (3.11),
−2ϕzz̄λ = eψ◦h
(
|hz|2 − |hz̄|2
) h̄z
h̄z̄
µ = eψ◦h
(
hzh̄z − hzh̄z
|hz̄|2
|hz|2
)
µ = −KϕΦ(h)µ.
This completes the proof. ■
Remark 3.7. When X and Y are the same Riemann surface, the harmonic diffeomorphism
h : X → Y is induced by J−1
1 : Ω1 → 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,Ω1) ≃ T(Γ1). This may sug-
gest that the second variation formula for S[h] would be simpler at the origin X = Y than other
points in D(Γ,Ω1). 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]).
Liouville Action for Harmonic Diffeomorphisms 11
4 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
S[ϕ] = Š[ϕ] +
∫
Γ\Ω
eϕd2z,
where Š[ϕ] is defined in (2.4). First we deal with the variation of Š = Š[ϕ].
For a harmonic Beltrami differential µ ∈ B−1,1(Γ), let f ε = f εµ : X → Xε denote the quasi-
conformal map satisfying the Beltrami equation (2.1).
Theorem 4.1. For a smooth family of conformal metrics eϕ
εµ(zε)|dzε|2 on Xε,
LµŠ[ϕ] =
∫
Γ\Ω
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
d2z,
where λ = ϕ̇+ ϕz ḟ + ḟz with ϕ̇ = d
dε |ε=0ϕ
εµ.
Most of the remaining part of this section is a proof of Theorem 4.1. By definition,
LµŠ[ϕ] =
i
2
(
⟨Lµω̌, F1 − F2⟩ −
〈
Lµθ̌, L1 − L2
〉
+ ⟨Lµǔ,W1 −W2⟩
)
. (4.1)
To deal with the first term on the right hand side of (4.1), we start with some lemmas.
Lemma 4.2. The following equality holds
Lµω̌ =
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄ − d(ϕzλ dz) + d(ϕz̄λ dz̄)− dξ, (4.2)
where λ = ϕ̇+ ϕz ḟ + ḟz, ξ = 2ϕz ḟz̄dz̄ − ϕ dḟz.
Proof. The proof is just a straightforward computation as follows.
Lµω̌ =
∂
∂ε
∣∣∣∣
ε=0
(
(ϕεµ)z ◦ f εµdf εµ ∧ (ϕεµ)z̄ ◦ f εµdf̄ εµ
)
=
(
ϕ̇z + ϕzz ḟ + ϕz ḟz
)
ϕz̄ dz ∧ dz̄ + ϕz
(
ϕ̇z̄ + ϕzz̄ ḟ
)
dz ∧ dz̄
=
(
ϕ̇+ ϕz ḟ + ḟz
)
z
ϕz̄ dz ∧ dz̄ + ϕz
(
ϕ̇+ ϕz ḟ + ḟz
)
z̄
dz ∧ dz̄
−
(
ϕz̄ ḟzz + ϕz
(
ϕz ḟz̄ + ḟzz̄
))
dz ∧ dz̄
=
(
2ϕzz − ϕ2z
)
µdz ∧ dz̄ − d
(
2ϕz ḟz̄dz̄ − ϕ dḟz
)
− d(ϕzλ dz) + d(ϕz̄λ dz̄)
− 2ϕzz̄λ dz ∧ dz̄. ■
Remark 4.3. The term λ = ϕ̇+ ϕz ḟ + ḟz in Lemma 4.2 vanishes when the metrics eϕ(z
ε)|dzε|2
are the hyperbolic metrics on Xε by the work of Ahlfors in [1].
12 J. Park
By Lemma 4.2, the equality (4.1) can be rewritten as follows:
LµŠ[ϕ] =
i
2
(
⟨Lµω̌, F1 − F2⟩ −
〈
Lµθ̌, L1 − L2
〉
+ ⟨Lµǔ,W1 −W2⟩
)
=
i
2
(〈((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄, F1 − F2
〉
−⟨d(ϕzλ dz)− d(ϕz̄λ dz̄) + dξ, F1− F2⟩− ⟨Lµθ̌, L1 − L2⟩+ ⟨Lµǔ,W1−W2⟩
)
=
i
2
(〈((
2ϕzz− ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄, F1− F2
〉
− ⟨δ(ϕzλ dz− ϕz̄λ dz̄ + ξ), L1− L2⟩
− ⟨Lµθ̌, L1 − L2⟩+ ⟨Lµǔ,W1 −W2⟩
)
, (4.3)
where the third equality follows from ∂′Fi = ∂′′Li for i = 1, 2. To deal with terms in the last
line of (4.3) together, let us put
χ := δξ + δ(ϕzλ dz − ϕz̄λdz̄) + Lµθ̌. (4.4)
First we have
Lemma 4.4. The χ satisfies that dχ = 0 and δχ = Lµǔ on Ω.
Proof. The second equality follows easily by
δχ = δ
(
δξ + δ(ϕzλ dz − ϕz̄λ dz̄) + Lµθ̌
)
= δLµθ̌ = Lµδθ̌ = Lµǔ.
To show the first equality dχ = 0, we start with some equalities. For the following equality
ϕεµ ◦ γεµ + log(γεµ)′ + log(γ̄εµ)′ = ϕεµ, (4.5)
we take derivative with respect to ε to obtain
ϕ̇ ◦ γ + ϕz ◦ γγ̇ +
γ̇′
γ′
= ϕ̇. (4.6)
We also take derivative with respect to z and put ε = 0 for the equality (4.5) to get
ϕz ◦ γγ′ +
γ′′
γ′
= ϕz. (4.7)
Similarly taking derivative with respect to ε for f εµ ◦ γ = γεµ ◦ f εµ, we have
ḟ ◦ γ = γ̇ + γ′ḟ . (4.8)
Using (4.6), (4.7), and (4.8), we observe that λ satisfies
λ ◦ γ = ϕ̇ ◦ γ + ϕz ◦ γ · ḟ ◦ γ + ḟz ◦ γ
= ϕ̇−
(
ϕz −
γ′′
γ′
)
γ̇
γ′
− γ̇′
γ′
+
ϕz − γ′′
γ′
γ′
(γ̇ + γ′ḟ) +
(
ḟz +
γ̇′
γ′
+
γ′′
γ′
ḟ
)
= ϕ̇+ ϕz ḟ + ḟz = λ.
Hence, λ is Γ-invariant and this implies((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
◦ γ|γ′|2 dz ∧ dz̄ =
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄. (4.9)
Recalling the definition of χ in (4.4) and using the equalities (4.2) and (4.9),
dχ = δ
(
dξ + d(ϕzλ dz − ϕz̄λ dz̄)
)
+ Lµdθ̌
= δ
(
−Lµω̌ +
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄
)
+ Lµdθ̌ = −δLµω̌ + Lµδω̌ = 0.
This completes the proof. ■
Liouville Action for Harmonic Diffeomorphisms 13
Lemma 4.5.
δξγ−1 = −2
(
ḟzz̄ ◦ γγ′ − ḟzz̄
)
dz̄ − ϕ d
(
ḟz ◦ γ − ḟz
)
+ log |γ′|2d
(
ḟz ◦ γ
)
.
Proof. From the equality (4.8),
ḟzz̄ ◦ γγ′(γ′) = γ′′ḟz̄ + γ′ḟzz̄. (4.10)
Then, from ξ = 2ϕz ḟz̄dz̄ − ϕ dḟz, using (4.10) we have
δξγ−1 =
(
2ϕz ḟz̄
)
◦ γγ′dz̄ −
(
ϕ dḟz
)
◦ γ − 2ϕz ḟz̄ dz̄ + ϕ dḟz
= 2
(
ϕz −
γ′′
γ′
)
ḟz̄ dz̄ −
(
ϕ− log |γ′|2
)
dḟz ◦ γ − 2ϕz ḟz̄ dz̄ + ϕ dḟz
= −2
(
ḟzz̄ ◦ γγ′ − ḟzz̄
)
dz̄ − ϕ
(
dḟz ◦ γ − dḟz
)
+ log |γ′|2dḟz ◦ γ.
This completes the proof since
(
dḟz
)
◦ γ = d
(
ḟz ◦ γ
)
. ■
Now, for the term Lµθ̌, we have
Lemma 4.6.
Lµθ̌γ−1 =
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)
d
(
ḟz ◦ γ − ḟz
)
− 2
(
ḟzz ◦ γγ′ − ḟzz − λ
γ′′
γ′
)
dz +
(
1
2
(
ḟz ◦ γ + ḟz
)
+
ċ(γ)
c(γ)
− λ
)
d log |γ′|2.
Proof. Recall
θ̌γ−1 [ϕ] =
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)(
γ′′
γ′
dz − γ′′
γ′
dz̄
)
.
For this, we observe the following equalities:
∂
∂ε
∣∣∣∣
ε=0
(
log
∣∣(γεµ)′ ◦ f εµ∣∣2) = γ̇′
γ′
+
γ′′
γ′
ḟ = ḟz ◦ γ − ḟz,
∂
∂ε
∣∣∣∣
ε=0
(
(γεµ)′′
(γεµ)′
◦ f εµ
)
= ḟzz ◦ γγ′ − ḟzz −
γ′′
γ′
ḟz.
For c(γ), we also have
−2c(γ) =
γ′′(z)
(γ′(z))
3
2
.
Then
ċ(γ)
c(γ)
=
ḟzz ◦ γγ′ − ḟzz − γ′′
γ′ ḟz
γ′′
γ′
− 1
2
(
ḟz ◦ γ − ḟz
)
=
ḟzz ◦ γγ′ − ḟzz
γ′′
γ′
− 1
2
(
ḟz ◦ γ + ḟz
)
. (4.11)
Hence,
Lµ
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)
= λ− ḟz −
1
2
(
ḟz ◦ γ − ḟz
)
− ċ(γ)
c(γ)
= λ− 1
2
(
ḟz ◦ γ + ḟz
)
− ċ(γ)
c(γ)
= λ− ḟzz ◦ γγ′− ḟzz
γ′′
γ′
.
14 J. Park
Moreover,
Lµ
(
γ′′
γ′
dz − γ′′
γ′
dz̄
)
= Lµd log |γ′|2 = d
(
ḟz ◦ γ − ḟz
)
.
Hence,
Lµθ̌γ−1 =
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)
d
(
ḟz ◦ γ − ḟz
)
−
(
1
2
(
ḟz ◦ γ + ḟz
)
+
ċ(γ)
c(γ)
− λ
)(
2
γ′′
γ′
dz − d log |γ′|2
)
=
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)
d
(
ḟz ◦ γ − ḟz
)
− 2
(
ḟzz ◦ γγ′ − ḟzz − λ
γ′′
γ′
)
dz +
(
1
2
(
ḟz ◦ γ + ḟz
)
+
ċ(γ)
c(γ)
− λ
)
d log |γ′|2.
This completes the proof. ■
Proposition 4.7. For χγ−1, there is an exact form lγ−1 such that χγ−1 = dlγ−1, where
lγ−1 =
1
2
log |γ′|2
(
ḟz ◦ γ + ḟz + 2
ċ(γ)
c(γ)
)
−
(
log |c(γ)|2 + 2 + 2 log 2
)(
ḟz ◦ γ − ḟz
)
.
Proof. Recall
χ = δξ + δ(ϕzλ dz − ϕz̄λ dz̄) + Lµθ̌.
Then, by Lemmas 4.5 and 4.6, we have
χγ−1 = − 2
(
ḟzz̄ ◦ γγ′ − ḟzz̄
)
dz̄ − ϕd
(
ḟz ◦ γ − ḟz
)
+ log |γ′|2d
(
ḟz ◦ γ
)
+
(
ϕ− 1
2
log |γ′|2 − 2 log 2− log |c(γ)|2
)
d
(
ḟz ◦ γ − ḟz
)
− 2
(
ḟzz ◦ γγ′ − ḟzz − λ
γ′′
γ′
)
dz +
(
1
2
(
ḟz ◦ γ + ḟz
)
+
ċ(γ)
c(γ)
− λ
)
d log |γ′|2
− λ
(
γ′′
γ′
dz − γ′′
γ′
dz̄
)
.
On the right hand side of the above equality, the terms involving λ cancel each other and the
terms involving ϕ also cancel each other. Now let us rewrite χγ−1 changing the order of terms
as follows.
χγ−1 =
1
2
d log |γ′|2
(
ḟz ◦ γ + ḟz
)
+
1
2
log |γ′|2d
(
ḟz ◦ γ + ḟz
)
−
(
log |c(γ)|2 + 2 log 2
)
d
(
ḟz ◦ γ − ḟz
)
− 2
(
ḟzz̄ ◦ γγ′ − ḟzz̄
)
dz̄ − 2
(
ḟzz ◦ γγ′ − ḟzzd
)
z +
ċ(γ)
c(γ)
d log |γ′|2
=
1
2
d log |γ′|2
(
ḟz ◦ γ + ḟz
)
+
1
2
log |γ′|2d
(
ḟz ◦ γ + ḟz
)
−
(
log |c(γ)|2 + 2 log 2
)
d
(
ḟz ◦ γ − ḟz
)
− 2ḟz̄
γ′′
γ′
dz̄ −
(
ḟz ◦ γ + ḟz
)γ′′
γ′
dz − ċ(γ)
c(γ)
(
γ′′
γ′
dz − γ′′
γ′
dz̄
)
,
Liouville Action for Harmonic Diffeomorphisms 15
where we used the equalities (4.10) and (4.11). Finally we can check that the exact form lγ−1
satisfying χγ−1 = dlγ−1 is given by
lγ−1 =
1
2
log |γ′|2
(
ḟz ◦ γ + ḟz + 2
ċ(γ)
c(γ)
)
−
(
log |c(γ)|2 + 2 + 2 log 2
)(
ḟz ◦ γ − ḟz
)
.
For this, we use the following equality
γ̇′ + γ′′ḟ = ḟz ◦ γγ′ − γ′ḟz,
which follows from γεµ ◦ f εµ = f εµ ◦ γ. ■
By Proposition 4.7, we have〈
δξ + δ(ϕzλ dz − ϕz̄λdz̄) + Lµθ̌, L1 − L2⟩ = ⟨dl, L1 − L2
〉
= ⟨l, ∂′L1 − ∂′L2⟩. (4.12)
Since
Lµǔ = Lµδθ̌ = δLµθ̌ = δχ = δdl = dδl,
we have
⟨Lµǔ,W1 −W2⟩ = ⟨δl, ∂′W1 − ∂′W2⟩ = ⟨δl, V1 − V2⟩ = ⟨l, ∂′′V1 − ∂′′V2⟩. (4.13)
From (4.3), (4.12), and (4.13), it follows that
LµŠ[ϕ] =
i
2
(〈((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄, F1 − F2
〉
− ⟨δ(ϕzλ dz − ϕz̄λdz̄ − ξ), L1 − L2⟩ − ⟨Lµθ̌, L1 − L2⟩+ ⟨Lµǔ,W1 −W2⟩
)
=
i
2
(〈((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄, F1 − F2
〉
− ⟨l, ∂′L1 − ∂′L2 − ∂′′V1 + ∂′′V2⟩
)
=
i
2
〈((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
dz ∧ dz̄, F1 − F2
〉
.
This completes the proof of Theorem 4.1.
Theorem 4.8. For a smooth family of conformal metrics eϕ
εµ(zε)|dzε|2 on Xε,
LµS[ϕ] =
∫
Γ\Ω
((
2ϕzz − ϕ2z
)
µ+ (1 +Kϕ)e
ϕλ
)
d2z,
where λ = ϕ̇+ ϕz ḟ + ḟz with ϕ̇ = d
dε |ε=0ϕ
εµ and Kϕ = −2ϕzz̄e
−ϕ.
Proof. We proved the formula for LµŠ in Theorem 4.1. For the remaining part, it is easy to see
∂
∂ε
∣∣∣∣
ε=0
(
eϕ
εµ◦fεµ i
2
df εµ ∧ df̄ εµ
)
= eϕλ d2z.
Hence,
LµS[ϕ] =
∫
Γ\Ω
((
2ϕzz − ϕ2z
)
µ− 2ϕzz̄λ
)
d2z +
∫
Γ\Ω
eϕλ d2z
=
∫
Γ\Ω
((
2ϕzz − ϕ2z
)
µ+
(
1− 2ϕzz̄e
−ϕ)eϕλ) d2z.
This completes the proof. ■
16 J. Park
Acknowledgements
This work was partially supported by Samsung Science and Technology Foundation under
Project Number SSTF-BA1701-02. The author thank referees for their helpful comments and
suggestions which improve the exposition of the paper.
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1 Introduction
2 Liouville action for quasi-Fuchsian groups
2.1 Homology construction
2.2 Cohomology construction
3 Liouville action for harmonic diffeomorphisms
4 Variation of Liouville action
References
|
| id | nasplib_isofts_kiev_ua-123456789-211430 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T10:46:37Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Park, Jinsung 2026-01-02T08:31:31Z 2021 Liouville Action for Harmonic Diffeomorphisms. Jinsung Park. SIGMA 17 (2021), 097, 16 pages 1815-0659 2020 Mathematics Subject Classification: 14H60; 32G15; 53C43; 58E20 arXiv:2105.11074 https://nasplib.isofts.kiev.ua/handle/123456789/211430 https://doi.org/10.3842/SIGMA.2021.097 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 ≥ 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. This work was partially supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-02. The author thanks referees for their helpful comments and suggestions, which improve the exposition of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Liouville Action for Harmonic Diffeomorphisms Article published earlier |
| spellingShingle | Liouville Action for Harmonic Diffeomorphisms Park, Jinsung |
| title | Liouville Action for Harmonic Diffeomorphisms |
| title_full | Liouville Action for Harmonic Diffeomorphisms |
| title_fullStr | Liouville Action for Harmonic Diffeomorphisms |
| title_full_unstemmed | Liouville Action for Harmonic Diffeomorphisms |
| title_short | Liouville Action for Harmonic Diffeomorphisms |
| title_sort | liouville action for harmonic diffeomorphisms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211430 |
| work_keys_str_mv | AT parkjinsung liouvilleactionforharmonicdiffeomorphisms |