Equivalent Descriptions of the Loewner Energy

Theorem 1.1. When γ is a chord from 0 to infinity in Σ with finite Loewner energy, then IΣ,0,∞(γ) = 1 π Z Σ\γ ∇log h′(z) 2 dz2 = 1 π Z Σ\γ

Theorem 1.2 (see Theorem 6.1). If γ is a loop passing through ∞with finite 3

Theorem 6.1 then opens the door to a number of connections with other ideas, which we then investigate in Section 7 and Section 8 and that we now describe. Relation to zeta-regularized determinants The first approach involves zeta-regularized determinants of Laplacians for smooth loops. Our main result in this direction is Theorem 7.3, which can be summarized by:

Theorem 1.3. For C∞loops, one has the identity IL(γ) = 12 log det′ ζN(γ, g) −12 log lg(γ) −(12 log det′ ζN(S1, g) −12 log lg(S1)), where g is any metric on the Riemann sphere conformally equivalent to the spherical metric, lg(γ) the arclength of γ, and det′ ζN(γ, g) the zeta-regularized determinant of the Neumann jump operator across γ. 4

Theorem 1.4. A Jordan curve γ has finite Loewner energy if and only if [γ] ∈T0(1) and IL(γ) = S1([γ])/π, where we identify γ with its welding function which lies in QS(S1). This provides therefore another characterization of T0(1) and a new viewpoint on its Kähler potential (or alternatively a way to look at the Loewner energy). Again the root-invariance (and also the reversibility) of the loop energy can be viewed as a corollary of this result, because there is no more parametrization involved i

Proposition 2.1. Let γ be a finite energy simple curve in (Σ, 0, ∞), I(γ[0, T]) = 1 π Z Σ\γ[0,T]

Proposition 2.1 is a rephrasing of Theorem 1.1 for γ. However, we will explain how it is in fact possible to deduce Theorem 1.1 from Proposition 2.1 by letting T →∞ in Section 6 while proving the more general Theorem 6.1 for simple loops. We will therefore aim at establishing Proposition 2.1 which is completed in Section 5. In the sequel we will denote the right-hand side of Proposition 2.1 by J(hT ). Note that J(hT ) = 1 π Z Σ\γ

Corollary 2.2. If T < ∞, 0 < α ≤1 and γ[0, T] is C1,α. Then γ is C1,β, where β = α if α < 1/2, and β can take any value less than 1/2 if α ≥1/2. 9

Lemma 3.1 (Extension of Stokes’ formula). For a C1,α domain H as above and all smooth and compactly supported functions g ∈C∞ c (H), Z H ∇g(z) · ∇σφ(z) dz2 = − Z R g(Γ(s)) dτ(s), (4) 10

Lemma 3.1. Let Hε = φ−1(H + iε) be the domain with boundary Γε = φ−1(R + iε) parametrized by arclength: s →Γε(s). We choose the parametrization such that Γε(0) →Γ(0) as ε →0. Since Γε is analytic, the remark above applies and one gets Z Hε ∇g(z) · ∇σφ(z) dz2 = Z Γε g(z)∂nσφ(z) dz = Z R g(Γε(s))∂sνφ(Γε(s)) ds = Z R

Proposition 3.2. If a finite capacity curve γ in (Σ, 0, ∞) satisfies: • γ ∪R+ is C1,α for some α > 0, • σh is in D∞(Σ \ γ). Then for all g ∈D∞(Σ), Z Σ\γ ∇g(z) · ∇σh(z) dz2 = 0. 12

Proposition 3.3 (Weak J-Additivity). If γ is a simple curve in (Σ, 0, ∞) such that γ ∪R+ is C1,α. For 0 ≤s < t ≤T, if both J(hs) and J(ht,s) are finite, then J(ht) = J(hs) + J(ht,s).

Proposition 4.1. Proposition 2.1 holds when γ is driven by a linear function. First notice that the function W(t) = λt for t ≥0 and W(t) = 0 for t ≤0 is C0,1. Therefore, γ ∪R+ is C1,α for α < 1/2 by Theorem A. Once we have shown that J(hε) < ∞for some ε > 0, the weak J-additivity (Proposition 3.3) applies. We can note that the J-additivity and the I-additivity imply that J(hT ) and I(γ[0, T]) are both linear in T, so that it suffices to check that I(γ[0, T]) ∼J(hT ) as T →0.

Corollary 4.2. Proposition 2.1 holds when γ is driven by a piecewise linear function. 17

Lemma 5.1. If T < ∞, (W (n))n≥1 is a sequence of driving functions defined on [0, T], that converges uniformly to W. Then J(h) ≤lim inf n→∞J(h(n)), where h(n) := h(n) T is the Loewner flow generated by W (n) at time T, and h is generated by W.

Corollary 5.2. If γ driven by W : [0, T] →R has finite Loewner energy in (Σ, 0, ∞), then J(h) ≤I(γ). In particular, σh ∈D∞(Σ \ γ).

Proposition 3.2 by assuming only that γ has finite Loewner energy.

Lemma 5.3. If γ is a Loewner chain in (Σ, 0, ∞) with finite Loewner energy and finite total capacity. Then for all g ∈D∞(Σ), Z Σ\γ ∇g(z) · ∇σh(z) dz2 = 0. (9)

Corollary 5.4 (Strong J-additivity). If γ has finite Loewner energy, then J(ht) = J(hs) + J(ht,s) for 0 ≤s ≤t ≤T.

Theorem 6.1. If γ has finite Loewner energy, then IL(γ, ∞) = 1 π Z C\γ |∇σh(z)|2 dz2 ! , where h|H1 (resp. h|H2) maps H1 (resp. H2) conformally onto a half-plane and fixes ∞. Notice that the expression J(h) on the right-hand side does not depend on the orientation of the loop, but does a priori depend on the special point ∞which is the root of γ. We have mentioned in the introduction that the loop energy is a generalization of the chordal energy. In fact, consider the loop γ = R+ ∪η, where η is a

Theorem 1.1 follows immediately from Theorem 6.1. As we described above, loops can be understood as embedded arcs with T = +∞. For arcs which do not make it all the way back to its root (T < ∞), the mapping-out function hT is a natural choice for the uniformizing function h. Let us first prove the analogous identity for an embedded arc.

Lemma 6.2. If γ is a simple arc in ˆC such that γ(0) = ∞with finite arc energy. Then J(h) = IA(γ, ∞), where h = hT is a mapping-out function of γ. Γ(T) Γ(0) Γ(t) Γ(−∞) Γ(0) Γ(T) Lt 0 ∞ ϕt h

Lemma 6.2 shows that σh−1 n has finite Dirichlet energy bounded by the arc Loewner energy of Γ[−∞, n] hence by IL(Γ, ∞). On the other hand, the inequality J(h) ≤ IL(γ) that we have proved above gives us the finiteness of the Dirichlet energy of σψn: For every ε > 0, there exists n0 large enough, such that ∀n ≥n0, J(ψn) ≤IR+(hn(Γ[n, ∞])) = Z ∞ n W ′2(t)/2 dt ≤ε. By the Cauchy-Schwarz inequality, the cross term in (13) converges to 0 as n →∞, and J(hn) converges to IL(Γ, ∞). Hence J(h) ≥IL(γ). 7 Zet

Proposition 7.1. The functional H (·, g) is invariant under Weyl-scalings.

Corollary 7.2. H (·, g) is conformally invariant: let µ be a conformal map from S2 onto S2, then H (γ, g) = H (µ(γ), g).

Theorem 7.3. If g = e2ϕg0 is a metric conformally equivalent to the spherical metric g0 on S2, then: (i) Circles minimize H (·, g) among all smooth Jordan curves. (ii) Let γ be a smooth Jordan curve on S2. We have the identity IL(γ, γ(0)) = 12H (γ, g) −12H (S1, g) = 12 log detζ(−∆D1,g)detζ(−∆D2,g) detζ(−∆D1,g)detζ(−∆D2,g), where D1 and D2 are the two connected components of the complement of S1. Let us make two remarks: • The right-hand side in (ii) does not depend on the root, so that the root-

Theorem 1.4). Let us start with some background material on the universal Teichmüller space T(1) and the the Weil-Petersson Teichmüller space T0(1). We follow here the notations of [40]. We define D = {z ∈C, |z| < 1}, D∗= {z ∈C, |z| > 1}, and let S1 = ∂D be the unit circle. Let QS(S1) be the group of sense-preserving quasisymmetric homeomorphisms of the unit circle (see e.g. [20]), Möb(S1) ≃ PSL(2, R) the group of Möbius transformations of S1 and Rot(S1) the rotation group of S1. The universal Te

Theorem 8.1. Let γ be a (bounded) Jordan curve, then γ has finite Loewner energy if and only if γ is a Weil-Petersson quasicircle. Moreover, IL(γ) = S1(γ)/π. (18) It is worth mentioning other characterizations of T0(1) due to Cui, Shen, Takhtajan and Teo, from which one obtains immediately other analytic characterizations of finite energy loops given Theorem 8.1: Theorem H ([8, 36, 40]). With the same notation as in Theorem F, ϕ is in the Weil-Petersson class if and only if one of the following eq

Lemma 8.2. If a sequence (γn : [0, 1] →ˆC)n≥0 of simple loops converges uniformly to a bounded loop γ, then lim inf n→∞S1(γn) ≥S1(γ). 37