The Loewner Energy via the Renormalised Energy of Moving Frames

We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a Kähler potential for the essentially unique Kähler metric on the Weil–Petersson universal Teichmüller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.


Background on Weil-Petersson quasicircles
In [40,30], the second author and Steffen Rohde introduced the Möbius-invariant Loewner energy to measure the roundness of Jordan curves on the Riemann sphere C∪{∞} using the Loewner transform [25].The original motivation comes from the probabilistic theory of Schramm-Loewner evolutions, see, e.g., [42] for an overview.It was proved in [41] that the Loewner energy is proportional to the universal Liouville action introduced by Takhtajan and Teo [14].In particular, the class of finite energy curves corresponds exactly to the Weil-Petersson class of quasicircles which has already been studied extensively by both physicists and mathematicians since the eighties, see, e.g., [8,43,28,27,31,11,14,32,15,33,16,4,37,38,20], and is still an active research area.See the introduction of [4] (see also the companion papers [5] and [6] for more on this topic) for a summary and a list of equivalent definitions of very different nature.
In this article, we sometimes view Jordan curves as curves on S 2 ⊂ R 3 and give new characterisations of the Loewner energy in terms of the moving frames on S 2 .Note that in this article, S 2 refers to the sphere of radius 1 centred at the origin in R 3 equipped with the induced round metric g 0 from its embedding into R 3 .Therefore, S 2 is isometric to C = C ∪ {∞} endowed with the metric by the stereographic projection.To distinguish the two setups, we will let γ denote a Jordan curve in C and let Γ denote a Jordan curve in S 2 .Let us first list a few equivalent definitions of Weil-Petersson quasicircles that are relevant to this work.Theorem 1.1 (Cui,[11], Tahktajan-Teo, [14], Shen, [33], Bishop,[4]).Let γ ⊂ C be a Jordan curve, Ω be the bounded connected component of C \ γ, and let f : D → Ω and g : C \ D → C \ Ω be biholomorphic maps such that g(∞) = ∞.The following conditions are equivalent: (1) There exists a quasiconformal extension of g to C such that the Beltrami coefficient ( ( (4) The (conformal) welding function ϕ = g −1 • f | S 1 satisfies log ϕ belongs to the Sobolev space H 1/2 (S 1 ).( 5) The curve γ is chord-arc and the unit tangent τ : γ → S 1 belongs to H 1/2 (γ).( 6) Every minimal surface Σ ⊂ H 3 C × R * + with asymptotic boundary γ has finite renormalised area, i.e., where for all ε > 0, Σ ε = Σ ∩ {(z, t) : t > ε} and ∂Σ ε = Σ ∩ {(z, t) : t = ε}.
If γ satisfies any of those conditions, γ is called a Weil-Petersson quasicircle.
The equivalences (1) and ( 2) are due to Cui, and independently to Takhtajan and Teo who proved the equivalences (1), (2), (3).In (4), the continuous extension of f, g to S 1 is well-defined by a classical theorem of Carathéodory [9].The equivalence between (1) and ( 4) is proved by Shen.The second condition is perhaps the simplest one since it corresponds to the condition log |f | ∈ W 1,2 (D), the Sobolev space of functions with squared-integrable weak derivatives.
For (5), we recall that a Jordan curve is chord-arc if there exists K < ∞ such that for all x, y ∈ γ, we have (x, y) ≤ K|x − y|, where (x, y) is the length of the shortest arc joining x to y.We mention that Weil-Petersson quasicircles are not only chord-arc but even asymptotically smooth, namely, the ratio (x, y)/|x − y| tends to 1 as x tends to y.These curves are not necessarily C 1 for they allow certain types of infinite spirals.See Section 5.1 for an explicit construction of such spirals.Recall that for any Jordan chord-arc curve γ, a function u : γ → C belongs to the Sobolev space H 1/2 (γ) if and only if where |dz| is the arc-length measure.
The equivalence between (1) and ( 5) was proven by Y. Shen and L. Wu ( [36]; see also [18,33,34,35]), and also by Christopher Bishop [4].The last characterisation (6) due to Bishop [4] using the notion of renormalised area was first investigated for Willmore surfaces by S. Alexakis and R. Mazzeo ( [1], [2]) which has strong motivations arising from string theory [17].The integral of the squared trace-free second fundamental form Å in (6) is the Willmore energy of Σ which is of particular interest for being conformally invariant.Amongst the important previous contribution that inspired this work, we should mention Epstein's work ( [12], [13]).
Not only we can characterize this class of curves qualitatively, as listed above, there is an important quantity associated with each element of the class.Indeed, after appropriate normalisation, the class of Weil-Petersson quasicircles can be identified with the Weil-Petersson universal Teichmüller space T 0 (1) via conformal welding.Takhtajan and Teo [14] showed that T 0 (1) carries an essentially unique homogeneous Kähler metric and introduced the universal Liouville action S 1 .They showed that S 1 is a Kähler potential on T 0 (1) which is of critical importance for the Kähler geometry.We take an analytic instead of a Teichmüller theoretic viewpoint, so we will consider S 1 as defined for Weil-Petersson quasicircles instead of their welding functions.Explicitly, for a Weil-Petersson quasicircle γ, (1.3) Theorem 1.2 (Y.Wang, [41]).A Jordan curve γ has finite Loewner energy I L (γ) if and only if γ is a Weil-Petersson quasicircle.Furthermore, we have (1.4) We will therefore use interchangeably the terms "Jordan curve of finite Loewner energy," "Weil-Petersson quasicircle," or simply "Weil-Petersson curve."As we did not define explicitly the Loewner energy I L (γ), readers may consider (1.4) as its definition.It may not be obvious from the expression of S 1 that it is invariant under Möbius transformations, such as the inversion i : z → 1/z.However, it would follow directly from the definition using Loewner transform in [30].Provided that γ separates 0 from ∞, we may choose the biholomorphic functions f and g as in Theorem 1.1 and assume further that f (0) = 0. Applying the invariance of the Loewner energy under i, we get

Moving Frames and the Ginzburg-Landau Equations
Moving frames, first introduced by Darboux in the late 19th century to study curves and surfaces, were later generalised by Élie Cartan and permit one to reformulate astutely a wide class of differentialgeometric problems.One of the rather recent such use of this theory is found in the work of Frédéric Hélein on harmonic maps ( [19]), where the moving frame pave the way towards new regularity results.
In [23], Paul Laurain and Romain Petrides suggest a new approach to relate the Loewner energy to the renormalised energy of moving frames using the Ginzburg-Landau energy in a minimal regularity setting (which is of independent interest).Although the Ginzburg-Landau is normally used to construct harmonic maps with values into S 1 under topological constraints where no smooth solutions exist ( [3]), it should be seen-although we will not use this functional here-more generally as a way to construct (singular) moving frames on surfaces.Through this approach, one may hope to link quantatively the Loewner energy and the Willmore energy that can also be written in terms of moving frames ( [26]).
Let Ω ⊂ C be a simply connected domain, and γ = ∂Ω.In [23], they show that the Bethuel-Brezis-Hélein ([3]) analysis carries on for general chorc-arc curves and H 1/2 boundary data.Using this delicate analysis, they obtain the following result, which is the most relevant one in this article.
The other main result of [23] is to identify the renormalised energy in the sense of Bethuel-Brezis-Hélein as an explicit term involving (1.7).

Remark 1.4. The harmonic function µ is explicitly given by
The last identities follow from the conformality of f .We note that in [23], the point p is a special point such that any biholomorphic map f with f (0) = p maximizes |f (0)| amongst all biholomorphic maps D → Ω.
We see that the frame energy (1.7) coincides with the first term in (1.3).To obtain the second half of the Loewner energy involving we cannot easily use the Ginzburg-Landau equation to construct the moving frames since that would force us to work on the non-compact domain C \ Ω.Using the inversion i will not suffice either.If we choose the biholomorphic map g : which is in general different from (1.8).To overcome this technicality, we work directly on S 2 to obtain a formula of the Loewner energy in terms of moving frames.

Theorem
Then, for any p j ∈ Ω j , there exists harmonic moving frames ( u j , v j ) : Ω j \ {p j } → U Ω j × U Ω j such that the Cartan form ω j = u j , d v j admits the decomposition where G Ωj : Ω j \ {p j } → R is the Green's function of the Laplacian ∆ g0 on Ω j with Dirichlet boundary condition, and where k g0 is the geodesic curvature on Γ = ∂Ω j .Define the functional E (that we call the renormalised energy associated to the frames ( u 1 , v 1 ) and ( u 2 , v 2 )) by Then there exists conformal maps (1.12)

D
Figure 1: Harmonic moving frames on the sphere associated to a Weil-Petersson quasicircle.
Remark 1.10.(1) In the theorem above, we wrote U Ω j (j = 1, 2) for the unit tangent bundle.The function µ j , explicitly given by correspond to the conformal parameter of the conformal maps (2) The constant term 4π in the definition of E is arranged so that E (S 1 ) = 0 (see Remark 2.8).Furthermore, the name renormalised energy is justified by the following identity where no constant term is involved.
(3) The solution (1.10) to the Dirichlet problem is unique, and so are the moving frames once the singularities (p 1 , p 2 ) ∈ Ω 1 × Ω 2 are fixed.See Theorem 3.2 and 3.5.Notice that the geodesic curvature is understood in the distributional sense here (see Section 5.1 from the appendix for more details).
This theorem corresponds to Theorem 2.6 in the article for smooth curves and to Theorem 4.5 for general Weil-Petersson quasicircles.The general case follows essentially from the following result which can also be viewed as a restatement of Theorem A without any mention of moving frames.
Theorem B (See Theorem 4.5).Let Γ ⊂ S 2 be a Weil-Petersson quasicircle and Ω 1 , Ω 2 ⊂ S 2 \ Γ be the two connected components of S 2 \ Γ.For all conformal maps f 1 : D → Ω 1 and f 2 : D → Ω 2 , we have Acknowledgements.This paper is part of a common project between Paul Laurain and Romain Petrides and the two authors on the various characterisations of Weil-Petersson quasicircles.We thank Paul Laurain and Romain Petrides for useful discussions and kindly sharing their manuscript with us.We also thank Christopher Bishop for allowing us into his topic class which helped us understand his recent work [4]. A. M. is supported by the Early Postdoc.Mobility Variational Methods in Geometric Analysis P2EZP2_191893.Y. W. is partially supported by NSF grant DMS-1953945.

Moving Frame Energy via Zeta-Regularised Determinants for Smooth Curves
The following expression of the Loewner energy will prove crucial in this section.
We now use the formula (2.1) expressing the Loewner energy in terms of zeta-regularised determinants to link the Loewner energy to the renormalised energy of moving frames on S 2 .First, let g 0 = g S 2 be the standard round metric on S 2 .Let Γ ⊂ S 2 be a simple smooth * curve, and let Ω 1 , Ω 2 ⊂ S 2 the two disjoint open connected components of S 2 \ Γ.Since we are working on a curved manifold, we cannot directly use the result of [23] to construct moving frames with the Ginzburg-Landau method.However, we will construct them directly in Section 3 (see Theorem 3.2).Therefore, let us assume that ( u 1 , v 1 ) : Ω 1 \ {p 1 } → U S 2 × U S 2 are harmonic vector fields such that u 1 = τ on ∂Ω 1 = Γ (where τ is the unit tangent on Γ), and the 1- where G Ω1 : Ω 1 \{p 1 } → R is the Green's function for the Laplacian on Ω 1 \{p 1 } with Dirichlet boundary condition.Namely, G Ω1 satisfies and where k g0 is the geodesic curvature with respect to the round metric g 0 , and the normal derivative is taken with respect to the g 0 .
To fix notations, we recall the following result.
Remark 2.3.The existence of a Green's function follows from its conformal invariance and the uniformisation theorem.Indeed, if Ω is a Jordan domain, and f : D → Ω is a biholomorphic map such that f (0) = p, and We assume that ∂Ω is chord-arc so that the trace theorems apply as in [22,39].The passage from C to S 2 is easy using a stereographic projection and the conformal invariance of Green's functions.Now, following Proposition 5.1 of [23], it is not hard to see that their proof using the Froebenius theorem also works for domains of the sphere, and we get a conformal diffeomorphism ϕ : (−∞, 0) × ∂B(0, ρ) → Ω 1 \ {p 1 } for some ρ > 0 such that Notice that the Proposition 5.1 of [23] gives a privileged p 1 ∈ Ω 1 , but we will show in Theorem 3.2 that p 1 can be taken arbitrarily (see also Theorem 4.5).However, the proof works for an arbitrary harmonic moving frame whose Cartan form admits an expansion as in (2.2) where µ 1 solves (2.4).Since µ 1 is defined up to an additive constant, we can assume that ρ = 1 in the following.We define the conformal map f 1 : D → Ω 1 using the polar coordinates by Therefore, we have (2.5) which shows that the conformal parameter of f is , which implies that In particular, we have where p 1 ∈ Ω 1 is the singularity of the moving frame ( u 1 , v 1 ) : We can relate the change of metric by As f 1 is conformal, we have where and π −1 : C → S 2 \ {N } is the inverse stereographic projection.Writing for simplicity we deduce by (2.8) that so that (by an abuse of notation for the last identity) where Remark 2.4.To summarize, the above discussion shows that the moving frame ( u 1 , v 1 ) satisfying the boundary condition u 1 = τ on Γ, (2.2), and (2.4) is tightly related to a conformal map f 1 : D → Ω 1 using Froebenius theorem as in [23], in the way that the moving frame satisfies (2.5).However, we can start directly with any conformal map f 1 and (2.5) gives a moving frame ( u 1 , v 1 ) which satisfies (2.2) and (2.4).This is the approach we take in Section 3 which allows us to relax the regularity assumption of ∂Ω 1 = Γ.

Definition 2.5. Define the open subsets
Then, for all p j ∈ Ω j and for all harmonic moving frames ( u j , v j ) : Ω j \ {p j } → U Ω j × U Ω j such that the Cartan form ω j = u j , d v j admits the decomposition where G Ωj : Ω j \ {p j } → R is the Green's function of the Laplacian ∆ g0 on Ω j with Dirichlet boundary condition, and where k g0 is the geodesic curvature on Γ = ∂Ω j .Define the functional E (that we call the renormalised energy associated to the frames ( u 1 , v 1 ) and ( u 2 , v 2 )) by Then there exists conformal maps Remark 2.7.If W j (j = 1, 2) is the renormalised energy in the sense of Bethuel-Brezis-Hélein associated to the moving frame ( u j , v j )-or more precisely, to its boundary data-(see [3] and [23]), we have and S 2 − is the southern hemisphere, we deduce that and by the Alvarez-Polyakov formula (see (1.17) of [29]) and (2.9), we have if we choose Γ to be oriented as ∂Ω 1 , and Therefore, using subscripts with evident notations, we deduce by Theorem 2.1 with g = g 0 that Notice that provided that Γ be given with the same orientation of ∂Ω 1 , we have 2 (2.12) by (2.4).Therefore, we deduce that 2 Now, by conformal invariance of the Dirichlet energy, we have (2.17) Therefore, we get (2.14), (2.15), (2.16) and (2.17) , and ψ(z) = 0 for all z ∈ S 1 , we have θ 1 = 0 on Γ.Therefore, we have where we used the Dirichlet condition G Ω1 = 0 on ∂Ω 1 = Γ.Therefore, (2. 19) and (2.20) imply that Gathering (2.18) and (2.21) yields We also have Indeed, we have by the boundary conditions (2.10) We also have by the conformal invariance of the Dirichlet energy Therefore, we finally get by (2.24) and (2.25) Finally, we deduce by (2.11), (2.22) and (2.23) that Recalling the identity (2.7), we finally deduce that Now we introduce the functional We deduce that This concludes the proof of the theorem.
Remark 2.8.We check that equality (2.29) holds for the equator S 1 .Using the definition (1.3) with the conformal maps f, g being the identity maps, we see that I L (S 1 ) vanishes.
For this, since K g0 = 1 on Ω 1 = S 2 − , after making a stereographic projection π : S 2 \ {N } → C sending S to 0, we find (2.30) We take We deduce by a direct verification that µ(z) = − log(1 + |z| 2 ).(This is easy to guess since by (2.6), µ can be computed from the conformal factor of f 1 .)Therefore, we have by the conformal invariance of the Dirichlet energy Finally, by (2.30) and (2.31), we have Applying the same computation to (2.32) Since the inverse stereographic projection is given by we compute directly that which concludes the proof of (2.32) and shows the identity (2.29) for the circle S 1 .

Construction of Harmonic Moving Frames for Weil-Petersson Curves
In the previous section, we showed that in the case of smooth curves, the Loewner energy was equal to a renormalised Dirichlet energy of a specific harmonic moving frame.In this section, we will directly construct harmonic moving frames satisfying appropriate boundary conditions for arbitrary Weil-Petersson quasicircles.In the next section, we will show that Theorem 2.6 holds for non-smooth curves.
Before stating the main theorem of this section, recall an easy lemma on harmonic vector fields.
Lemma 3.1.Let Σ ⊂ R 3 be a smooth surface, n : Σ → S 2 its unit normal, and g = ι * g R 3 be the induced metric.Assume that u : Σ → S 2 is a smooth critical point of the Dirichlet energy amongst S 2 -valued maps such that u, n = 0. Then u satisfies the following Euler-Lagrange equation: Proof.We proceed as in [19] in Lemme (1.4.10), taking variations X that also satisfy X, n = 0.
The following result is the same as Theorem 2.6, but for a general Weil-Petersson quasicircle.
Theorem 3.2.Let Γ ⊂ S 2 a Weil-Petersson quasicircle and let Ω 1 , Ω 2 be the two open connected components of S 2 \ Γ.For j = 1, 2 and for all p j ∈ Ω j , there exists a harmonic moving frame ( u j , v j ) : Ω j \ {p j } → U Ω j × U Ω j such that the Cartan form ω j = u j , d v j admits the decomposition where G Ωj = G Ωj ,pj : Ω j \ {p j } → R is the Green's function of the Laplacian ∆ g0 on Ω j with Dirichlet boundary condition and singularity p j ∈ Ω j , and where k g0 is the geodesic curvature on Γ = ∂Ω j .
Remark 3.3.The Neumann condition for µ j (1 ≤ j ≤ 2) is understood in the sense of distributions, since the geodesic curvature is only in H −1/2 (Γ) in general (see the appendix for more details).
Proof.Rather than using the moving frame that comes from a Ginzburg-Landau type minimisation as in [23]-that would have had to be carried in the geometric setting of domains of S 2 -we directly use the uniformisation theorem and the geometric formula of [41] (that does not require any regularity on the curve Γ) to construct the relevant moving frame.We now construct the moving frame on Ω 1 .The construction for Ω 2 is similar.
Let π : S 2 \ {N } → C be the standard stereographic projection and assume without loss of generality that N ∈ Ω 2 .Let Ω = π(Ω 1 ) ⊂ C be the image domain and γ = π(Γ) ⊂ C be the image curve.Thanks to the Uniformisation Theorem, there exists a univalent holomorphic map f : A direct computation show that Now, by analogy with the construction in Section 2 (see also [23], Proposition 5.1), define Then, we have from direct computations . we deduce that More generally, if ϕ : C → C is a smooth complex function, we have Notice that is a holomorphic null vector, i.e.F (z), F (z) = 0, so we see directly since Step 2. Verification of the system (3.3).
Part 1. Equation on Ω 1 for µ 1 . Since Thanks to the explicit expression in (3.4), and by harmonicity of log |f |, we have Recalling that Therefore, (3.7) can be rewritten as Part 2. Boundary conditions.
If h : C → R is a smooth function, we have and on ∂D in the distributional sense.We will comment on it in Remark 4.6.Recall that the geodesic curvature on ∂Ω 1 is given (see [10]) by (3.9) where (3.11) We also get (3.12) Since ψ, ψ = 0, we have ∂ z ψ, ψ = ∂ z ψ, ψ = 0.In particular, we have Therefore, we deduce by (3.12), (3.13) and (3.14) that (3.15) The identities (3.11) and (3.15) imply that and since |χ| 2 = 1 and |ψ| 2 = 2, we have which concludes the proof of the system (3.3) by the conformal invariance of the Green's function (we denoted for simplicity Step 3. Verification that ( u 1 , v 1 ) is a harmonic moving frame.Now, thanks to Lemma 3.1 and (3.1), the maps u 1 and v 1 are unit harmonic moving frames if and only if they satisfy in the distributional sense (see Theorem 3.5) the system (writing (3.17) where n : D → S 2 is the same map as f 1 but viewed as the Gauss map associated to the branched minimal immersion of the disk from D into R 3 with Weierstrass data (f, dz).It is given by By a direct computation, we see that the Gauss map satisfies the following equations In particular, the previous equation (3.17) must reduce to to show that u and v satisfy the equations (3.18).Recall from (3.10) that u = Re (ϕ) and v = Im (ϕ), we deduce that and we have which implies in particular that Im ( ∆ϕ, ϕ ) = 0.Then, we compute Therefore, we have where we used ∂ z ϕ = ∂ z ϕ.By (3.23), we deduce that Therefore, we deduce that (3.21) holds, which implies that u and v solve the equations (3.18).
Recall that u = u 1 and v = v 1 , and let which is equivalent to the identity We have by (3.11) and (3.12) Therefore, using (3.13), (3.14), (3.15), and u, v = 0, we deduce that and this concludes the proof of (3.25) since by (3.4) This last identity concludes the proof of the theorem.
Finally, we will establish the uniqueness of distributional solutions of the system (3.2) with appropriate boundary conditions (3.3).This is the exact analogous of Remark I.1 of [3].First, we need to define explicit maps that yield trivialisations of vector fields on simply connected domains of the sphere.Let Ω 1 ⊂ S 2 be as Theorem 3.2.Using the stereographic projection π : S 2 \ {N } → C, we have one holomorphic chart z on S 2 \ {N }, and for a domain More explicitly, let X : Ω → R 3 be a vector field such that X, n = 0, where n : Ω → S 2 is the unit normal given by Now, we introduce the function ψ : C → C 3 , given by and we easily check that Therefore, we deduce that ( u 1 , u 2 ) defined as follows is a tangent unit moving frame (orthogonal to n ) The trivialisation map on Ω 1 ⊂ S 2 \ {N } is then given by while the trivialisation map of sections is given by Notice that for all tangent vector field X, we have X, n = 0, which implies that there exists real functions λ 1 , λ 2 : Ω 1 → R such that Remark 3.4.Using the next Theorem 3.5, it is easy to check that ( u j , v j ) (j = 1, 2) are harmonic vector fields since by (3.10) and (3.11), we have In particular, we have Summing those equations and substracting the first one to the second one yields the system Im e 2iϕ ∆ψ, ψ = 0 We will show that for all smooth real-valued which will imply that ( u, v) solves the system (3.30) if and only if ∆ϕ = 0, or ϕ is harmonic.

Now, we compute
We have Then we have which implies as ψ, ψ = 0 and by (3.32) that ∆ψ, ψ = 0. (3.33) Now, we have (3.34) We now compute Therefore, we have In particular, the function , one can rewrite the equation distributionally as In particular, we deduce as u 0 is harmonic that By Theorem I.5 and Remark I.1 of [3], we deduce that u 0 is the unique harmonic function with a singularity at p 1 such that u 0 = h on ∂Ω 1 .This concludes the proof of the theorem.

Proof of the Main Theorems for Non-Smooth Curves
In order to extend Theorem 2.6 to the non-smooth setting, we will obtain another formula for E 0 in terms of conformal maps and that holds true for any closed simple curve of finite Loewner energy.Using this additional formula, the convergence result will be easily obtained.
Under the preceding notations, if Γ ⊂ S 2 Weil-Petersson quasicircle, from Remark 2.4, thanks to Theorem 3.2, there exists harmonic moving frames ( u 1 , v 1 ) and ( u 2 , v 2 ) on Ω 1 and Ω 2 with arbitrary singularities p 1 and p 2 respectively, such that where , and µ j satisfies (2.10).We saw in Theorem 2.6 that in the case of smooth curves, there exists conformal maps f 1 : D → Ω 1 and f 2 : D → Ω 2 such that In this section, we generalise this result for curves of finite Loewner energy.Now, if π : S 2 \ {p 2 } → C is a stereographic projection, since f j : D → Ω j is conformal and π is also conformal, we deduce that π•f j : D → π(Ω j ) ⊂ R is also conformal.Therefore, these maps are biholomorphic or anti-biholomorphic, so up to a complex conjugate (which is an isometry), we can assume that they are holomorphic.Notice that we have Indeed, since f 2 (0) = p 2 , we have g(∞) = ∞, so that the functions f , g satisfy the needed conditions to apply Theorem 1.1.Now, with the previous notations, define the functional Remark 4.2.(1) One may wonder from where this definition comes from.It will be made clear in the proof of the next theorem where we explicitly rewrite E 0 with the help of the conformal maps f and g defined above.
(2) We call this quantity S 3 since a functional called S 2 was defined in [14] as the log-determinant of the Grunsky operator associated with the curve γ (up to a factor − 1  12 ).
The goal of this section is to show the identity The third equality is straightforward and is proved in Theorem 4.3, and the proof of the whole identity is completed in Theorem 4.5.Theorem 4.3.Let Γ ⊂ S 2 be a simple curve of finite Loewner energy.Then we have is a curve of finite Loewner energy Ω 1 and Ω 2 the two connected components of S 2 \ Γ, and f 1 : D → Ω 1 , f 2 : D → Ω 2 are the conformal maps associated to Γ in the definition of E with f j (0) = p j for j = 1, 2. Now, recall from (2.6) that We have by conformal invariance of the Dirichlet energy Since f 1 is conformal and f 1 (0) = p 1 , we have Finally, we deduce by (4.3) and (4.4) that Up to a rotation of S 2 , we can assume that p 2 = N and if π : S 2 \ {N } → C is the standard stereographic projection, let which we assume without loss of generality to be biholomorphic (up to a complex conjugation).Now, since a computation shows that We deduce that Therefore, we have Since Ω = f (D) is compact, we have Therefore, (4.8) implies that ∇ log |∇f 1 | ∈ L 2 (D) and while (4.6) implies that which is finite by (4.8) and the smoothness of f in D. Since the function g : D → C \ Ω is unbounded at 0, we do not see trivially that For this, as g is univalent and g(0) = ∞, we deduce that g admits the following meromorphic expansion at z = 0 for some a ∈ C \ {0} and a 0 , a 1 ∈ C Therefore, we have by a direct computation Since Γ is a Weil-Petersson quasicircle, we deduce by estimates similar to (4.8) and (4.9) that ∇ log | g | ∈ L 2 (D \ D(0, ε)) and g ∈ L 2 (D \ D(0, ε)) for all ε > 0 and we finally deduce that Furthermore, we directly get Now, notice that which concludes the proof of the theorem.
Remark 4.4.If Γ = S 1 , then we can take f = Id D and g = Id C\D , and we compute In the next theorem, we finally complete the proof of (4.2) by showing that πI L (Γ) = S 3 (Γ).
Theorem 4.5.Let Γ ⊂ S 2 be a closed simple curve of finite Loewner energy.Then we have where E 0 is defined in (1.12).Furthermore, if Ω 1 , Ω 2 ⊂ S 2 \ Γ are the two connected components of S 2 \ Γ, for all conformal maps f 1 : D → Ω 1 and f 2 : D → Ω 2 , we have Proof.By Theorem 2.6, we have the identity I L (Γ) = 1 π E 0 (Γ) for all smooth Γ, and by the preceding Theorem 4.3, we have E 0 (Γ) = S 3 (Γ) for any Jordan curve Γ of finite Loewner energy.Therefore, we will prove that I L = 1 π S 3 which will imply our result.We now let Ω 1 , Ω 2 ⊂ S 2 \ Γ be the two connected components of S 2 \ Γ, and f 1 : D → Ω 1 , f 2 : D → Ω 2 be the two conformal maps associated to Ω 1 and Ω 2 , and let p 1 = f 1 (0) and p 2 = f 2 (0).Up to a rotation on S 2 (which does not change any of the energies considered), we can assume that p 2 = N .If π : S 2 \ {N } → C is the standard stereographic projection, let γ = π(Γ), and Ω the bounded component of C \ γ and define Now, by Corollary A.4 of [14] and Theorem 8.1 [41], if {γ n } n∈N is a sequence of smooth curves converging uniformly to a simple curve γ and such that for a sequence of maps where Ω n is the bounded component of C \ γ n , and satisfies where f : D → Ω is a univalent function such that f (0) = 0 and f (0) = 1, we have In particular, for any sequence of holomorphic maps for all univalent function g : , and define Then ) is smooth and uniformly converges to γ.Furthermore, we have which implies that By Brezis-Lieb lemma ( [24]), since f n −→ Therefore, we also get the convergence which finally shows by (4.20) that As previously, we have Therefore, (4.24) and (4.26) imply that Remarks 4.6.Notice that we can also directly express the Loewner energy using moving frames.First, we trivially have Alternatively, we have which is (up to the second line involving the conformal maps f 1 and f 2 ) very reminiscent of the Ginzburg-Landau renormalised energy ([3, Chapter VIII]).
To see this equality, since u 1 , v 1 and n are unitary, we have Then, integrating by parts and using that G Ω1 = 0 on ∂Ω 1 , we deduce by Stokes theorem-and the equation (that follows from (3.8)) where K g0 = 1 is the Gauss curvature of the sphere-that which implies since Area g0 (S 2 ) = 4π that Notice that it gives another explanation for the factor 4π in the definition of E .

Appendix
In this appendix, we provide more details on the geodesic curvature for Weil-Petersson quasicircles and show a consequence of Theorem 4.5 which is an identity on univalent functions associated to a Weil-Petersson quasicircle.

Properties of the Geodesic
Proof.The geodesic curvature is given by is the Cartesian frame given by (in the following formulae, f is seen as a R 2 -valued function) Therefore, we deduce that where we used Therefore, we have and which concludes the proof of the lemma.
Proof.We compute f (z) = e i log log(z) + i log(z) e i log log(z) = 1 + i log(z) e i log log(z) f (z) = − i z log 2 (z) e i log log(z) + i z log(z) 1 + i log(z) e i log log (z) .
Therefore, we have .
Notice that we have Remark 5.5.In particular, we see that there exists curves whose geodesic curvature is a distribution of order 1.This curve is an example of spiral mentioned earlier in the introduction.

A consequence of Theorem 4.5
The new identity of π I L = S 3 from Theorem 4.3 and Theorem 4.5 provides a new identity about holomorphic univalent maps of the plane.* Beware that the log function here is defined as the trace of our continuous determination of the logarithm on the upper-half plane and is not the standard log function on (0, ∞).

Definition 4 . 1 .
Let γ be a Jordan curve with finite Loewner energy.Let f : D → Ω, g : C \ D → C \ Ω be biholomorphic maps such that g(∞) = ∞, we define the third universal Liouville action S 3 by

Figure 2 :
Figure 2: Spherical formula for the Loewner energy with respect to conformal maps.
Now, we need to show the result f n −→ n→∞ f in L 2 (D) strongly.Notice that since f is smooth in D, we have by construction f n −→ n→∞ f almost everywhere.Furthermore, a linear change of variable shows that

Curvature for Weil-Petersson Quasicircles Lemma 5 . 1 .
Let H = C ∩ {z : Im (z) > 0} be the Poincaré half-plane, and f : H → C a univalent holomorphic map, Ω = f (H), and assume that γ = ∂Ω is a simple curve of finite Loewner energy.Then the geodesic curvature k g0 of γ is given in the distributional sense by
1is a harmonic map with values into S 1 .=