Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SLE}_{8/3}$$\end{document}SLE8/3 loops touching the Jordan curve under quasiconformal deformations.


Introduction
One hundred and one years ago, Loewner introduced [14] a method to encode a simple planar curve by a family of uniformizing maps (called the Loewner chain) which satisfies a differential equation driven by a real-valued function.This method has become a powerful tool in geometric function theory.It was instrumental in the proof of Bieberbach conjecture by De Branges [5] (which was also the original motivation of Loewner) and was revived around 2000 as a fundamental building block in the definition of the Schramm-Loewner Evolution [18].On the other hand, quasiconformal mapping is one of the fundamental concepts in geometric function theory and Teichmüller theory.Thus, we find it natural to investigate the interplay between quasiconformal maps and the Loewner transform.We will further comment on the motivation of this work and discuss follow-up questions in Section 4. We mention that analytic properties of the Loewner driving function have been investigated in, e.g., [6,12,13,[15][16][17].
Our first result shows how quasiconformal deformations of the ambient domain H = {z ∈ C : Im(z) > 0} affect the driving function of a simple chord in H connecting 0 to ∞. Theorem 1.1.Let η be a simple chord in (H; 0, ∞) under capacity parametrization and ν ∈ L ∞ (H) be an infinitesimal Beltrami differential whose support is compact and disjoint from η.For ε ∈ R such that ∥εν∥ ∞ < 1, let ψ εν be the unique quasiconformal self-map of H with Beltrami coefficient εν such that ψ εν (0) = 0 and ψ εν (z) − z = O(1) as z → ∞.Denote the capacity and driving functions of the parametrized chord ψ εν • η in (H, 0, ∞) by a εν • and λ εν • , respectively.Then, and where d 2 z is the Euclidean area measure, λ • is the driving function of η, g • is the Loewner chain of η.
Our proof relies on the simple but crucial observation that the Loewner driving function and the capacity parametrization of the curve can be expressed by the pre-Schwarzian and Schwarzian derivatives, respectively, of well-chosen maps (Lemma 2.1).
We extend our considerations to the Loewner driving function associated with a Jordan curve γ ⊂ Ĉ = C ∪ {∞}, now defined on R instead of R + .The loop driving function was defined in [21] and can be thought of as a consistent family of chordal Loewner driving functions.See Section 3.1 for the precise definition.We point out that for a given Jordan curve, there are a few choices we make to define its driving function t → λ t : • the orientation of γ; • a point on γ called the root, which we denote by γ(−∞) = γ(+∞) (we also use γ(±∞) when we do not emphasize the difference between the start point and the end point of the parametrization); • another point on γ, which we call γ(0); • a conformal map H 0 : Ĉ ∖ γ[−∞, 0] → C ∖ R + , such that H 0 (γ(0)) = 0 and H 0 (γ(+∞)) = ∞, where γ[−∞, 0] denotes the closed subinterval of γ (as a set) going from the root to γ(0) following the orientation of the curve.
If the orientation and the root of γ are fixed, different choices of γ(0) and H 0 result in changes to the driving function of the form for some c > 0 and s ∈ R. Such transformations do not change the Dirichlet energy of λ.
Rather surprisingly, the Dirichlet energy of the loop driving function does not depend on the choice of the root or the orientation either, as shown in [17,20].These symmetries are further explained by the following theorem.
Theorem 1.2 (See [21]).The Loewner energy of γ, defined as equals 1/π times the universal Liouville action S introduced by Takhtajan and Teo in [19], defined as Using Theorem 1.1, we obtain in Section 3 the following first variation formula of the Loewner energy.This formula coincides with that of the universal Liouville action S in [19,Ch. 2,Thm. 3.8] divided by π, thus giving another explanation of the identity I L = S/π.This variational formula was crucial in [19] to show that S is a Kähler potential of the Weil-Petersson Teichmüller space.
where f and g are the conformal maps in Theorem 1.2 and is the Schwarzian derivative of φ.
In the language of conformal field theory, this theorem states that the holomorphic stress-energy tensor of the Loewner energy is given by a multiple of the Schwarzian derivative of the uniformizing map on each complementary component of the curve.
Remark 1.5.The Loewner energy of γ εµ does not depend on the choice of the solution ω εµ to the Beltrami equation, as all such solutions are equivalent up to post-compositions by Möbius transformations of Ĉ.In [19], µ is an L2 -harmonic Beltrami differential supported on only one side of the curve γ.Here, we allow the support of µ to be on both sides of γ but require it to be disjoint from γ.
Since the support of µ is away from γ, there exists a (not necessarily simply connected) domain D containing γ such that D ∩ supp(µ) = ∅.In particular, ω εµ is conformal in D. In [22,Thm. 4.1], the second author showed that the change of the Loewner energy under a conformal map in the neighborhood of the curve could be expressed in terms of the SLE 8/3 loop measure introduced in [26], which is the induced measure obtained by taking the outer boundary of a loop under Brownian loop measure [9,11].Combining this with Theorem 1.4, we immediately obtain the following variational formula for the SLE 8/3 loop measure.
• Let g εν t : H ∖ η εν (0, t] → H be the Loewner chain associated with the deformed curve η εν [0, t]. • Denote the driving function of η εν by λ εν t := g εν t (η εν (t)).• Note that η εν is not necessarily parametrized by its half-plane capacity.Denote the capacity function of η εν [0, t] by a εν t , so that This section aims to prove Theorem 1.1.For this, we first express λ εν t and a εν t in terms of the pre-Schwarzian and Schwarzian derivatives of an appropriately conjugated Loewner chain (Lemma 2.1).We then find the first variations of these derivatives using the measurable Riemann mapping theorem (Proposition 2.2).
The centered Loewner chain satisfies f t (η(t)) = 0 and Let ι(z) := −1/z be the inversion map.Define the inverted Loewner chain by ft (z Similarly, define A calculation analogous to (2.5) using the series expansion of g εν t at ∞ leads to By the Schwarz reflection principle, ft and f εν t extend respectively to conformal maps on C ∖ ι(η(0, t] ∪ η(0, t]) and C ∖ ι(η εν (0, t] ∪ η εν (0, t]), where • denotes the complex conjugate.In particular, they are conformal in some neighborhood of 0.
Recall that the pre-Schwarzian (also known as non-linearity) and Schwarzian derivatives of a conformal map φ are, respectively, (2.8) The chain rules for the pre-Schwarzian and Schwarzian derivatives are for any conformal maps f and g such that f • g is well-defined.
Lemma 2.1.Consider ft and f εν t as conformal maps extended by reflection to a neighborhood of 0. Then, Proof.The lemma follows from inspecting the coefficients of (2.5) and (2.7).
Proposition 2.2.Let νt be the Beltrami differential defined above.Then, (2.12) (2.13) Proof of Proposition 2.2.We can extend ψενt to a quasiconformal self-map of the Riemann sphere Ĉ = C ∪ {∞} by reflecting it with respect to the real axis.The Beltrami coefficient for this extension of ψενt is εν t where νt (z) := (2.14) Then, by the measurable Riemann mapping theorem, locally uniformly in ζ ∈ C as ε → 0.Moreover, since ∂ ε commutes with ∂ ζ when applied to ψενt and νt has a compact support in C ∖ {0}, we have and ψενt (z), ft (z), and ψεν 0 (z) all behave as z + o(z) as z → 0, we have from Lemma 2.1 and the chain rules (2.9) that Combining these with (2.16) and (2.17), we obtain the first equalities in (2.12) and (2.13).

Variation of chordal Loewner energy
Let η : (0, T + ) → H be a simple chord from 0 to ∞ parametrized by half-plane capacity (i.e., a t = t) and t → λ t be its Loewner driving function.The Loewner energy of η (resp.the partial Loewner energy of η up to time T ∈ (0, T + )) is if λ is absolutely continuous and λ is its almost everywhere defined derivative with respect to t.We set We also define the Loewner energy of a simple chord η in a simply connected domain D connecting two distinct prime ends a, b as where and Proof.From the Loewner equation ∂ t g t (z) = 2/(g t (z) − λ t ), we have and ) and a εν t − t = − 1 12 (S ψενt (0) − S ψεν 0 (0)) are continuously differentiable in the same variables [1].
We can check directly from the integral representations of ∂ ε N ψενt (0) and ∂ ε S ψενt (0) that they are continuously differentiable in t.If λ t is absolutely continuous, then we have from (2.12) that for almost every t, Similarly, (2.13) implies Using the formulas for ∂ t f t and ∂ t f ′ t above, we have This completes the proof.
Proposition 2.3 allows us to compute the first variation of the chordal Loewner energy for a finite portion of the chord η εν .
Since ν is compactly supported in H ∖ η, the integral on the right-hand side is continuous in t and hence bounded on [0, T ].Using the Leibniz integral rule, we conclude that as ε → 0.

Deformation of Weil-Petersson quasicircles
In this section, we consider the Loewner chain for a Jordan curve by conjugating the chordal Loewner chain by z → z 2 .This simple operation relates the integrand in (2.26) to a Schwarzian derivative (Proposition 3.8) which leads to the proof of Theorem 1.4.
Convention.We take the branch of the complex square root function √ z (or z 1/2 ) on C to be the one with the image in H ∪ R + .

Loop driving function
We first recall the definition of the Loewner driving function for a Jordan curve.See Figure 2 for an illustration of the maps used in Section 3.
Let γ : [−∞, +∞] → Ĉ be a Jordan curve where γ(−∞) = γ(+∞).We choose a family of uniformizing maps H t : Ĉ ∖ γ[−∞, t] → C ∖ R + such that H t (γ(t)) = 0 and H t (γ(+∞)) = ∞ for each t ∈ R. Note that each H t is unique up to a real multiplicative factor.We fix a consistent normalization such that for every s < t, H t • H −1 s (z) = z + o(z) as z → ∞.Note that it suffices to fix the map H 0 and then normalize H t for t ̸ = 0 so that H t • H −1 0 has the correct asymptotic behavior.This is possible since if we write We always have λ 0 = a 0 = 0 and t → a t is continuous and strictly increasing.If a t → ±∞ when t → ±∞, then we can reparametrize γ such that a t = t for every t ∈ R. In this case, we say that γ is capacity-parametrized by R. 3 We remind the reader that the capacity parametrization and the corresponding driving function depend on the choices of the orientation of γ, γ(±∞), γ(0), and H 0 .
The reader may wonder about the different behaviors of the map f t depending on the sign of t, which seem to give a different meaning to the term "capacity."This difference is not fundamental as the designation of the point of zero capacity on γ is artificial.We shall view the capacity given by f t as the "relative capacity" with respect to our choice of the part γ[−∞, 0].Precisely, it means the following.
For general s ∈ R, we note that (f t+s • f −1 s ) t≥0 is a centered Loewner chain associated with the curve H s • γ(• + s).Moreover, (3.2) implies for all t ≥ 0. Thus, t → λ t+s − λ s and t → a t+s − a s are respectively the driving function and the capacity function corresponding to the chain (f t+s • f −1 s ) t≥0 .In particular, the assumption a t = t for all t ∈ R means that the chain (f t+s • f −1 s ) t≥0 is also in capacity parametrization.

Remark 3.3. The loop driving function generalizes the chordal Loewner driving function.
If η is a simple chord in (H; 0, ∞) with driving function λ : R + → R, then the Jordan curve γ := η(•) 2 ∪ R + with the same orientation as η (from 0 to ∞), root ∞, γ(0) = 0, and H 0 (z) = z, has the driving function ( λt ) t∈R where λt = λ t if t ≥ 0 and λt = 0 if t ≤ 0. Remark 3.4.Let γ be a Jordan curve capacity-parametrized by R using the conformal map H 0 .If A : Ĉ → Ĉ is a Möbius transformation, then the loop t → γ(t) := A(γ(t)) has the same driving function as γ when we choose its capacity parametrization using the conformal map H0 = H 0 • A −1 .Moreover, the conformal maps corresponding to H t and h t for γ are Ht = Hence, the map f t remains unchanged, and so are the capacity and driving functions.Definition 3.5.The Loewner energy of a Jordan curve γ is where (λ t ) t∈R is the driving function of γ described above.See also Lemma A.1.Theorem 1.2 shows that this energy does not depend on the parametrization of the curve (but we will not use this fact in our proof).
The next corollary is immediate after Lemma 3.2.

Variation of Loewner energy for a part of the quasicircle
We now consider deformations of a Jordan curve γ.Let µ ∈ L ∞ ( Ĉ) be a complex-valued function with compact support in Ĉ ∖ γ.For ε ∈ R with ∥εµ∥ ∞ < 1, let ω εµ : Ĉ → Ĉ be a quasiconformal homeomorphism which satisfies the Beltrami equation Denote the deformation of γ under the quasiconformal map ω εµ as Again we choose a family of uniformizing maps We define analogously the chains (h εµ t ) t∈R and (f εµ t ) t∈R , the driving function (λ εµ t ) t∈R , and the capacity function (a εµ t ) t∈R .That is, Then, by our choice of normalization, as z → ∞.We define λ εµ t and a εµ t from the expansion Remark 3.7.The map ω εµ , and hence the Jordan curve γ εµ , is unique only up to a post-composition by some Möbius transformation.The choice of ω εµ does not affect our analysis, because the first step in it is always to apply the appropriately normalized uniformizing map In this subsection, we translate Corollary 2.4 into an analogous formula for the Weil-Peterson curve γ and its deformation γ εµ .The following is the main result.
Proposition 3.8.Suppose s < t and The following lemma is a straightforward calculation used in the proof of Proposition 3.8.
Lemma 3.9.Suppose λ t is absolutely continuous.Then, for each z ∈ H 0 ( Ĉ ∖ γ), the Schwarzian S h t (z) is absolutely continuous in t.Moreover, for almost every t, Proof.From the relation Hence, it suffices to show that To see this, note that fu := f t+u • f −1 t solves the Loewner equation (see Lemma 3.2) Then, because h0 (z) = z, Replacing z in (3.6) with h t (z), we obtain (3.5).This completes the proof.
Proof of Proposition 3.8.Let us first consider the case s = 0. Letting ).Let εν be the Beltrami coefficient corresponding to the quasiconformal map ψ ε : H → H. Let εµ 0 denote the Beltrami coefficient of

Substituting this ν into (2.26) and letting
Applying Lemma 3.9, we have Recalling our definition of εµ 0 , we have Therefore, the case s = 0 holds.In fact, this implies (3.3) for any s ∈ R because the parametrization of γ is arbitrary up to translations as discussed in Lemma 3.2.

Variation of the loop Loewner energy
The goal of this subsection is to prove Theorem 1.4.Let H +∞ : Ĉ ∖ γ → C ∖ R be any conformal map which maps Ω → H and Ω * → H * .Note that the map H +∞ restricted to Ω coincides with f −1 (as in Theorem 1.4) post-composed by a Möbius transformation, so S . In view of Corollary 3.6 and Proposition 3.8, it suffices to show that as s → −∞ and t → +∞, we have For this, we need a few lemmas.
Proof.Given any R > 0, there exists a large negative s R such that for s ≤ s R , we have for every z ∈ RD.The right-hand side of the inequality tends uniformly to 0 as R → +∞ on any compact subset of C. Lemma 3.11.Suppose γ(±∞) = ∞.Then, S H t → S H +∞ locally uniformly on C ∖ γ as t → +∞.
Proof.Choose either component of C ∖ γ and call it U +∞ .Let us denote by γ U (s) the prime end of γ(s) as viewed from U +∞ .
Let γ t = γ[−∞, t] and denote by γ U t := s∈(−∞,t) γ U (s) the prime ends of γ t accessible from U +∞ .Let Γ t be the hyperbolic geodesic in C ∖ γ t connecting γ(t) and γ(+∞).Let U t be the component of C ∖ (γ t ∪ Γ t ) such that the prime ends of γ t as viewed from U t comprise γ U t .Observe that if hm(z, D; •) is the harmonic measure on the domain By the Beurling projection theorem, This completes the proof of the claim.
Note that H t is a conformal map which sends C ∖ (γ t ∪ Γ t ) onto C ∖ R.Then, by the Carathéodory kernel theorem, H t post-composed with an appropriate Möbius transformation converges locally uniformly on U +∞ to H +∞ as t → ∞.Consequently, S H t → S H +∞ locally uniformly on U +∞ as t → ∞.An analogous argument applies to the other component of Ĉ ∖ γ.
To show (3.8), it suffices to prove that we can switch between the integral over t ∈ (−∞, +∞) and the derivative in ε.To this end, we prove that the following integral is absolutely convergent: Recall from the proof of Corollary 2.4 that for t ≥ 0, where ν is the push-forward of the Beltrami differential µ under the map √ H 0 as displayed in Figure 2. Using Lemma 3.2 and the composition rule (2.20)  Letting s → −∞ and t → +∞, by the dominated convergence theorem and (3.9), we conclude (3.8).This proves Theorem 1.4 as explained at the beginning of the subsection.

Remarks and open questions
While Loewner chains have been studied extensively due to their applications in the study of Schlicht functions and, more recently, in that of Schramm-Loewner evolution (SLE), their infinitesimal variations seem to have been overlooked.Our investigation of this topic is motivated by an effort to understand the large deviations of SLE at a deeper level.Roughly speaking, an SLE curve is a non-self-intersecting curve whose Loewner chain is driven by a constant multiple of Brownian motion.We refer the reader to the textbooks [2,7,10,25] for detailed introductions to SLE and its applications.
Theorem 1.2 implies that the Loewner energy is at the same time the action functional of an SLE loop [4,20] and the Kähler potential on the Weil-Petersson Teichmüller space T 0 (1) [19,21], thus building a bridge between two fundamentally different perspectives on the geometry of the space of Jordan curves in Ĉ.On the SLE side, Jordan curves are viewed as dynamically growing slits, which are naturally described through the language of Loewner chains.The link from SLE to Loewner energy comes from stochastic analysis and the fact that the action functional of a Brownian motion is its Dirichlet energy.
On the Kähler geometry side, there are no dynamics.Instead, all geometric structures are expressed infinitesimally on the tangent spaces of T 0 (1).The reason behind the fact that the action functional of SLE coincides with the Kähler potential of the unique homogeneous Kähler metric on T 0 (1) remains a mystery.See [24] for an expository article on this link.
Our motivation lies in building tools that can be used to reconcile these two distinct viewpoints.The current work serves as the first step in one of the possible directions to this end by elucidating which infinitesimal variations of the Loewner driving function correspond to those on T 0 (1).Natural questions going forward are how the complex structure, the symplectic form, the Weil-Petersson metric, and the group structure on T 0 (1) are encoded in the Loewner driving function.Through these identifications, there is hope to build a more robust connection between random conformal geometry and Teichmüller theory and shed new light on their relationship.
There are yet other possible avenues in the study of quasiconformal deformations of SLE.For example, there is an interesting open question [3,Conj. 7.1] to identify the conformal dimension of SLE, which is the minimal Hausdorff dimension of the image of an SLE curve under quasiconformal mappings.The relevance of our work in this direction is that some of our results are sufficiently general to be applied in this context.For instance, the variational formula for the Loewner driving function (Theorem 1.1) does not assume any regularity on the driving function, so it can be applied even when the driving function is a Brownian motion.On the other hand, there is room for improvement in our results.The most obvious limitation is that we require the support of the Beltrami differential to be away from the curve so as not to deal with improper integrals.The deformations we considered in this work are thus conformal in a neighborhood of the curve, and in this regard, we are still in the same setup as in the conformal restriction of SLE considered in [9].To understand general quasiconformal deformations of SLE, we need to allow the supports of Beltrami differentials to touch the curve.We think this is an interesting question that may require taking into account the stochastic nature of SLE.

A Large time behavior of finite-energy Loewner chains
A priori, a Jordan curve may have finite total capacity.However, we now show that if a Jordan curve has finite Loewner energy, then its capacity must be infinite at both its initial and terminal parts.
Consequently, the capacity parametrization of a finite-energy Jordan curve must take all of R as its domain, which justifies the formula in Definition 3.5.
Proof.If the Loewner energy of the chord γ(0, T + ) traversing the domain Ĉ ∖ γ[T − , 0] is finite, then its total capacity T + must be infinite by [23,Thm. 2.4].By definition, this chordal Loewner energy is bounded above by the loop Loewner energy I L (γ).
Completing the square in the above inequality, we obtain By assumption, the right-hand side of (A.2) is a finite quantity independent of T ∈ (T − , 0).This is a contradiction since t → x t maps [T − , 0] onto [0, +∞].
The following lemma is used in the proof of Theorem 1.4 to switch the order between the derivative and the integral in (3.8).Note that if λ t = 0 for all t, then f t (z) = √ z 2 + 4t.In particular, f t (z) 2 /(4t) → 1 and 2|t|  Since the choice of ε was arbitrary, we conclude Re(f t (z) 2 )/(4t) → 1 as t → +∞.
As for Im(f t (z) 2 ) = 2u t v t , note that (A.3) implies v t is monotonically decreasing.Hence, (A.5) implies Im(f t (z) 2 )/t → 0 as t → +∞.Combining the limits of the real and imaginary parts, we obtain f t (z) 2 /(4t) → 1 as t → +∞.Moreover, since f T 0 (z) = u T 0 + iv T 0 depends continuously on z whereas T 0 was chosen independently of z, the limit we proved converges uniformly on each compact subset of H 0 ( Ĉ ∖ γ).
Let us now consider the t → −∞ limit.This time, since the Loewner energy of γ is finite, we can find a large negative T0 such that T0 −∞ λ2 t dt < ε.Hence, (A.4) implies that there exists a T1 ≤ T0 such that for all T ≤ T1 , , and choosing ε to be arbitrarily small, we obtain Re(f t (z) 2 )/(4t) → 1 as t → −∞.
For the imaginary part of f t (z) Hence, using the Cauchy-Schwarz inequality as above, we have as T → −∞.Therefore, f t (z) 2 /(4t) → 1 as t → −∞.Again, this limit converges uniformly on compact subsets of H since f T0 (z) depends continuously on z and T0 can be chosen independently of z on a compact set.

. 5 )Remark 1 . 3 .
Here, f : D → Ω and g : D * → Ω * are conformal maps such that g(∞) = ∞, Ω and Ω * are respectively the bounded and unbounded connected components of C ∖ γ, and g ′ (∞) = lim z→∞ g ′ (z).If γ passes through ∞, we replace γ by A(γ) where A is any Möbius transformation of Ĉ sending γ to a bounded curve.Although it may not be so apparent from (1.5), it will follow immediately from the definition of the loop driving function and (1.4) that I L is invariant under Möbius transformations of Ĉ. See Remark 3.4.A Jordan curve for which S is finite is called a Weil-Petersson quasicircle.

Fig. 2 A
Fig. 2 A commutative diagram illustrating the quasiconformal maps and related conformal mapping-out functions in Section 3. The gray shaded areas denote the support of the Beltrami differentials.The arrows in red are quasiconformal maps, and those in black are conformal maps.