We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE κ as κ → 0 and the known reversibility of the SLE κ curves for small κ, we show that the energy of a deterministic curve from one boundary point a of a simply connected domain D to another boundary point b, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from b to a in D.
Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map.This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil-Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil-Petersson metric on the Weil-Petersson Teichmüller space.
Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a C 1,β curve (differentiable parametrization with β-Hölder continuous derivative) is in the class C 1,β−1/2 if 1/2 < β ≤ 1, and in the class C 0,β+1/2 if 0 ≤ β ≤ 1/2. This is the converse of a result of Carto Wong [27] and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. Recently, in [6] and [23] the chordal Loewner energy ∞ 0Ẇ (t) 2 /2 dt of η was introduced independently, and basic properties (such as rectifiability) of curves with finite energy were obtained. The chordal Loewner energy a priori depends on the orientation of η, namely viewed as a curve from 0 to ∞ or from ∞ to 0. However, the second author [23] proved the direction-independence (or reversibility).In this paper, we generalize the definition of Loewner energy to simple loops on the Riemann sphere γ : R →Ĉ where γ is continuous, 1-periodic and injective on [0, 1) : We just observe that the limit when ε → 0 of the chordal energy of γ [ε, 1] in the simply connected domainĈ \ γ[0, ε] exists in [0, ∞] (Proposition 2.8), and define it as the loop Loewner energy of γ rooted at γ(0). Note that circles have loop energy 0. Intuitively, the loop energy measures how much the Jordan curve γ[0, 1] differs from a circle seen from the root γ(0), in a Möbius invariant fashion. The loop Loewner energy generalizes * the chordal Loewner energy: Indeed, if we apply z → z 2 to a chord η from 0 to ∞ in H, the positive real line together with the image of η forms a loop γ through ∞. It is clear that its loop energy rooted at ∞ (i.e. we parametrize the loop such that γ(0) = ∞) equals the chordal energy of η.Note also that the loop energy neither depends on any increasing reparametrization of γ fixing the root, nor on the direction of parametrization. The latter fact basically comes from the chordal reversibility, which can be used to show thatγ(t) = γ(1 − t) has the same energy as γ (see Section 2.2 for details). But it depends a priori on the root γ(0) where the limit is taken, not only on the Jordan curve γ[0, 1]. However, our first main result states:Theorem 1.1. The loop Loewner energy is root-invariant.In particular, this result shows that the loop Loewner energy is a conformal invariant on the set of unrooted loops on the Riemann sphere, which attains its minimum 0 only on circles.In our proof of the root-invariance, we approximate the curve by well-chosen regular curves and are led to the following question: What can we say about the relation between the regularity of the driving function and the regularity of the curve? Prior to this work, only one direction was well understood. Slightly imprecisely, the following results state that C α driving functions generate C α+1/2 curves for α > 1/2, where C α is understood with the usual convention as C n,β , where n is the integer part of α and β =...
The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.
We derive the large deviation principle for radial Schramm-Loewner evolution (SLE) on the unit disk with parameter κ → ∞. Restricting to the time interval [0, 1], the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures {φ 2 t (ζ) dζ} t∈[0,1] on the unit circle andOur proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.
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